2. Review of Vectors and Matrices

 

 

VECTORS

 

1. Definition

 

For the purposes of this course, a vector is an object which has magnitude and direction.  Examples include forces, electric fields, and the normal to a surface.  A vector is often represented pictorially as an arrow and symbolically by an underlined letter a _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaamaaabaGaam yyaaaaaaa@372F@  or using bold type a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggaaaa@3723@ .  Its magnitude is denoted | a _ | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaWaaW aaaeaacaWGHbaaaaGaay5bSlaawIa7aaaa@3A51@  or | a | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC yyaaGaay5bSlaawIa7aaaa@3A45@ .  There are two special cases of vectors: the unit vector n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah6gaaaa@3730@  has | n |=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC OBaaGaay5bSlaawIa7aiabg2da9iaaigdaaaa@3C13@ ; and the null vector 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahcdaaaa@36F2@  has | 0 |=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC imaaGaay5bSlaawIa7aiabg2da9iaaicdaaaa@3BD4@ .

 

2. Vector Operations

 

 Addition

 

Let a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggaaaa@3723@  and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahkgaaaa@3724@  be vectors.  Then c=a+b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpcaWHHbGaey4kaSIaaCOyaaaa@3AE2@  is also a vector.  The vector c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogaaaa@3725@  may be shown diagramatically by placing arrows representing a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggaaaa@3723@  and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahkgaaaa@3724@  head to tail, as shown in the figure.

 

 Multiplication

 

1.      Multiplication by a scalar. Let a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggaaaa@3723@  be a vector, and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37D8@  a scalar.  Then b=αa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahkgacqGH9a qpcqaHXoqycaaMc8UaaCyyaaaa@3C3E@  is a vector.  The direction of b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahkgaaaa@3724@  is parallel to a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggaaaa@3723@  and its magnitude is given by | b |=α| a | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC OyaaGaay5bSlaawIa7aiabg2da9iabeg7aHnaaemaabaGaaCyyaaGa ay5bSlaawIa7aaaa@40F7@ .

Note that you can form a unit vector n which is parallel to a by setting n= a | a | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah6gacqGH9a qpdaWcaaqaaiaahggaaeaadaabdaqaaiaahggaaiaawEa7caGLiWoa aaaaaa@3D3C@ .

2.      Dot Product (also called the scalar product). Let a and b be two vectors.  The dot product of a and b is a scalar denoted by α=ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjabg2 da9iaahggacqGHflY1caWHIbaaaa@3CFD@ , and is defined by

ab=| a || b |cosθ(a,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHIbGaeyypa0ZaaqWaaeaacaWHHbaacaGLhWUaayjcSdWaaqWa aeaacaWHIbaacaGLhWUaayjcSdGaci4yaiaac+gacaGGZbGaeqiUde NaaiikaiaahggacaGGSaGaaCOyaiaacMcaaaa@4BDE@ ,

where θ(a,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHHbGaaiilaiaahkgacaGGPaaaaa@3BCD@  is the angle subtended by a and b. Note that ab=ba MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHIbGaeyypa0JaaCOyaiabgwSixlaahggaaaa@3F7D@ , and aa= | a | 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHHbGaeyypa0ZaaqWaaeaacaWHHbaacaGLhWUaayjcSdWaaWba aSqabeaacaaIYaaaaaaa@4052@ .  If | a |0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC yyaaGaay5bSlaawIa7aiabgcMi5kaaicdaaaa@3CC6@  and | b |0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC OyaaGaay5bSlaawIa7aiabgcMi5kaaicdaaaa@3CC7@  then ab=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHIbGaeyypa0JaaGimaaaa@3C18@  if and only if cosθ(a,b)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacogacaGGVb Gaai4CaiabeI7aXjaacIcacaWHHbGaaiilaiaahkgacaGGPaGaeyyp a0JaaGimaaaa@4060@ ; i.e. a and b are perpendicular.

3.      Cross Product (also called the vector product).  Let a and b be two vectors.  The cross product of a and b is a vector denoted by c=a×b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpcaWHHbGaey41aqRaaCOyaaaa@3C17@ The direction of c is perpendicular to a and b, and is chosen so that (a,b,c) form a right handed triad, Fig. 3.  The magnitude of c is given by

| c |=| a×b |=| a || b |sinθ(a,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC 4yaaGaay5bSlaawIa7aiabg2da9maaemaabaGaaCyyaiabgEna0kaa hkgaaiaawEa7caGLiWoacqGH9aqpdaabdaqaaiaahggaaiaawEa7ca GLiWoadaabdaqaaiaahkgaaiaawEa7caGLiWoaciGGZbGaaiyAaiaa c6gacqaH4oqCcaGGOaGaaCyyaiaacYcacaWHIbGaaiykaaaa@53E6@

Note that a×b=b×a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHxd aTcaWHIbGaeyypa0JaeyOeI0IaaCOyaiabgEna0kaahggaaaa@4004@  and a(a×b)=b(a×b)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caGGOaGaaCyyaiabgEna0kaahkgacaGGPaGaeyypa0JaaCOyaiab gwSixlaacIcacaWHHbGaey41aqRaaCOyaiaacMcacqGH9aqpcaaIWa aaaa@49F2@ .

 

 Some useful vector identities

a(b×c)=b(c×a)=c(a×b) a×(b×c)=(ac)b(ab)c (a×b)(c×d)=(ac)(bd)(bc)(ad) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCyyai abgwSixlaacIcacaWHIbGaey41aqRaaC4yaiaacMcacqGH9aqpcaWH IbGaeyyXICTaaiikaiaahogacqGHxdaTcaWHHbGaaiykaiabg2da9i aahogacqGHflY1caGGOaGaaCyyaiabgEna0kaahkgacaGGPaaabaGa aCyyaiabgEna0kaacIcacaWHIbGaey41aqRaaC4yaiaacMcacqGH9a qpcaGGOaGaaCyyaiabgwSixlaahogacaGGPaGaaCOyaiabgkHiTiaa cIcacaWHHbGaeyyXICTaaCOyaiaacMcacaWHJbaabaGaaiikaiaahg gacqGHxdaTcaWHIbGaaiykaiabgwSixlaacIcacaWHJbGaey41aqRa aCizaiaacMcacqGH9aqpcaGGOaGaaCyyaiabgwSixlaahogacaGGPa GaaiikaiaahkgacqGHflY1caWHKbGaaiykaiabgkHiTiaacIcacaWH IbGaeyyXICTaaC4yaiaacMcacaGGOaGaaCyyaiabgwSixlaahsgaca GGPaaaaaa@8D74@

 

3. Cartesian components of vectors

 

Let ( e 1 , e 2 , e 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWHLb WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaGOm aaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIZaaabeaakiaacMcaaa a@3E92@  be three mutually perpendicular unit vectors which form a right handed triad, Fig. 4.  Then { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  are said to form and orthonormal basis. The vectors satisfy

| e 1 |=| e 2 |=| e 3 |=1 e 1 × e 2 = e 3 , e 1 × e 3 = e 2 e 2 × e 3 = e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaGaaCyzamaaBaaaleaacaaIXa aabeaaaOGaay5bSlaawIa7aiabg2da9maaemaabaGaaCyzamaaBaaa leaacaaIYaaabeaaaOGaay5bSlaawIa7aiabg2da9maaemaabaGaaC yzamaaBaaaleaacaaIZaaabeaaaOGaay5bSlaawIa7aiabg2da9iaa igdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaCyzamaaBaaaleaacaaIXaaabeaakiabgEna 0kaahwgadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWHLbWaaSbaaS qaaiaaiodaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaahwgadaWgaaWcbaGaaGymaaqabaGccqGHxdaTcaWHLb WaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaeyOeI0IaaCyzamaaBaaa leaacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaC yzamaaBaaaleaacaaIYaaabeaakiabgEna0kaahwgadaWgaaWcbaGa aG4maaqabaGccqGH9aqpcaWHLbWaaSbaaSqaaiaaigdaaeqaaaaa@7F6B@

We may express any vector a as a suitable combination of the unit vectors e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGymaaqabaaaaa@380E@ , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGOmaaqabaaaaa@380F@  and e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaG4maaqabaaaaa@3810@ .  For example, we may write

a= a 1 e 1 + a 2 e 2 + a 3 e 3 = i=1 3 a i e i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGH9a qpcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaCyzamaaBaaaleaacaaI XaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaWHLb WaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaI ZaaabeaakiaahwgadaWgaaWcbaGaaG4maaqabaGccqGH9aqpdaaeWb qaaiaadggadaWgaaWcbaGaamyAaaqabaGccaWHLbWaaSbaaSqaaiaa dMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaaG4maaqdcqGHri s5aaaa@4FD0@

where ( a 1 , a 2 , a 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaGOm aaqabaGccaGGSaGaamyyamaaBaaaleaacaaIZaaabeaakiaacMcaaa a@3E7A@  are scalars, called the components of a in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@ .   The components of a have a simple physical interpretation.  For example, if we evaluate the dot product a e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHLbWaaSbaaSqaaiaaigdaaeqaaaaa@3B42@  we find that

a e 1 =( a 1 e 1 + a 2 e 2 + a 3 e 3 ) e 1 = a 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHLbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaaiikaiaadgga daWgaaWcbaGaaGymaaqabaGccaWHLbWaaSbaaSqaaiaaigdaaeqaaO Gaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiaahwgadaWgaaWc baGaaGOmaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaiodaaeqaaO GaaCyzamaaBaaaleaacaaIZaaabeaakiaacMcacqGHflY1caWHLbWa aSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaaIXa aabeaaaaa@5193@

in view of the properties of the three vectors e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGymaaqabaaaaa@380E@ , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGOmaaqabaaaaa@380F@  and e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaG4maaqabaaaaa@3810@ .  Recall that

a e 1 =| a || e 1 |cosθ(a, e 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHLbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaqWaaeaacaWH HbaacaGLhWUaayjcSdWaaqWaaeaacaWHLbWaaSbaaSqaaiaaigdaae qaaaGccaGLhWUaayjcSdGaci4yaiaac+gacaGGZbGaeqiUdeNaaiik aiaahggacaGGSaGaaCyzamaaBaaaleaacaaIXaaabeaakiaacMcaaa a@4EBA@

 

Then, noting that | e 1 |=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC yzamaaBaaaleaacaaIXaaabeaaaOGaay5bSlaawIa7aiabg2da9iaa igdaaaa@3CFB@ , we have

a 1 =a e 1 =| a |cosθ(a, e 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWHHbGaeyyXICTaaCyzamaaBaaa leaacaaIXaaabeaakiabg2da9maaemaabaGaaCyyaaGaay5bSlaawI a7aiGacogacaGGVbGaai4CaiabeI7aXjaacIcacaWHHbGaaiilaiaa hwgadaWgaaWcbaGaaGymaaqabaGccaGGPaaaaa@4C96@

 

Thus, a 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaaGymaaqabaaaaa@3806@  represents the projected length of the vector a  in the direction of e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGymaaqabaaaaa@380E@ , as illustrated in the figure.  Similarly, a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaaGOmaaqabaaaaa@3807@  and a 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaaG4maaqabaaaaa@3808@  may be shown to represent the projection of a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggaaaa@3723@  in the directions e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGOmaaqabaaaaa@380F@  and e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaG4maaqabaaaaa@3810@ , respectively.

 

The advantage of representing vectors in a Cartesian basis is that vector addition and multiplication can be expressed as simple operations on the components of the vectors.  For example, let a, b and c be vectors, with components ( a 1 , a 2 , a 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaGOm aaqabaGccaGGSaGaamyyamaaBaaaleaacaaIZaaabeaakiaacMcaaa a@3E7A@ , ( b 1 , b 2 , b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGIb WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadkgadaWgaaWcbaGaaGOm aaqabaGccaGGSaGaamOyamaaBaaaleaacaaIZaaabeaakiaacMcaaa a@3E7D@  and ( c 1 , c 2 , c 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGJb WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadogadaWgaaWcbaGaaGOm aaqabaGccaGGSaGaam4yamaaBaaaleaacaaIZaaabeaakiaacMcaaa a@3E80@ , respectively.  Then, it is straightforward to show that

c=a+b c 1 = a 1 + b 1 ; c 2 = a 2 + b 2 ; c 3 = a 3 + b 3 ab= i=1 3 a i b i c=a×b c 1 = a 2 b 3 a 3 b 2 ; c 2 = a 3 b 1 a 1 b 3 ; c 3 = a 1 b 2 a 2 b 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaiqabaGaaC4yai abg2da9iaahggacqGHRaWkcaWHIbGaaGPaVlaaykW7caaMc8UaaGPa VlabgsDiBlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqaai aaigdaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaaIXaaabeaakiab gUcaRiaadkgadaWgaaWcbaGaaGymaaqabaGccaGG7aGaaGPaVlaayk W7caaMc8UaaGPaVlaadogadaWgaaWcbaGaaGOmaaqabaGccqGH9aqp caWGHbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamOyamaaBaaale aacaaIYaaabeaakiaacUdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaadogadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWGHbWaaSbaaS qaaiaaiodaaeqaaOGaey4kaSIaamOyamaaBaaaleaacaaIZaaabeaa aOqaaiaahggacqGHflY1caWHIbGaeyypa0ZaaabCaeaacaWGHbWaaS baaSqaaiaadMgaaeqaaOGaamOyamaaBaaaleaacaWGPbaabeaaaeaa caWGPbGaeyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoaaOqaaiaaho gacqGH9aqpcaWHHbGaey41aqRaaCOyaiaaykW7caaMc8UaaGPaVlaa ykW7cqGHuhY2caaMc8UaaGPaVlaaykW7caaMc8Uaam4yamaaBaaale aacaaIXaaabeaakiabg2da9iaadggadaWgaaWcbaGaaGOmaaqabaGc caWGIbWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaamyyamaaBaaale aacaaIZaaabeaakiaadkgadaWgaaWcbaGaaGOmaaqabaGccaGG7aGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqaaiaaik daaeqaaOGaeyypa0JaamyyamaaBaaaleaacaaIZaaabeaakiaadkga daWgaaWcbaGaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaaig daaeqaaOGaamOyamaaBaaaleaacaaIZaaabeaakiaacUdacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaadogadaWgaaWcbaGaaG4maaqaba GccqGH9aqpcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamOyamaaBaaa leaacaaIYaaabeaakiabgkHiTiaadggadaWgaaWcbaGaaGOmaaqaba GccaWGIbWaaSbaaSqaaiaaigdaaeqaaaaaaa@C537@

 

 

4. Change of basis

 

Let a be a vector, and let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis.  Suppose that the components of a in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  are known to be ( a 1 , a 2 , a 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadggadaWgaaWcbaGaaGOm aaqabaGccaGGSaGaamyyamaaBaaaleaacaaIZaaabeaakiaacMcaaa a@3E7A@ .  Now, suppose that we wish to compute the components of a in a second Cartesian basis, { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yBamaaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F82@ .  This means we wish to find components ( α 1 , α 2 , α 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacqaHXo qydaWgaaWcbaGaaGymaaqabaGccaGGSaGaeqySde2aaSbaaSqaaiaa ikdaaeqaaOGaaiilaiabeg7aHnaaBaaaleaacaaIZaaabeaakiaacM caaaa@40A5@ , such that

a= α 1 m 1 + α 2 m 2 + α 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGH9a qpcqaHXoqydaWgaaWcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaaiaa igdaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaaC yBamaaBaaaleaacaaIYaaabeaakiabgUcaRiabeg7aHnaaBaaaleaa caaIZaaabeaakiaah2gadaWgaaWcbaGaaG4maaqabaaaaa@474E@

To do so, note that

α 1 =a m 1 = a 1 e 1 m 1 + a 2 e 2 m 1 + a 3 e 3 m 1 α 2 =a m 2 = a 1 e 1 m 2 + a 2 e 2 m 2 + a 3 e 3 m 2 α 3 =a m 3 = a 1 e 1 m 3 + a 2 e 2 m 3 + a 3 e 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqySde 2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaCyyaiabgwSixlaah2ga daWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaaig daaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgwSixlaah2ga daWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaik daaeqaaOGaaCyzamaaBaaaleaacaaIYaaabeaakiabgwSixlaah2ga daWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaio daaeqaaOGaaCyzamaaBaaaleaacaaIZaaabeaakiabgwSixlaah2ga daWgaaWcbaGaaGymaaqabaaakeaacqaHXoqydaWgaaWcbaGaaGOmaa qabaGccqGH9aqpcaWHHbGaeyyXICTaaCyBamaaBaaaleaacaaIYaaa beaakiabg2da9iaadggadaWgaaWcbaGaaGymaaqabaGccaWHLbWaaS baaSqaaiaaigdaaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaaIYaaa beaakiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaWHLbWaaS baaSqaaiaaikdaaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaaIYaaa beaakiabgUcaRiaadggadaWgaaWcbaGaaG4maaqabaGccaWHLbWaaS baaSqaaiaaiodaaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaaIYaaa beaaaOqaaiabeg7aHnaaBaaaleaacaaIZaaabeaakiabg2da9iaahg gacqGHflY1caWHTbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaamyy amaaBaaaleaacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqaba GccqGHflY1caWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyy amaaBaaaleaacaaIYaaabeaakiaahwgadaWgaaWcbaGaaGOmaaqaba GccqGHflY1caWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyy amaaBaaaleaacaaIZaaabeaakiaahwgadaWgaaWcbaGaaG4maaqaba GccqGHflY1caWHTbWaaSbaaSqaaiaaiodaaeqaaaaaaa@9FE8@

This transformation is conveniently written as a matrix operation

[ α ]=[ Q ][ a ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeq ySdegacaGLBbGaayzxaaGaeyypa0ZaamWaaeaacaWGrbaacaGLBbGa ayzxaaWaamWaaeaacaWGHbaacaGLBbGaayzxaaaaaa@4070@ ,

where [ α ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeq ySdegacaGLBbGaayzxaaaaaa@39CA@  is a matrix consisting of the components of a in the basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yBamaaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F82@ , [ a ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yyaaGaay5waiaaw2faaaaa@3911@  is a matrix consisting of the components of a in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@ , and [ Q ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yuaaGaay5waiaaw2faaaaa@3901@  is a `rotation matrix’ as follows

[ α ]=[ α 1 α 2 α 3 ][ a ]=[ a 1 a 2 a 3 ][ Q ]=[ m 1 e 1 m 1 e 2 m 1 e 3 m 2 e 1 m 2 e 2 m 2 e 3 m 3 e 1 m 3 e 2 m 3 e 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeq ySdegacaGLBbGaayzxaaGaeyypa0ZaamWaaqaabeqaaiabeg7aHnaa BaaaleaacaaIXaaabeaaaOqaaiabeg7aHnaaBaaaleaacaaIYaaabe aaaOqaaiabeg7aHnaaBaaaleaacaaIZaaabeaaaaGccaGLBbGaayzx aaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVpaadmaabaGaamyyaaGaay5waiaaw2faaiabg2da 9maadmaaeaqabeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaam yyamaaBaaaleaacaaIYaaabeaaaOqaaiaadggadaWgaaWcbaGaaG4m aaqabaaaaOGaay5waiaaw2faaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaa dmaabaGaamyuaaGaay5waiaaw2faaiabg2da9maadmaaeaqabeaaca WHTbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaa caaIXaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aah2gadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWHLbWaaSbaaSqa aiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaCyBamaaBaaaleaacaaIXaaabeaakiabgwSixlaahwgadaWgaaWc baGaaG4maaqabaaakeaacaWHTbWaaSbaaSqaaiaaikdaaeqaaOGaey yXICTaaCyzamaaBaaaleaacaaIXaaabeaakiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGOmaaqabaGccq GHflY1caWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIYaaabeaaki abgwSixlaahwgadaWgaaWcbaGaaG4maaqabaaakeaacaWHTbWaaSba aSqaaiaaiodaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaIXaaabe aakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWg aaWcbaGaaG4maaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaikdaae qaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyBamaa BaaaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaG4maa qabaaaaOGaay5waiaaw2faaaaa@E5C5@

Note that the elements of [ Q ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yuaaGaay5waiaaw2faaaaa@3901@  have a simple physical interpretation.  For example, m 1 e 1 =cosθ( m 1 , e 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa WcbaGaaGymaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaigdaaeqa aOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiUdeNaaiikaiaah2gada WgaaWcbaGaaGymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIXaaa beaakiaacMcaaaa@47A7@ , where θ( m 1 , e 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaahwgadaWgaaWc baGaaGymaaqabaGccaGGPaaaaa@3DBE@  is the angle between the m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa WcbaGaaGymaaqabaaaaa@3816@  and e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGymaaqabaaaaa@380E@  axes.  Similarly m 1 e 2 =cosθ( m 1 , e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa WcbaGaaGymaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiUdeNaaiikaiaah2gada WgaaWcbaGaaGymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaa beaakiaacMcaaaa@47A9@  where θ( m 1 , e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaahwgadaWgaaWc baGaaGOmaaqabaGccaGGPaaaaa@3DBF@  is the angle between the m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa WcbaGaaGymaaqabaaaaa@3816@  and e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGOmaaqabaaaaa@380F@  axes.  In practice, we usually know the angles between the axes that make up the two bases, so it is simplest to assemble the elements of [ Q ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yuaaGaay5waiaaw2faaaaa@3901@  by putting the cosines of the known angles in the appropriate places.

 

Index notation provides another convenient way to write this transformation:

α i = Q ij a j , Q ij = e i m j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaWgaaWcbaGaamyAaaqaba GccqGH9aqpcaWGrbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadgga daWgaaWcbaGaamOAaaqabaGccaGGSaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGrbWaaSbaaSqaaiaadMgacaWGQbaabeaaki abg2da9iaahwgadaWgaaWcbaGaamyAaaqabaGccqGHflY1caWHTbWa aSbaaSqaaiaadQgaaeqaaaaa@5C28@

You don’t need to know index notation in detail to understand this MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  all you need to know is that

Q ij a j j=1 3 Q ij a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGrbWaaSbaaSqaaiaadMgacaWGQb aabeaakiaadggadaWgaaWcbaGaamOAaaqabaGccaaMc8UaeyyyIO7a aabCaeaacaWGrbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadggada WgaaWcbaGaamOAaaqabaaabaGaamOAaiabg2da9iaaigdaaeaacaaI ZaaaniabggHiLdaaaa@459B@

 

The same approach may be used to find an expression for a i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaa aa@34C6@  in terms of α i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaWgaaWcbaGaamyAaaqaba aaaa@357F@ .  If you work through the details, you will find that

[ a 1 a 2 a 3 ]=[ m 1 e 1 m 2 e 1 m 3 e 1 m 1 e 2 m 2 e 2 m 3 e 2 m 1 e 3 m 2 e 3 m 3 e 3 ][ α 1 α 2 α 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaabaeqabaGaamyyamaaBaaale aacaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGaaGOmaaqabaaakeaa caWGHbWaaSbaaSqaaiaaiodaaeqaaaaakiaawUfacaGLDbaacaaMc8 UaaGPaVlaaykW7cqGH9aqpdaWadaqaauaabeqadmaaaeaacaWHTbWa aSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaIXa aabeaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWH LbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaCyBamaaBaaaleaacaaIZa aabeaakiabgwSixlaahwgadaWgaaWcbaGaaGymaaqabaaakeaacaWH TbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaaca aIYaaabeaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1 caWHLbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaCyBamaaBaaaleaaca aIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaGOmaaqabaaakeaa caWHTbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaale aacaaIZaaabeaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWHLbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaaCyBamaaBaaale aacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaG4maaqabaaa aaGccaGLBbGaayzxaaGaaGPaVlaaykW7daWadaabaeqabaGaeqySde 2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqySde2aaSbaaSqaaiaaikda aeqaaaGcbaGaeqySde2aaSbaaSqaaiaaiodaaeqaaaaakiaawUfaca GLDbaaaaa@855C@

Comparing this result with the formula for α i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaWgaaWcbaGaamyAaaqaba aaaa@357F@  in terms of a i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaa aa@34C6@ , we see that

[ a ]= [ Q ] T [ α ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadggaaiaawUfacaGLDb aacqGH9aqpdaWadaqaaiaadgfaaiaawUfacaGLDbaadaahaaWcbeqa aiaadsfaaaGcdaWadaqaaiabeg7aHbGaay5waiaaw2faaaaa@3E0C@

where the superscript T denotes the transpose (rows and columns interchanged). The transformation matrix [ Q ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yuaaGaay5waiaaw2faaaaa@3901@  is therefore orthogonal, and satisfies

[ Q ] 1 = [ Q ] T [ Q ] [ Q ] T = [ Q ] T [ Q ]=[ I ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpdaWadaqaaiaa dgfaaiaawUfacaGLDbaadaahaaWcbeqaaiaadsfaaaGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7daWadaqaaiaadgfaaiaawUfacaGLDbaadaWada qaaiaadgfaaiaawUfacaGLDbaadaahaaWcbeqaaiaadsfaaaGccqGH 9aqpdaWadaqaaiaadgfaaiaawUfacaGLDbaadaahaaWcbeqaaiaads faaaGcdaWadaqaaiaadgfaaiaawUfacaGLDbaacqGH9aqpdaWadaqa aiaadMeaaiaawUfacaGLDbaaaaa@60DB@

where [I] is the identity matrix.

 

 

 

 

 

 

 

 

 

5. Useful vector operations

 Calculating areas

The area of a triangle bounded by vectors a, b¸and b-a is

A= 1 2 |a×b| MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0JaaGPaVpaalaaaba GaaGymaaqaaiaaikdaaaGaaiiFaiaahggacqGHxdaTcaWHIbGaaiiF aaaa@3D8F@

The area of the parallelogram shown in the picture is 2A.

 

 Calculating angles

The angle between two vectors a and b is

θ= cos 1 ( ab/| a || b | ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCcqGH9aqpcaaMc8Uaci4yai aac+gacaGGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaeWaaeaa caWHHbGaeyyXICTaaCOyaiaac+cadaabdaqaaiaahggaaiaawEa7ca GLiWoadaabdaqaaiaahkgaaiaawEa7caGLiWoaaiaawIcacaGLPaaa aaa@4A32@

 

 Calculating the normal to a surface.

If two vectors a and b can be found which are known to lie in the surface, then the unit normal to the surface is

n=± a×b | a×b | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyypa0JaaGPaVlabgglaXo aalaaabaGaaCyyaiabgEna0kaahkgaaeaadaabdaqaaiaahggacqGH xdaTcaWHIbaacaGLhWUaayjcSdaaaaaa@4345@

If the surface is specified by a parametric equation of the form r=r(s,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaaGPaVlaahkhaca GGOaGaam4CaiaacYcacaWG0bGaaiykaaaa@3B46@ , where s and t are two parameters and r is the position vector of a point on the surface, then two vectors which lie in the plane may be computed from

a= r s ,b= r t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHHbGaeyypa0ZaaSaaaeaacqGHci ITcaWHYbaabaGaeyOaIyRaam4CaaaacaGGSaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaahkgacqGH9aqpdaWcaaqaai abgkGi2kaahkhaaeaacqGHciITcaWG0baaaaaa@4BC2@

 

 Calculating Volumes

The volume of the parallelopiped defined by three vectors a, b, c is

V=|c( a×b )| MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0JaaiiFaiaaykW7ca WHJbGaeyyXIC9aaeWaaeaacaWHHbGaey41aqRaaCOyaaGaayjkaiaa wMcaaiaacYhaaaa@40DC@

The volume of the tetrahedron shown outlined in red is V/6.

 

 

 

 

 

 VECTOR FIELDS AND VECTOR CALCULUS

 

 

1. Scalar field.

 

Let  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis with origin O in three dimensional space.  Let

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaaGPaVlaadIhada WgaaWcbaGaaGymaaqabaGccaWHLbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamiEamaaBaaaleaacaaIYaaabeaakiaahwgadaWgaaWcba GaaGOmaaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGa aCyzamaaBaaaleaacaaIZaaabeaaaaa@4378@

denote the position vector of a point in space.  A scalar field is a scalar valued function of position in space.  A scalar field is a function of the components of the position vector, and so may be expressed as ϕ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcaGGOaGaamiEamaaBaaale aacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGa aiilaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3D13@ . The value of ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  at a particular point in space must be independent of the choice of basis vectors.  A scalar field may be a function of time (and possibly other parameters) as well as position in space.

 

 

2. Vector field

 

Let  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis with origin O in three dimensional space.  Let

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaaGPaVlaadIhada WgaaWcbaGaaGymaaqabaGccaWHLbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamiEamaaBaaaleaacaaIYaaabeaakiaahwgadaWgaaWcba GaaGOmaaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGa aCyzamaaBaaaleaacaaIZaaabeaaaaa@4378@

denote the position vector of a point in space.  A vector field is a vector valued function of position in space.  A vector field is a function of the components of the position vector, and so may be expressed as v( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@3C4A@ .  The vector may also be expressed as components in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaGaaGPaVd aa@3D81@

v( x 1 , x 2 , x 3 )= v 1 ( x 1 , x 2 , x 3 ) e 1 + v 2 ( x 1 , x 2 , x 3 ) e 2 + v 3 ( x 1 , x 2 , x 3 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiabg2da9iaadA hadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEamaaBaaaleaacaaI XaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiilai aadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaCyzamaaBaaaleaa caaIXaaabeaakiabgUcaRiaadAhadaWgaaWcbaGaaGOmaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqaba GccaGGPaGaaCyzamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadAha daWgaaWcbaGaaG4maaqabaGccaGGOaGaamiEamaaBaaaleaacaaIXa aabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaCyzamaaBaaaleaaca aIZaaabeaaaaa@6403@

The magnitude and direction of v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahAhaaaa@3738@  at a particular point in space is independent of the choice of basis vectors.   A vector field may be a function of time (and possibly other parameters) as well as position in space.

 

 

3. Change of basis for scalar fields.

Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis with origin O in three dimensional space. Express the position vector of a point relative to O in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  as

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaaGPaVlaadIhada WgaaWcbaGaaGymaaqabaGccaWHLbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamiEamaaBaaaleaacaaIYaaabeaakiaahwgadaWgaaWcba GaaGOmaaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGa aCyzamaaBaaaleaacaaIZaaabeaaaaa@4378@

and let ϕ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcaGGOaGaamiEamaaBaaale aacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGa aiilaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaGPaVdaa@3E9E@  be a scalar field.

Let { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaGaaGPaVd aa@3D99@  be a second Cartesian basis, with origin P.  Let c OP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHJbGaeyyyIO7aa8HaaeaacaWGpb GaamiuaaGaay51GaGaaGPaVdaa@3A62@  denote the position vector of  P relative to O. Express the position vector of a point relative to P in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaGaaGPaVd aa@3D99@  as

p= ξ 1 m 1 + ξ 2 m 2 + ξ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHWbGaeyypa0JaeqOVdG3aaSbaaS qaaiaaigdaaeqaaOGaaCyBamaaBaaaleaacaaIXaaabeaakiabgUca Riabe67a4naaBaaaleaacaaIYaaabeaakiaah2gadaWgaaWcbaGaaG OmaaqabaGccqGHRaWkcqaH+oaEdaWgaaWcbaGaaG4maaqabaGccaWH TbWaaSbaaSqaaiaaiodaaeqaaaaa@4455@

 

To find ϕ( ξ 1 , ξ 2 , ξ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcaGGOaGaeqOVdG3aaSbaaS qaaiaaigdaaeqaaOGaaiilaiabe67a4naaBaaaleaacaaIYaaabeaa kiaacYcacqaH+oaEdaWgaaWcbaGaaG4maaqabaGccaGGPaGaaGPaVd aa@40F0@ , use the following procedure.  First, express  p as components in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@ , using the procedure outlined in Section 1.4:

p= p 1 e 1 + p 2 e 2 + p 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHWbGaeyypa0JaamiCamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWGWbWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadchadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@41D3@

where

p 1 = ξ 1 e 1 m 1 + ξ 2 e 2 m 1 + ξ 3 e 3 m 1 p 2 = ξ 1 e 1 m 2 + ξ 2 e 2 m 2 + ξ 3 e 3 m 2 p 3 = ξ 1 e 1 m 3 + ξ 2 e 2 m 3 + ξ 3 e 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadchadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcqaH+oaEdaWgaaWcbaGaaGymaaqabaGccaWHLbWa aSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaaIXa aabeaakiabgUcaRiabe67a4naaBaaaleaacaaIYaaabeaakiaahwga daWgaaWcbaGaaGOmaaqabaGccqGHflY1caWHTbWaaSbaaSqaaiaaig daaeqaaOGaey4kaSIaeqOVdG3aaSbaaSqaaiaaiodaaeqaaOGaaCyz amaaBaaaleaacaaIZaaabeaakiabgwSixlaah2gadaWgaaWcbaGaaG ymaaqabaaakeaacaWGWbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Ja eqOVdG3aaSbaaSqaaiaaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXa aabeaakiabgwSixlaah2gadaWgaaWcbaGaaGOmaaqabaGccqGHRaWk cqaH+oaEdaWgaaWcbaGaaGOmaaqabaGccaWHLbWaaSbaaSqaaiaaik daaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaaIYaaabeaakiabgUca Riabe67a4naaBaaaleaacaaIZaaabeaakiaahwgadaWgaaWcbaGaaG 4maaqabaGccqGHflY1caWHTbWaaSbaaSqaaiaaikdaaeqaaaGcbaGa amiCamaaBaaaleaacaaIZaaabeaakiabg2da9iabe67a4naaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHflY1 caWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaeqOVdG3aaSbaaS qaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYaaabeaakiabgwSi xlaah2gadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcqaH+oaEdaWgaa WcbaGaaG4maaqabaGccaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaeyyX ICTaaCyBamaaBaaaleaacaaIZaaabeaaaaaa@8FD5@

or, using index notation

p i = Q ij ξ j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamyuamaaBaaaleaacaWGPbGaamOAaaqabaGccqaH+oaE daWgaaWcbaGaamOAaaqabaaaaa@3BAB@

where the transformation matrix Q ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3918@  is defined in Sect 1.4.

Now, express c as components in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@ , and note that

r=p+c x 1 e 1 + x 2 e 2 + x 3 e 3 = p 1 e 1 + p 2 e 2 + p 3 e 3 + c 1 e 1 + c 2 e 2 + c 3 e 3 x 1 = p 1 + c 1 , x 2 = p 2 + c 2 , x 3 = p 3 + c 3 x i = Q ij ξ j + c i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaCOCaiabg2da9iaahchacqGH RaWkcaWHJbaabaGaeyO0H4TaaGPaVlaaykW7caWG4bWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaa dIhadaWgaaWcbaGaaGOmaaqabaGccaWHLbWaaSbaaSqaaiaahkdaae qaaOGaey4kaSIaamiEamaaBaaaleaacaaIZaaabeaakiaahwgadaWg aaWcbaGaaG4maaqabaGccqGH9aqpcaWGWbWaaSbaaSqaaiaaigdaae qaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadchadaWg aaWcbaGaaGOmaaqabaGccaWHLbWaaSbaaSqaaiaahkdaaeqaaOGaey 4kaSIaamiCamaaBaaaleaacaaIZaaabeaakiaahwgadaWgaaWcbaGa aG4maaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaigdaaeqaaOGaaC yzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadogadaWgaaWcbaGa aGOmaaqabaGccaWHLbWaaSbaaSqaaiaahkdaaeqaaOGaey4kaSIaam 4yamaaBaaaleaacaaIZaaabeaakiaahwgadaWgaaWcbaGaaG4maaqa baaakeaacqGHshI3caWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0 JaamiCamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadogadaWgaaWc baGaaGymaaqabaGccaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqp caWGWbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaam4yamaaBaaale aacaaIYaaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaamiEamaaBaaaleaacaaIZaaabeaakiabg2da9i aadchadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGJbWaaSbaaSqa aiaaiodaaeqaaaGcbaGaeyO0H4TaamiEamaaBaaaleaacaWGPbaabe aakiabg2da9iaadgfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeqOV dG3aaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaam4yamaaBaaaleaaca WGPbaabeaaaaaa@B0DA@

so that

ϕ( x 1 , x 2 , x 3 )=ϕ( p 1 + c 1 , p 2 + c 2 , p 3 + c 3 ) =ϕ( Q 1j ξ j + c 1 , Q 2j ξ j + c 2 , Q 3j ξ j + c 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabew9aMjaacIcacaWG4bWaaS baaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqa baGccaGGSaGaamiEamaaBaaaleaacaaIZaaabeaakiaacMcacqGH9a qpcqaHvpGzcaGGOaGaamiCamaaBaaaleaacaaIXaaabeaakiabgUca RiaadogadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiCamaaBaaale aacaaIYaaabeaakiabgUcaRiaadogadaWgaaWcbaGaaGOmaaqabaGc caGGSaGaamiCamaaBaaaleaacaaIZaaabeaakiabgUcaRiaadogada WgaaWcbaGaaG4maaqabaGccaGGPaaabaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7cqGH9aqpcqaHvpGzcaGGOaGaamyuamaaBaaaleaacaaI XaGaamOAaaqabaGccqaH+oaEdaWgaaWcbaGaamOAaaqabaGccqGHRa WkcaWGJbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadgfadaWgaaWc baGaaGOmaiaadQgaaeqaaOGaeqOVdG3aaSbaaSqaaiaadQgaaeqaaO Gaey4kaSIaam4yamaaBaaaleaacaaIYaaabeaakiaacYcacaWGrbWa aSbaaSqaaiaaiodacaWGQbaabeaakiabe67a4naaBaaaleaacaWGQb aabeaakiabgUcaRiaadogadaWgaaWcbaGaaG4maaqabaGccaGGPaaa aaa@9BE1@

 

 

 

4. Change of basis for vector fields.

Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis with origin O in three dimensional space. Express the position vector of a point relative to O in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  as

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@41ED@

and let v( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@3C4A@  be a vector  field, with components

v( x 1 , x 2 , x 3 )= v 1 ( x 1 , x 2 , x 3 ) e 1 + v 2 ( x 1 , x 2 , x 3 ) e 2 + v 3 ( x 1 , x 2 , x 3 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiabg2da9iaadA hadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEamaaBaaaleaacaaI XaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiilai aadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaCyzamaaBaaaleaa caaIXaaabeaakiabgUcaRiaadAhadaWgaaWcbaGaaGOmaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqaba GccaGGPaGaaCyzamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadAha daWgaaWcbaGaaG4maaqabaGccaGGOaGaamiEamaaBaaaleaacaaIXa aabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaCyzamaaBaaaleaaca aIZaaabeaaaaa@6403@

Let { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@  be a second Cartesian basis, with origin P.  Let c OP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHJbGaeyyyIO7aa8HaaeaacaWGpb GaamiuaaGaay51Gaaaaa@38D7@  denote the position vector of  P relative to O. Express the position vector of a point relative to P in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yBamaaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F82@  as

p= ξ 1 m 1 + ξ 2 m 2 + ξ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHWbGaeyypa0JaeqOVdG3aaSbaaS qaaiaaigdaaeqaaOGaaCyBamaaBaaaleaacaaIXaaabeaakiabgUca Riabe67a4naaBaaaleaacaaIYaaabeaakiaah2gadaWgaaWcbaGaaG OmaaqabaGccqGHRaWkcqaH+oaEdaWgaaWcbaGaaG4maaqabaGccaWH TbWaaSbaaSqaaiaaiodaaeqaaaaa@4455@

 

To express the vector field as components in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@  and as a function of the components of p, use the following procedure.  First, express ( v 1 , v 2 , v 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamODamaaBaaaleaacaaIXa aabeaakiaacYcacaWG2bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dAhadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B45@  in terms of ( ξ 1 , ξ 2 , ξ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaeqOVdG3aaSbaaSqaaiaaig daaeqaaOGaaiilaiabe67a4naaBaaaleaacaaIYaaabeaakiaacYca cqaH+oaEdaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3D9D@  using the procedure outlined for scalar fields in the preceding section

v k ( x 1 , x 2 , x 3 )= v k ( p 1 + c 1 , p 2 + c 2 , p 3 + c 3 ) = v k ( Q 1j ξ j + c 1 , Q 2j ξ j + c 2 , Q 3j ξ j + c 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadAhadaWgaaWcbaGaam4Aaa qabaGccaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG 4bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG 4maaqabaGccaGGPaGaeyypa0JaamODamaaBaaaleaacaWGRbaabeaa kiaacIcacaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaam4yam aaBaaaleaacaaIXaaabeaakiaacYcacaWGWbWaaSbaaSqaaiaaikda aeqaaOGaey4kaSIaam4yamaaBaaaleaacaaIYaaabeaakiaacYcaca WGWbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaam4yamaaBaaaleaa caaIZaaabeaakiaacMcaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa Vlabg2da9iaadAhadaWgaaWcbaGaam4AaaqabaGccaGGOaGaamyuam aaBaaaleaacaaIXaGaamOAaaqabaGccqaH+oaEdaWgaaWcbaGaamOA aaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaigdaaeqaaOGaaiilai aadgfadaWgaaWcbaGaaGOmaiaadQgaaeqaaOGaeqOVdG3aaSbaaSqa aiaadQgaaeqaaOGaey4kaSIaam4yamaaBaaaleaacaaIYaaabeaaki aacYcacaWGrbWaaSbaaSqaaiaaiodacaWGQbaabeaakiabe67a4naa BaaaleaacaWGQbaabeaakiabgUcaRiaadogadaWgaaWcbaGaaG4maa qabaGccaGGPaaaaaa@9CEC@

for k=1,2,3.  Now, find the components  of v in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@  using the procedure outlined in Section 1.4.  Using index notation, the result is

v= Q 1i v i ( Q 1j ξ j + c 1 , Q 2j ξ j + c 2 , Q 3j ξ j + c 3 ) e 1 + Q 2i v i ( Q 1j ξ j + c 1 , Q 2j ξ j + c 2 , Q 3j ξ j + c 3 ) e 2 + Q 2i v i ( Q 1j ξ j + c 1 , Q 2j ξ j + c 2 , Q 3j ξ j + c 3 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahAhacqGH9aqpcaWGrbWaaS baaSqaaiaaigdacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqa baGccaGGOaGaamyuamaaBaaaleaacaaIXaGaamOAaaqabaGccqaH+o aEdaWgaaWcbaGaamOAaaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaa igdaaeqaaOGaaiilaiaadgfadaWgaaWcbaGaaGOmaiaadQgaaeqaaO GaeqOVdG3aaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaam4yamaaBaaa leaacaaIYaaabeaakiaacYcacaWGrbWaaSbaaSqaaiaaiodacaWGQb aabeaakiabe67a4naaBaaaleaacaWGQbaabeaakiabgUcaRiaadoga daWgaaWcbaGaaG4maaqabaGccaGGPaGaaCyzamaaBaaaleaacaaIXa aabeaaaOqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab gUcaRiaadgfadaWgaaWcbaGaaGOmaiaadMgaaeqaaOGaamODamaaBa aaleaacaWGPbaabeaakiaacIcacaWGrbWaaSbaaSqaaiaaigdacaWG Qbaabeaakiabe67a4naaBaaaleaacaWGQbaabeaakiabgUcaRiaado gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyuamaaBaaaleaacaaI YaGaamOAaaqabaGccqaH+oaEdaWgaaWcbaGaamOAaaqabaGccqGHRa WkcaWGJbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadgfadaWgaaWc baGaaG4maiaadQgaaeqaaOGaeqOVdG3aaSbaaSqaaiaadQgaaeqaaO Gaey4kaSIaam4yamaaBaaaleaacaaIZaaabeaakiaacMcacaWHLbWa aSbaaSqaaiaaikdaaeqaaaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaey4kaSIaamyuamaaBaaaleaacaaIYaGaamyAaaqa baGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadgfadaWgaa WcbaGaaGymaiaadQgaaeqaaOGaeqOVdG3aaSbaaSqaaiaadQgaaeqa aOGaey4kaSIaam4yamaaBaaaleaacaaIXaaabeaakiaacYcacaWGrb WaaSbaaSqaaiaaikdacaWGQbaabeaakiabe67a4naaBaaaleaacaWG QbaabeaakiabgUcaRiaadogadaWgaaWcbaGaaGOmaaqabaGccaGGSa GaamyuamaaBaaaleaacaaIZaGaamOAaaqabaGccqaH+oaEdaWgaaWc baGaamOAaaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaiodaaeqaaO GaaiykaiaahwgadaWgaaWcbaGaaG4maaqabaaaaaa@B08C@

 

 

5. Time derivatives of vectors

 

Let a(t) be a vector whose magnitude and direction vary with time, t.  Suppose that { i,j,k } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yAaiaacYcacaWHQbGaaiilaiaahUgaaiaawUhacaGL9baaaaa@3CA3@  is a fixed basis, i.e. independent of time.  We may express a(t) in terms of components ( a x , a y , a z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGHb WaaSbaaSqaaiaadIhaaeqaaOGaaiilaiaadggadaWgaaWcbaGaamyE aaqabaGccaGGSaGaamyyamaaBaaaleaacaWG6baabeaakiaacMcaaa a@3F40@  in the basis { i,j,k } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yAaiaacYcacaWHQbGaaiilaiaahUgaaiaawUhacaGL9baaaaa@3CA3@  as

a(t)= a x i+ a y j+ a z k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacaGGOa GaamiDaiaacMcacqGH9aqpcaWGHbWaaSbaaSqaaiaadIhaaeqaaOGa aCyAaiabgUcaRiaadggadaWgaaWcbaGaamyEaaqabaGccaWHQbGaey 4kaSIaamyyamaaBaaaleaacaWG6baabeaakiaahUgaaaa@4566@ .

The time derivative of a is defined using the usual rules of calculus

a ˙ (t)= d dt a(t)= lim 0 a(t+)a(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqahggagaGaai aacIcacaWG0bGaaiykaiabg2da9maalaaabaGaamizaaqaaiaadsga caWG0baaaiaahggacaGGOaGaamiDaiaacMcacqGH9aqpdaWfqaqaai GacYgacaGGPbGaaiyBaaWcbaGaeyicI4SaeyOKH4QaaGimaaqabaGc daWcaaqaaiaahggacaGGOaGaamiDaiabgUcaRiabgIGiolaacMcacq GHsislcaWHHbGaaiikaiaadshacaGGPaaabaGaeyicI4maaaaa@543E@ ,

or in component form as

a ˙ (t)= a ˙ x i+ a ˙ y j+ a ˙ z k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqahggagaGaai aacIcacaWG0bGaaiykaiabg2da9iqadggagaGaamaaBaaaleaacaWG 4baabeaakiaahMgacqGHRaWkceWGHbGbaiaadaWgaaWcbaGaamyEaa qabaGccaWHQbGaey4kaSIabmyyayaacaWaaSbaaSqaaiaadQhaaeqa aOGaaC4Aaaaa@458A@

The definition of the time derivative of a vector may be used to show the following rules

d dt [ α(t)a(t) ]= α ˙ (t)a(t)+α(t) a ˙ (t) d dt [ a(t)b(t) ]= a ˙ (t)b(t)+a(t) b ˙ (t) d dt [ a(t)×b(t) ]= a ˙ (t)×b(t)+a(t)× b ˙ (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaSaaae aacaWGKbaabaGaamizaiaadshaaaWaamWaaeaacqaHXoqycaGGOaGa amiDaiaacMcacaWHHbGaaiikaiaadshacaGGPaaacaGLBbGaayzxaa Gaeyypa0JafqySdeMbaiaacaGGOaGaamiDaiaacMcacaWHHbGaaiik aiaadshacaGGPaGaey4kaSIaeqySdeMaaiikaiaadshacaGGPaGabC yyayaacaGaaiikaiaadshacaGGPaaabaWaaSaaaeaacaWGKbaabaGa amizaiaadshaaaWaamWaaeaacaWHHbGaaiikaiaadshacaGGPaGaey yXICTaaCOyaiaacIcacaWG0bGaaiykaaGaay5waiaaw2faaiabg2da 9iqahggagaGaaiaacIcacaWG0bGaaiykaiabgwSixlaahkgacaGGOa GaamiDaiaacMcacqGHRaWkcaWHHbGaaiikaiaadshacaGGPaGaeyyX ICTabCOyayaacaGaaiikaiaadshacaGGPaaabaWaaSaaaeaacaWGKb aabaGaamizaiaadshaaaWaamWaaeaacaWHHbGaaiikaiaadshacaGG PaGaey41aqRaaCOyaiaacIcacaWG0bGaaiykaaGaay5waiaaw2faai abg2da9iqahggagaGaaiaacIcacaWG0bGaaiykaiabgEna0kaahkga caGGOaGaamiDaiaacMcacqGHRaWkcaWHHbGaaiikaiaadshacaGGPa Gaey41aqRabCOyayaacaGaaiikaiaadshacaGGPaaaaaa@9416@

 

 

6. Using a rotating basis

 

It is often convenient to express position vectors as components in a basis which rotates with time.  To write equations of motion one must evaluate time derivatives of rotating vectors.

 

Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a basis which rotates with instantaneous angular velocity Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHPoaaaa@33FA@ .  Then,

d e 1 dt =Ω× e 1 , d e 2 dt =Ω× e 2 , d e 3 dt =Ω× e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHLbWaaSbaaS qaaiaaigdaaeqaaaGcbaGaamizaiaadshaaaGaeyypa0JaaCyQdiab gEna0kaahwgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGaamizaiaa hwgadaWgaaWcbaGaaGOmaaqabaaakeaacaWGKbGaamiDaaaacqGH9a qpcaWHPoGaey41aqRaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaads gacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadshaaaGa eyypa0JaaCyQdiabgEna0kaahwgadaWgaaWcbaGaaG4maaqabaaaaa@68F1@

 

 

7. Gradient of a scalar field.

 

Let ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  be a scalar field in three dimensional space.  The gradient of ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  is a vector field denoted by grad(ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabEgacaqGYb GaaeyyaiaabsgacaGGOaGaeqy1dyMaaiykaaaa@3D04@  or ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqaHvpGzaaa@3613@ , and is defined so that

(ϕ)a= lim 0 ϕ(r+a)ϕ(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaey4bIeTaeqy1dyMaaiykai abgwSixlaahggacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWc baGaeyicI4SaeyOKH4QaaGimaaqabaGcdaWcaaqaaiabew9aMjaacI cacaWHYbGaey4kaSIaeyicI4SaaGPaVlaahggacaGGPaGaeyOeI0Ia eqy1dyMaaiikaiaahkhacaGGPaaabaGaeyicI4maaaaa@5278@

for every position r in space and for every vector a.

 

Let  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis with origin O in three dimensional space.  Let

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@41ED@

denote the position vector of a point in space.  Express ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  as a function of the components of r ϕ=ϕ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpcqaHvpGzcaGGOa GaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqa aiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqabaGcca GGPaaaaa@3FE1@ .  The gradient of  ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  in this basis is then given by

ϕ= ϕ x 1 e 1 + ϕ x 2 e 2 + ϕ x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqaHvpGzcqGH9aqpdaWcaa qaaiabgkGi2kabew9aMbqaaiabgkGi2kaadIhadaWgaaWcbaGaaGym aaqabaaaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRmaala aabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaamiEamaaBaaaleaacaaI YaaabeaaaaGccaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSYaaS aaaeaacqGHciITcqaHvpGzaeaacqGHciITcaWG4bWaaSbaaSqaaiaa iodaaeqaaaaakiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@522C@

 

8. Gradient of a vector field

 

Let v be a vector field in three dimensional space.  The gradient of v is a tensor field denoted by grad(v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqGNbGaaeOCaiaabggacaqGKbGaai ikaiaahAhacaGGPaaaaa@38C7@  or v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0caWH2baaaa@354A@ , and is defined so that

(v)a= lim 0 v(r+a)v(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaey4bIeTaaCODaiaacMcacq GHflY1caWHHbGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqa aiabgIGiolabgkziUkaaicdaaeqaaOWaaSaaaeaacaWH2bGaaiikai aahkhacqGHRaWkcqGHiiIZcaaMc8UaaCyyaiaacMcacqGHsislcaWH 2bGaaiikaiaahkhacaGGPaaabaGaeyicI4maaaaa@501D@

for every position r in space and for every vector a.

 

Let  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  be a Cartesian basis with origin O in three dimensional space.  Let

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@41ED@

denote the position vector of a point in space.  Express v as a function of the components of r, so that v=v( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaCODaiaacIcaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIZaaabeaakiaacM caaaa@3E4F@ .  The gradient of  v in this basis is then given by

v=[ v 1 x 1 v 1 x 2 v 1 x 3 v 2 x 1 v 2 x 2 v 2 x 3 v 3 x 1 v 3 x 2 v 3 x 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0caWH2bGaeyypa0ZaamWaae aafaqabeWadaaabaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaa igdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaa aakeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaaGymaaqabaaa keaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOqaamaala aabaGaeyOaIyRaamODamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi 2kaadIhadaWgaaWcbaGaaG4maaqabaaaaaGcbaWaaSaaaeaacqGHci ITcaWG2bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaa BaaaleaacaaIXaaabeaaaaaakeaadaWcaaqaaiabgkGi2kaadAhada WgaaWcbaGaaGOmaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaa ikdaaeqaaaaaaOqaamaalaaabaGaeyOaIyRaamODamaaBaaaleaaca aIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaa aaGcbaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaaiodaaeqaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaakeaadaWc aaqaaiabgkGi2kaadAhadaWgaaWcbaGaaG4maaqabaaakeaacqGHci ITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOqaamaalaaabaGaeyOa IyRaamODamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhada WgaaWcbaGaaG4maaqabaaaaaaaaOGaay5waiaaw2faaaaa@74D1@

Alternatively, in index notation

[ v ] ij v i x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabgEGirlaahAhaaiaawU facaGLDbaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyyyIO7aaSaa aeaacqGHciITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIy RaamiEamaaBaaaleaacaWGQbaabeaaaaaaaa@422B@

 

9. Divergence of a vector field

 

Let v be a vector field in three dimensional space.  The divergence of v is a scalar field denoted by div(v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqGKbGaaeyAaiaabAhacaGGOaGaaC ODaiaacMcaaaa@37E9@  or v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHflY1caWH2baaaa@3794@ .  Formally, it is defined as trace(grad(v)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzaiaabIcacaqGNbGaaeOCaiaabggacaqGKbGaaeikaiaahAhacaqG PaGaaeykaaaa@3EBA@  (the trace of a tensor is the sum of its diagonal terms). 

 

Let  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis with origin O in three dimensional space.  Let

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@41ED@

denote the position vector of a point in space.  Express v as a function of the components of r: v=v( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaCODaiaacIcaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIZaaabeaakiaacM caaaa@3E4F@ . The divergence of v is then

 

div(v)= v 1 x 1 + v 2 x 2 + v 3 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqGKbGaaeyAaiaabAhacaqGOaGaaC ODaiaabMcacaqG9aWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaa igdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaa GccqGHRaWkdaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaaGOmaaqa baaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabgU caRmaalaaabaGaeyOaIyRaamODamaaBaaaleaacaaIZaaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaaaa@4E89@

 

 

10. Curl of a vector field.

 

Let v be a vector field in three dimensional space.  The curl of  v  is a vector field denoted by curl(v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqGJbGaaeyDaiaabkhacaqGSbGaai ikaiaahAhacaGGPaaaaa@38DF@  or ×v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHxdaTcaWH2baaaa@3761@ .  It is best defined in terms of its components in a given basis, although its magnitude and direction are not dependent on the choice of basis.

 

Let  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  be a Cartesian basis with origin O in three dimensional space.  Let

r= x 1 e 1 + x 2 e 2 + x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHYbGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@41ED@

denote the position vector of a point in space.  Express v as a function of the components of r  v=v( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaCODaiaacIcaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIZaaabeaakiaacM caaaa@3E4F@ . The curl of  v in this basis is then given by

×v=| e 1 e 2 e 3 x 1 x 2 x 3 v 1 v 2 v 3 |=( v 3 x 2 v 2 x 3 ) e 1 +( v 1 x 3 v 3 x 1 ) e 2 +( v 2 x 1 v 1 x 2 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHxdaTcaWH2bGaeyypa0 ZaaqWaaeaafaqabeWadaaabaGaaCyzamaaBaaaleaacaaIXaaabeaa aOqaaiaahwgadaWgaaWcbaGaaGOmaaqabaaakeaacaWHLbWaaSbaaS qaaiaaiodaaeqaaaGcbaWaaSaaaeaacqGHciITaeaacqGHciITcaWG 4bWaaSbaaSqaaiaaigdaaeqaaaaaaOqaamaalaaabaGaeyOaIylaba GaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaaakeaadaWcaaqa aiabgkGi2cqaaiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaa GcbaGaamODamaaBaaaleaacaaIXaaabeaaaOqaaiaadAhadaWgaaWc baGaaGOmaaqabaaakeaacaWG2bWaaSbaaSqaaiaaiodaaeqaaaaaaO Gaay5bSlaawIa7aiabg2da9maabmaabaWaaSaaaeaacqGHciITcaWG 2bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaaIYaaabeaaaaGccqGHsisldaWcaaqaaiabgkGi2kaadAhadaWg aaWcbaGaaGOmaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaio daaeqaaaaaaOGaayjkaiaawMcaaiaahwgadaWgaaWcbaGaaGymaaqa baGccqGHRaWkdaqadaqaamaalaaabaGaeyOaIyRaamODamaaBaaale aacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqa baaaaOGaeyOeI0YaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaaio daaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaa kiaawIcacaGLPaaacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaS YaaeWaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaaGOmaaqa baaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgk HiTmaalaaabaGaeyOaIyRaamODamaaBaaaleaacaaIXaaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaay zkaaGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@8D7A@

Using index notation, this may be expressed as

[ ×v ] i = ijk v j x k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabgEGirlabgEna0kaahA haaiaawUfacaGLDbaadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcqGH iiIZdaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakmaalaaabaGaey OaIyRaamODamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaam4Aaaqabaaaaaaa@4719@

 

 

11. The Divergence Theorem.

Let V be a closed region in three dimensional space, bounded by an orientable surface S. Let n  denote the unit vector normal to S, taken so that n points out of V. Let u be a vector field which is continuous and has continuous first partial derivatives in some domain containing T.  Then

V div(u) dV= S un dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiaabsgacaqGPbGaaeODai aabIcacaWH1bGaaeykaaWcbaGaamOvaaqab0Gaey4kIipakiaaykW7 caWGKbGaamOvaiabg2da9maapefabaGaaCyDaiabgwSixlaah6gaaS qaaiaadofaaeqaniabgUIiYdGccaaMc8Uaamizaiaadgeaaaa@4A0F@

alternatively, expressed in index notation

V u i x i dV= S u i n i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaamaalaaabaGaeyOaIyRaam yDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaamyAaaqabaaaaOGaamizaiaadAfacqGH9aqpdaWdrbqaaiaadw hadaWgaaWcbaGaamyAaaqabaGccaWGUbWaaSbaaSqaaiaadMgaaeqa aOGaamizaiaadgeaaSqaaiaadofaaeqaniabgUIiYdaaleaacaWGwb aabeqdcqGHRiI8aaaa@48D5@

For a proof of this extremely useful theorem consult e.g. Kreyzig, Advanced Engineering Mathematics, Wiley, New York, (1998).

 

 

 

MATRICES

 

1. Definition

 

An (n×m) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGUb Gaey41aqRaamyBaiaacMcaaaa@3B8E@  matrix [A] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacUfacaWGbb Gaaiyxaaaa@38BF@  is a set of numbers, arranged in m rows and n columns

[ A ]=[ a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a m1 a m2 a m3 a mn ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiabg2da9maadmaaeaqabeaacaWGHbWaaSba aSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaadggada WgaaWcbaGaaGymaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8Uaamyy amaaBaaaleaacaaIXaGaaG4maaqabaGccaaMc8UaaGPaVlabl+Uimj aaykW7caaMc8UaamyyamaaBaaaleaacaaIXaGaamOBaaqabaaakeaa caWGHbWaaSbaaSqaaiaaikdacaaIXaaabeaakiaaykW7caaMc8UaaG PaVlaadggadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaGPaVlaaykW7 caaMc8UaamyyamaaBaaaleaacaaIYaGaaG4maaqabaGccaaMc8UaaG PaVlabl+UimjaaykW7caaMc8UaamyyamaaBaaaleaacaaIYaGaamOB aaqabaaakeaacaaMc8UaaGPaVlabl6UinjaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeSO7I0KaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqWIUlstcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaeSy8I8KaaGPaVlaaykW7caaMc8UaeSO7I0eabaGaamyyam aaBaaaleaacaWGTbGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caWG HbWaaSbaaSqaaiaad2gacaaIYaaabeaakiaaykW7caaMc8UaaGPaVl aadggadaWgaaWcbaGaamyBaiaaiodaaeqaaOGaaGPaVlaaykW7cqWI VlctcaaMc8UaaGPaVlaadggadaWgaaWcbaGaamyBaiaad6gaaeqaaa aakiaawUfacaGLDbaaaaa@C9EF@

 

 A square matrix has equal numbers of rows and columns

 A diagonal matrix is a square matrix with elements such that a ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaGimaaaa@3AF2@  for ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMgacqGHGj sUcaWGQbaaaa@39DD@

 The identity matrix [ I ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam ysaaGaay5waiaaw2faaaaa@38F9@  is a diagonal matrix for which all diagonal elements a ii =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadMgaaeqaaOGaeyypa0JaaGymaaaa@3AF2@

 A symmetric matrix is a square matrix with elements such that a ij = a ji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaWG QbGaamyAaaqabaaaaa@3D27@

 A skew symmetric matrix is a square matrix with elements such that a ij = a ji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaamyyamaaBaaa leaacaWGQbGaamyAaaqabaaaaa@3E14@

 

 

2. Matrix operations

 

 Addition  Let  [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  and [ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam OqaaGaay5waiaaw2faaaaa@38F2@  be two matrices of order (m×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGTb Gaey41aqRaamOBaiaacMcaaaa@3B8E@  with elements a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3928@  and b ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3929@ .  Then

[ C ]=[ A ]+[ B ] c ij = a ij + b ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4qaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyqaaGaay5waiaa w2faaiabgUcaRmaadmaabaGaamOqaaGaay5waiaaw2faaiabgsDiBl aadogadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0Jaamyyamaa BaaaleaacaWGPbGaamOAaaqabaGccqGHRaWkcaWGIbWaaSbaaSqaai aadMgacaWGQbaabeaaaaa@4D74@

 

 

 Multiplication by a scalar.  Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be a matrix with elements a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3928@ , and let k be a scalar.  Then

[ B ]=k[ A ] b ij =k a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam OqaaGaay5waiaaw2faaiabg2da9iaadUgadaWadaqaaiaadgeaaiaa wUfacaGLDbaacqGHuhY2caWGIbWaaSbaaSqaaiaadMgacaWGQbaabe aakiabg2da9iaadUgacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaa aaa@47DB@

 

 

 Multiplication by a matrix. Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be a matrix of order (m×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGTb Gaey41aqRaamOBaiaacMcaaaa@3B8E@  with elements a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3928@ , and let [ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam OqaaGaay5waiaaw2faaaaa@38F2@  be a matrix of order (p×q) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGWb Gaey41aqRaamyCaiaacMcaaaa@3B94@  with elements b ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3929@ .  The product [ C ]=[ A ][ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4qaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyqaaGaay5waiaa w2faamaadmaabaGaamOqaaGaay5waiaaw2faaaaa@3F6A@  is defined only if n=p, and is an (m×q) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGTb Gaey41aqRaamyCaiaacMcaaaa@3B91@  matrix such that

[ C ]=[ A ][ B ] c ij = k=1 n a ik b kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4qaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyqaaGaay5waiaa w2faamaadmaabaGaamOqaaGaay5waiaaw2faaiabgsDiBlaadogada WgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaabCaeaacaWGHbWa aSbaaSqaaiaadMgacaWGRbaabeaakiaadkgadaWgaaWcbaGaam4Aai aadQgaaeqaaaqaaiaadUgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGH ris5aaaa@518E@

Note that multiplication is distributive and associative, but not commutative, i.e.

[ A ]( [ B ]+[ C ] )=[ A ][ B ]+[ A ][ C ][ A ]( [ B ][ C ] )=( [ A ][ B ] )[ C ][ A ][ B ][ B ][ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaabmaabaWaamWaaeaacaWGcbaacaGLBbGa ayzxaaGaey4kaSYaamWaaeaacaWGdbaacaGLBbGaayzxaaaacaGLOa GaayzkaaGaeyypa0ZaamWaaeaacaWGbbaacaGLBbGaayzxaaWaamWa aeaacaWGcbaacaGLBbGaayzxaaGaey4kaSYaamWaaeaacaWGbbaaca GLBbGaayzxaaWaamWaaeaacaWGdbaacaGLBbGaayzxaaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8+aamWaaeaacaWGbbaacaGLBb GaayzxaaWaaeWaaeaadaWadaqaaiaadkeaaiaawUfacaGLDbaadaWa daqaaiaadoeaaiaawUfacaGLDbaaaiaawIcacaGLPaaacqGH9aqpda qadaqaamaadmaabaGaamyqaaGaay5waiaaw2faamaadmaabaGaamOq aaGaay5waiaaw2faaaGaayjkaiaawMcaamaadmaabaGaam4qaaGaay 5waiaaw2faaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8+aamWaaeaacaWGbbaacaGLBbGaayzxaaWaamWaaeaaca WGcbaacaGLBbGaayzxaaGaeyiyIK7aamWaaeaacaWGcbaacaGLBbGa ayzxaaWaamWaaeaacaWGbbaacaGLBbGaayzxaaaaaa@844B@

The multiplication of a vector by a matrix is a particularly important operation.  Let b and c be two vectors with n components, which we think of as (1×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaaIXa Gaey41aqRaamOBaiaacMcaaaa@3B57@  matrices.  Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be an (m×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGTb Gaey41aqRaamOBaiaacMcaaaa@3B8E@  matrix.  Thus

b=[ b 1 b 2 b 3 b n ]c=[ c 1 c 2 c 3 c n ][ A ]=[ a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a m1 a m2 a m3 a mn ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahkgacqGH9a qpdaWadaabaeqabaGaamOyamaaBaaaleaacaaIXaaabeaaaOqaaiaa dkgadaWgaaWcbaGaaGOmaaqabaaakeaacaWGIbWaaSbaaSqaaiaaio daaeqaaaGcbaGaaGPaVlaaykW7cqWIUlstaeaacaWGIbWaaSbaaSqa aiaad6gaaeqaaaaakiaawUfacaGLDbaacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa hogacqGH9aqpdaWadaabaeqabaGaam4yamaaBaaaleaacaaIXaaabe aaaOqaaiaadogadaWgaaWcbaGaaGOmaaqabaaakeaacaWGJbWaaSba aSqaaiaaiodaaeqaaaGcbaGaaGPaVlaaykW7cqWIUlstaeaacaWGJb WaaSbaaSqaaiaad6gaaeqaaaaakiaawUfacaGLDbaacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVpaadmaabaGaamyqaaGaay5waiaaw2faaiabg2da9maadmaa eaqabeaacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaaykW7ca aMc8UaaGPaVlaadggadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaaGPa VlaaykW7caaMc8UaamyyamaaBaaaleaacaaIXaGaaG4maaqabaGcca aMc8UaaGPaVlabl+UimjaaykW7caaMc8UaamyyamaaBaaaleaacaaI XaGaamOBaaqabaaakeaacaWGHbWaaSbaaSqaaiaaikdacaaIXaaabe aakiaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaaGOmaiaaikda aeqaaOGaaGPaVlaaykW7caaMc8UaamyyamaaBaaaleaacaaIYaGaaG 4maaqabaGccaaMc8UaaGPaVlabl+UimjaaykW7caaMc8Uaamyyamaa BaaaleaacaaIYaGaamOBaaqabaaakeaacaaMc8UaaGPaVlabl6Uinj aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eSO7I0KaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqWIUlstcaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaeSy8I8KaaGPaVlaaykW7caaMc8 UaeSO7I0eabaGaamyyamaaBaaaleaacaWGTbGaaGymaaqabaGccaaM c8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaad2gacaaIYaaabeaaki aaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaamyBaiaaiodaaeqa aOGaaGPaVlaaykW7cqWIVlctcaaMc8UaaGPaVlaadggadaWgaaWcba GaamyBaiaad6gaaeqaaaaakiaawUfacaGLDbaaaaa@0CFE@

Now,

c=[ A ]b c i = j=1 n a ij b j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpdaWadaqaaiaadgeaaiaawUfacaGLDbaacaWHIbGaaGPaVlaaykW7 cqGHuhY2caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqaaiaadMgaae qaaOGaeyypa0ZaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaa beaakiaadkgadaWgaaWcbaGaamOAaaqabaaabaGaamOAaiabg2da9i aaigdaaeaacaWGUbaaniabggHiLdaaaa@53C8@

i.e.

c 1 = a 11 b 1 + a 12 b 2 + a 13 b 3 + a 1n b n c 2 = a 21 b 1 + a 22 b 2 + a 23 b 3 + a 2n b n c m = a m1 b 1 + a m2 b 2 + a m3 b 3 + a mn b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaam4yam aaBaaaleaacaaIXaaabeaakiabg2da9iaadggadaWgaaWcbaGaaGym aiaaigdaaeqaaOGaamOyamaaBaaaleaacaaIXaaabeaakiabgUcaRi aadggadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaamOyamaaBaaaleaa caaIYaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGymaiaaiodaae qaaOGaamOyamaaBaaaleaacaaIZaaabeaakiabgUcaRiaaykW7caaM c8UaaGPaVlabl+UimjaaykW7caaMc8UaaGPaVlaadggadaWgaaWcba GaaGymaiaad6gaaeqaaOGaamOyamaaBaaaleaacaWGUbaabeaaaOqa aiaadogadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWGHbWaaSbaaS qaaiaaikdacaaIXaaabeaakiaadkgadaWgaaWcbaGaaGymaaqabaGc cqGHRaWkcaWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaakiaadkgada WgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaikda caaIZaaabeaakiaadkgadaWgaaWcbaGaaG4maaqabaGccqGHRaWkca aMc8UaaGPaVlaaykW7cqWIVlctcaaMc8UaaGPaVlaaykW7caWGHbWa aSbaaSqaaiaaikdacaWGUbaabeaakiaadkgadaWgaaWcbaGaamOBaa qabaaakeaacaaMc8UaaGPaVlabl6UinbqaaiaadogadaWgaaWcbaGa amyBaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaad2gacaaIXaaabe aakiaadkgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWaaSba aSqaaiaad2gacaaIYaaabeaakiaadkgadaWgaaWcbaGaaGOmaaqaba GccqGHRaWkcaWGHbWaaSbaaSqaaiaad2gacaaIZaaabeaakiaadkga daWgaaWcbaGaaG4maaqabaGccqGHRaWkcaaMc8UaaGPaVlaaykW7cq WIVlctcaaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaad2gacaWG UbaabeaakiaadkgadaWgaaWcbaGaamOBaaqabaaaaaa@A4CD@

 

 Transpose. Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be a matrix of order (m×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGTb Gaey41aqRaamOBaiaacMcaaaa@3B8E@  with elements a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3928@ .  The transpose of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  is denoted [ A ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaaCaaaleqabaGaamivaaaaaaa@39F7@ .  If [ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam OqaaGaay5waiaaw2faaaaa@38F2@  is an (n×m) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGUb Gaey41aqRaamyBaiaacMcaaaa@3B8E@  matrix such that [ B ]= [ A ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam OqaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyqaaGaay5waiaa w2faamaaCaaaleqabaGaamivaaaaaaa@3DB6@ , then b ij = a ji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaWG QbGaamyAaaqabaaaaa@3D28@ , i.e.

[ A ] T = [ a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a m1 a m2 a m3 a mn ] T =[ a 11 a 21 a 31 a n1 a 12 a 22 a 3 2 a n2 a 1m a 2m a 3m a nm ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaaCaaaleqabaGaamivaaaakiabg2da9maa dmaaeaqabeaacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaayk W7caaMc8UaaGPaVlaadggadaWgaaWcbaGaaGymaiaaikdaaeqaaOGa aGPaVlaaykW7caaMc8UaamyyamaaBaaaleaacaaIXaGaaG4maaqaba GccaaMc8UaaGPaVlabl+UimjaaykW7caaMc8UaamyyamaaBaaaleaa caaIXaGaamOBaaqabaaakeaacaWGHbWaaSbaaSqaaiaaikdacaaIXa aabeaakiaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaaGOmaiaa ikdaaeqaaOGaaGPaVlaaykW7caaMc8UaamyyamaaBaaaleaacaaIYa GaaG4maaqabaGccaaMc8UaaGPaVlabl+UimjaaykW7caaMc8Uaamyy amaaBaaaleaacaaIYaGaamOBaaqabaaakeaacaaMc8UaaGPaVlabl6 UinjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaeSO7I0KaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7cqWIUlstcaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaeSy8I8KaaGPaVlaaykW7ca aMc8UaeSO7I0eabaGaamyyamaaBaaaleaacaWGTbGaaGymaaqabaGc caaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaad2gacaaIYaaabe aakiaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaamyBaiaaioda aeqaaOGaaGPaVlaaykW7cqWIVlctcaaMc8UaaGPaVlaadggadaWgaa WcbaGaamyBaiaad6gaaeqaaaaakiaawUfacaGLDbaadaahaaWcbeqa aiaadsfaaaGccqGH9aqpdaWadaabaeqabaGaamyyamaaBaaaleaaca aIXaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqa aiaaikdacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaadggadaWgaa WcbaGaaG4maiaaigdaaeqaaOGaaGPaVlaaykW7cqWIVlctcaaMc8Ua aGPaVlaadggadaWgaaWcbaGaamOBaiaaigdaaeqaaaGcbaGaamyyam aaBaaaleaacaaIXaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caWG HbWaaSbaaSqaaiaaikdacaaIYaaabeaakiaaykW7caaMc8UaaGPaVl aadggadaWgaaWcbaGaaG4maaqabaGcdaWgaaWcbaGaaGOmaaqabaGc caaMc8UaaGPaVlabl+UimjaaykW7caaMc8UaamyyamaaBaaaleaaca WGUbGaaGOmaaqabaaakeaacaaMc8UaaGPaVlabl6UinjaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeSO7I0KaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7cqWIUlstcaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaeSy8I8KaaGPaVlaaykW7caaMc8UaeSO7I0ea baGaamyyamaaBaaaleaacaaIXaGaamyBaaqabaGccaaMc8UaaGPaVl aaykW7caWGHbWaaSbaaSqaaiaaikdacaWGTbaabeaakiaaykW7caaM c8UaaGPaVlaadggadaWgaaWcbaGaaG4maiaad2gaaeqaaOGaaGPaVl aaykW7cqWIVlctcaaMc8UaaGPaVlaadggadaWgaaWcbaGaamOBaiaa d2gaaeqaaaaakiaawUfacaGLDbaaaaa@5D43@

Note that

( [ A ][ B ] ) T = [ B ] T [ A ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaWaam WaaeaacaWGbbaacaGLBbGaayzxaaWaamWaaeaacaWGcbaacaGLBbGa ayzxaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaaaOGaeyypa0 ZaamWaaeaacaWGcbaacaGLBbGaayzxaaWaaWbaaSqabeaacaWGubaa aOWaamWaaeaacaWGbbaacaGLBbGaayzxaaWaaWbaaSqabeaacaWGub aaaaaa@46D0@

 

 Determinant  The determinant is defined only for a square matrix.  Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be a (2×2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaaIYa Gaey41aqRaaGOmaiaacMcaaaa@3B21@  matrix with components a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3928@ .  The determinant of  [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  is denoted by det[ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacsgacaGGLb GaaiiDamaadmaabaGaamyqaaGaay5waiaaw2faaaaa@3BBC@  or | A | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaam yqaaGaay5bSlaawIa7aaaa@3A21@  and is given by

| A |=| a 11 a 12 a 21 a 22 |= a 11 a 22 a 12 a 21 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaam yqaaGaay5bSlaawIa7aiabg2da9maaemaaeaqabeaacaWGHbWaaSba aSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaaGym aiaaikdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIYaGaaGymaaqaba GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaaaaGccaGLhW UaayjcSdGaeyypa0JaamyyamaaBaaaleaacaaIXaGaaGymaaqabaGc caWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgkHiTiaadggada WgaaWcbaGaaGymaiaaikdaaeqaaOGaamyyamaaBaaaleaacaaIYaGa aGymaaqabaaaaa@7097@

Now, let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be an ( n×n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam OBaiabgEna0kaad6gaaiaawIcacaGLPaaaaaa@3BBF@  matrix.  Define the minors M ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3914@  of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  as the determinant formed by omitting the ith row and jth column of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@ .  For example, the minors M 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGymaiaaigdaaeqaaaaa@38AD@  and M 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGymaiaaikdaaeqaaaaa@38AE@  for a ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG 4maiabgEna0kaaiodaaiaawIcacaGLPaaaaaa@3B53@  matrix are computed as follows.   Let

[ A ]=[ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiabg2da9maadmaaeaqabeaacaWGHbWaaSba aSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaadggada WgaaWcbaGaaGymaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPa VlaadggadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaamyyamaaBa aaleaacaaIYaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caWGHbWa aSbaaSqaaiaaikdacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaadg gadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaaGPaVdqaaiaadggadaWg aaWcbaGaaG4maiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8Uaamyyam aaBaaaleaacaaIZaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caWG HbWaaSbaaSqaaiaaiodacaaIZaaabeaaaaGccaGLBbGaayzxaaaaaa@7201@

Then

M 11 =| a 22 a 23 a 32 a 33 |= a 22 a 33 a 32 a 23 M 12 =| a 21 a 23 a 31 a 33 |= a 21 a 33 a 31 a 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaeyypa0ZaaqWaaqaabeqaaiaadgga daWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamyyamaaBaaaleaa caaIYaGaaG4maaqabaaakeaacaWGHbWaaSbaaSqaaiaaiodacaaIYa aabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadggadaWgaaWcbaGaaG4maiaaiodaaeqaaaaaki aawEa7caGLiWoacqGH9aqpcaWGHbWaaSbaaSqaaiaaikdacaaIYaaa beaakiaadggadaWgaaWcbaGaaG4maiaaiodaaeqaaOGaeyOeI0Iaam yyamaaBaaaleaacaaIZaGaaGOmaaqabaGccaWGHbWaaSbaaSqaaiaa ikdacaaIZaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGnbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da 9maaemaaeaqabeaacaWGHbWaaSbaaSqaaiaaikdacaaIXaaabeaaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaadggadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaamyyam aaBaaaleaacaaIZaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaai aaiodacaaIZaaabeaaaaGccaGLhWUaayjcSdGaeyypa0Jaamyyamaa BaaaleaacaaIYaGaaGymaaqabaGccaWGHbWaaSbaaSqaaiaaiodaca aIZaaabeaakiabgkHiTiaadggadaWgaaWcbaGaaG4maiaaigdaaeqa aOGaamyyamaaBaaaleaacaaIYaGaaG4maaqabaaaaa@B315@

Define the cofactors C ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@390A@  of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  as

C ij = ( 1 ) i+j M ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaeWaaeaacqGHsislcaaI XaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGPbGaey4kaSIaamOAaa aakiaad2eadaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@431C@

Then, the determinant of the ( n×n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam OBaiabgEna0kaad6gaaiaawIcacaGLPaaaaaa@3BBF@  matrix [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  is computed as follows

| A |= j=1 n a ij C ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaam yqaaGaay5bSlaawIa7aiabg2da9maaqahabaGaamyyamaaBaaaleaa caWGPbGaamOAaaqabaGccaWGdbWaaSbaaSqaaiaadMgacaWGQbaabe aaaeaacaWGQbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa@46CB@

The result is the same whichever row i is chosen for the expansion.  For the particular case of a ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG 4maiabgEna0kaaiodaaiaawIcacaGLPaaaaaa@3B53@  matrix

det[ A ]=det[ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ]= a 11 ( a 22 a 33 a 23 a 32 )+ a 12 ( a 23 a 31 a 21 a 33 )+ a 13 ( a 21 a 32 a 31 a 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacsgacaGGLb GaaiiDamaadmaabaGaamyqaaGaay5waiaaw2faaiabg2da9iGacsga caGGLbGaaiiDamaadmaaeaqabeaacaWGHbWaaSbaaSqaaiaaigdaca aIXaaabeaakiaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaaGym aiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaadggadaWgaa WcbaGaaGymaiaaiodaaeqaaaGcbaGaamyyamaaBaaaleaacaaIYaGa aGymaaqabaGccaaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaaik dacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGa aGOmaiaaiodaaeqaaOGaaGPaVdqaaiaadggadaWgaaWcbaGaaG4mai aaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaamyyamaaBaaaleaacaaI ZaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaai aaiodacaaIZaaabeaaaaGccaGLBbGaayzxaaGaeyypa0Jaamyyamaa BaaaleaacaaIXaGaaGymaaqabaGccaGGOaGaamyyamaaBaaaleaaca aIYaGaaGOmaaqabaGccaWGHbWaaSbaaSqaaiaaiodacaaIZaaabeaa kiabgkHiTiaadggadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaamyyam aaBaaaleaacaaIZaGaaGOmaaqabaGccaGGPaGaey4kaSIaamyyamaa BaaaleaacaaIXaGaaGOmaaqabaGccaGGOaGaamyyamaaBaaaleaaca aIYaGaaG4maaqabaGccaWGHbWaaSbaaSqaaiaaiodacaaIXaaabeaa kiabgkHiTiaadggadaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaamyyam aaBaaaleaacaaIZaGaaG4maaqabaGccaGGPaGaey4kaSIaamyyamaa BaaaleaacaaIXaGaaG4maaqabaGccaGGOaGaamyyamaaBaaaleaaca aIYaGaaGymaaqabaGccaWGHbWaaSbaaSqaaiaaiodacaaIYaaabeaa kiabgkHiTiaadggadaWgaaWcbaGaaG4maiaaigdaaeqaaOGaamyyam aaBaaaleaacaaIYaGaaGOmaaqabaGccaGGPaaaaa@A7E2@

The determinant may also be evaluated by summing over rows, i.e.

| A |= i=1 n a ij C ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaam yqaaGaay5bSlaawIa7aiabg2da9maaqahabaGaamyyamaaBaaaleaa caWGPbGaamOAaaqabaGccaWGdbWaaSbaaSqaaiaadMgacaWGQbaabe aaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa@46CA@

and as before the result is the same for each choice of column j.  Finally, note that

det [ A ] T =det[ A ]det( [ A ][ B ] )=det[ A ]det[ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacsgacaGGLb GaaiiDamaadmaabaGaamyqaaGaay5waiaaw2faamaaCaaaleqabaGa amivaaaakiabg2da9iGacsgacaGGLbGaaiiDamaadmaabaGaamyqaa Gaay5waiaaw2faaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlGacsgacaGGLbGaaiiDamaabmaabaWaam WaaeaacaWGbbaacaGLBbGaayzxaaWaamWaaeaacaWGcbaacaGLBbGa ayzxaaaacaGLOaGaayzkaaGaeyypa0JaciizaiaacwgacaGG0bWaam WaaeaacaWGbbaacaGLBbGaayzxaaGaciizaiaacwgacaGG0bWaamWa aeaacaWGcbaacaGLBbGaayzxaaaaaa@670A@

 

 Inversion.  Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be an ( n×n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam OBaiabgEna0kaad6gaaiaawIcacaGLPaaaaaa@3BBF@  matrix.  The inverse of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  is denoted by [ A ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@3AC6@  and is defined such that

[ A ] 1 [ A ]=[ I ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaa dmaabaGaamyqaaGaay5waiaaw2faaiabg2da9maadmaabaGaamysaa Gaay5waiaaw2faaaaa@414E@

The inverse of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  exists if and only if det[ A ]0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacsgacaGGLb GaaiiDamaadmaabaGaamyqaaGaay5waiaaw2faaiabgcMi5kaaicda aaa@3E3D@ .  A matrix which has no inverse is said to be singular.  The inverse of a matrix may be computed explicitly, by forming the cofactor matrix [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4qaaGaay5waiaaw2faaaaa@38F3@  with components c ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@392A@  as defined in the preceding section.  Then

[ A ] 1 = 1 det[ A ] [ C ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaakiab g2da9maalaaabaGaaGymaaqaaiGacsgacaGGLbGaaiiDamaadmaaba GaamyqaaGaay5waiaaw2faaaaadaWadaqaaiaadoeaaiaawUfacaGL DbaadaahaaWcbeqaaiaadsfaaaaaaa@45E4@

In practice, it is faster to compute the inverse of a matrix using methods such as Gaussian elimination. 

 

Note that

( [ A ][ B ] ) 1 = [ B ] 1 [ A ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaWaam WaaeaacaWGbbaacaGLBbGaayzxaaWaamWaaeaacaWGcbaacaGLBbGa ayzxaaaacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaO Gaeyypa0ZaamWaaeaacaWGcbaacaGLBbGaayzxaaWaaWbaaSqabeaa cqGHsislcaaIXaaaaOWaamWaaeaacaWGbbaacaGLBbGaayzxaaWaaW baaSqabeaacqGHsislcaaIXaaaaaaa@493D@

For a diagonal matrix, the inverse is

[ A ]=[ a 11 000 0 a 22 00 000 a nn ]=[ 1/ a 11 000 01/ a 22 00 0001/ a nn ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiabg2da9iaaykW7caaMc8+aamWaaqaabeqa aiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaeS47IWKaaGPaVlaaykW7caaMc8UaaGimaiaa ykW7aeaacaaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaakiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabl+UimjaaykW7caaMc8UaaGimaaqa aiaaykW7caaMc8UaeSO7I0KaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabl6UinjaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7cqWIUlstcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlablgVipjaa ykW7caaMc8UaaGPaVlabl6UinbqaaiaaykW7caaMc8UaaGimaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGim aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7cqWIVlctcaaMc8UaaGPa VlaadggadaWgaaWcbaGaamOBaiaad6gaaeqaaaaakiaawUfacaGLDb aacqGH9aqpcaaMc8UaaGPaVlaaykW7daWadaabaeqabaGaaGymaiaa c+cacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqWIVlctcaaMc8UaaGPaVlaaykW7 caaIWaGaaGPaVdqaaiaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaigdacaGG VaGaaGPaVlaadggadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaGPaVl aaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabl+UimjaaykW7caaMc8UaaGPaVlaaicdaaeaacaaMc8UaaG PaVlabl6UinjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeS O7I0KaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaeSO7I0KaaGPaVlaaykW7ca aMc8UaeSy8I8KaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabl6UinbqaaiaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPa VlaaykW7cqWIVlctcaaMc8UaaGPaVlaaykW7caaIXaGaai4laiaadg gadaWgaaWcbaGaamOBaiaad6gaaeqaaaaakiaawUfacaGLDbaaaaa@D2FF@

For a ( 2×2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG OmaiabgEna0kaaikdaaiaawIcacaGLPaaaaaa@3B51@  matrix, the inverse is

[ a 11 a 12 a 21 a 22 ]= 1 a 11 a 22 a 12 a 21 [ a 22 a 12 a 21 a 11 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaaeaqabe aacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadggada WgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaamyyamaaBaaaleaacaaI YaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaaikdacaaIYaaa beaaaaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaam yyamaaBaaaleaacaaIXaGaaGymaaqabaGccaWGHbWaaSbaaSqaaiaa ikdacaaIYaaabeaakiabgkHiTiaadggadaWgaaWcbaGaaGymaiaaik daaeqaaOGaamyyamaaBaaaleaacaaIYaGaaGymaaqabaaaaOWaamWa aqaabeqaaiaaykW7caaMc8UaamyyamaaBaaaleaacaaIYaGaaGOmaa qabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyOeI0IaaGPaVlaadggadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcba GaeyOeI0IaamyyamaaBaaaleaacaaIYaGaaGymaaqabaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca WGHbWaaSbaaSqaaiaaigdacaaIXaaabeaaaaGccaGLBbGaayzxaaaa aa@96BE@

 

 Eigenvalues and eigenvectors. Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be an ( n×n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam OBaiabgEna0kaad6gaaiaawIcacaGLPaaaaaa@3BBF@  matrix, with coefficients a ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@3928@ .  Consider the vector equation

[ A ]x=λx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiaahIhacqGH9aqpcqaH7oaBcaWH4baaaa@3DAD@                                                 (1)

where x is a vector with n components, and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@  is a scalar (which may be complex).  The n nonzero vectors x and corresponding scalars λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@  which satisfy this equation are the eigenvectors and eigenvalues of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@ .

 

Formally, eighenvalues and eigenvectors may be computed as follows.  Rearrange the preceding equation to

( [ A ]λ[ I ] )x=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaWaam WaaeaacaWGbbaacaGLBbGaayzxaaGaeyOeI0Iaeq4UdW2aamWaaeaa caWGjbaacaGLBbGaayzxaaaacaGLOaGaayzkaaGaaCiEaiabg2da9i aahcdaaaa@429B@                                      (2)

This has nontrivial solutions for x only if the determinant of the matrix ( [ A ]λ[ I ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaWaam WaaeaacaWGbbaacaGLBbGaayzxaaGaeyOeI0Iaeq4UdW2aamWaaeaa caWGjbaacaGLBbGaayzxaaaacaGLOaGaayzkaaaaaa@3FDB@  vanishes.  The equation

det( [ A ]λ[ I ] )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacsgacaGGLb GaaiiDamaabmaabaWaamWaaeaacaWGbbaacaGLBbGaayzxaaGaeyOe I0Iaeq4UdW2aamWaaeaacaWGjbaacaGLBbGaayzxaaaacaGLOaGaay zkaaGaeyypa0JaaGimaaaa@4466@

is an nth order polynomial which may be solved for λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@ .  In general the polynomial will have n roots, which may be complex.  The eigenvectors may then be computed using equation (2).  For example, a (2×2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaaIYa Gaey41aqRaaGOmaiaacMcaaaa@3B21@  matrix generally has two eigenvectors, which satisfy

| AλI |=| a 11 λ a 12 a 21 a 22 λ |=( a 11 λ)( a 22 λ) a 12 a 21 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaam yqaiabgkHiTiabeU7aSjaadMeaaiaawEa7caGLiWoacqGH9aqpdaab daabaeqabaGaamyyamaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsi slcqaH7oaBcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaaigdacaaIYaaabeaaaO qaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamyyamaaBaaaleaa caaIYaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWGHbWaaSbaaSqaaiaaikdacaaI YaaabeaakiabgkHiTiabeU7aSbaacaGLhWUaayjcSdGaeyypa0Jaai ikaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOeI0Iaeq4U dWMaaiykaiaacIcacaWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaaki abgkHiTiabeU7aSjaacMcacqGHsislcaWGHbWaaSbaaSqaaiaaigda caaIYaaabeaakiaadggadaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaey ypa0JaaGimaaaa@8ABD@

Solve the quadratic equation to see that

λ 1 = 1 2 ( a 11 + a 22 ) 1 2 { ( a 11 + a 22 ) 2 4( a 11 a 22 a 12 a 21 ) } 1/2 λ 2 = 1 2 ( a 11 + a 22 )+ 1 2 { ( a 11 + a 22 ) 2 4( a 11 a 22 a 12 a 21 ) } 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4UdW 2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGa aGOmaaaadaqadaqaaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaO Gaey4kaSIaamyyamaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawIca caGLPaaacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaamaacmaaba WaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUca RiaadggadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinamaabmaabaGaamyy amaaBaaaleaacaaIXaGaaGymaaqabaGccaWGHbWaaSbaaSqaaiaaik dacaaIYaaabeaakiabgkHiTiaadggadaWgaaWcbaGaaGymaiaaikda aeqaaOGaamyyamaaBaaaleaacaaIYaGaaGymaaqabaaakiaawIcaca GLPaaaaiaawUhacaGL9baadaahaaWcbeqaaiaaigdacaGGVaGaaGOm aaaaaOqaaiabeU7aSnaaBaaaleaacaaIYaaabeaakiabg2da9maala aabaGaaGymaaqaaiaaikdaaaWaaeWaaeaacaWGHbWaaSbaaSqaaiaa igdacaaIXaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGOmaiaaik daaeqaaaGccaGLOaGaayzkaaGaey4kaSYaaSaaaeaacaaIXaaabaGa aGOmaaaadaGadaqaamaabmaabaGaamyyamaaBaaaleaacaaIXaGaaG ymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaais dadaqadaqaaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaamyy amaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHsislcaWGHbWaaSbaaS qaaiaaigdacaaIYaaabeaakiaadggadaWgaaWcbaGaaGOmaiaaigda aeqaaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaWaaWbaaSqabeaaca aIXaGaai4laiaaikdaaaaaaaa@8BA0@

The two corresponding eigenvectors may be computed from (2), which shows that

[ a 11 λ i a 12 a 21 a 22 λ i ][ x 1 (i) x 2 (i) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaaeaqabe aacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHiTiabeU7a SnaaBaaaleaacaWGPbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadggadaWgaaWcbaGaaGym aiaaikdaaeqaaaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca WGHbWaaSbaaSqaaiaaikdacaaIXaaabeaakiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadggadaWgaa WcbaGaaGOmaiaaikdaaeqaaOGaeyOeI0Iaeq4UdW2aaSbaaSqaaiaa dMgaaeqaaaaakiaawUfacaGLDbaadaWadaabaeqabaGaamiEamaaDa aaleaacaaIXaaabaGaaiikaiaadMgacaGGPaaaaaGcbaGaamiEamaa DaaaleaacaaIYaaabaGaaiikaiaadMgacaGGPaaaaaaakiaawUfaca GLDbaacqGH9aqpcaaIWaaaaa@79AB@

so that, multiplying out the first row of the matrix (you can use the second row too, if you wish MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  since we chose λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@  to make the determinant of the matrix vanish, the two equations have the same solutions.  In fact, if a 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaigdacaaIYa aabeaakiabg2da9iaaicdaaaa@3719@ , you will need to do this, because the first equation will simply give 0=0 when trying to solve for one of the eigenvectors)

( 1 2 ( a 11 a 22 )+ 1 2 { ( a 11 + a 22 ) 2 4( a 11 a 22 a 12 a 21 ) } 1/2 ) x 1 (1) + a 12 x 2 (1) =0 ( 1 2 ( a 11 a 22 ) 1 2 { ( a 11 + a 22 ) 2 4( a 11 a 22 a 12 a 21 ) } 1/2 ) x 1 (2) + a 12 x 2 (2) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaabmaabaWaaSaaaeaacaaIXa aabaGaaGOmaaaadaqadaqaaiaadggadaWgaaWcbaGaaGymaiaaigda aeqaaOGaeyOeI0IaamyyamaaBaaaleaacaaIYaGaaGOmaaqabaaaki aawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaamaa cmaabaWaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabeaaki abgUcaRiaadggadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinamaabmaaba GaamyyamaaBaaaleaacaaIXaGaaGymaaqabaGccaWGHbWaaSbaaSqa aiaaikdacaaIYaaabeaakiabgkHiTiaadggadaWgaaWcbaGaaGymai aaikdaaeqaaOGaamyyamaaBaaaleaacaaIYaGaaGymaaqabaaakiaa wIcacaGLPaaaaiaawUhacaGL9baadaahaaWcbeqaaiaaigdacaGGVa GaaGOmaaaaaOGaayjkaiaawMcaaiaadIhadaqhaaWcbaGaaGymaaqa aiaacIcacaaIXaGaaiykaaaakiabgUcaRiaadggadaWgaaWcbaGaaG ymaiaaikdaaeqaaOGaamiEamaaDaaaleaacaaIYaaabaGaaiikaiaa igdacaGGPaaaaOGaeyypa0JaaGimaaqaamaabmaabaWaaSaaaeaaca aIXaaabaGaaGOmaaaadaqadaqaaiaadggadaWgaaWcbaGaaGymaiaa igdaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaaIYaGaaGOmaaqaba aakiaawIcacaGLPaaacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaa amaacmaabaWaaeWaaeaacaWGHbWaaSbaaSqaaiaaigdacaaIXaaabe aakiabgUcaRiaadggadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinamaabm aabaGaamyyamaaBaaaleaacaaIXaGaaGymaaqabaGccaWGHbWaaSba aSqaaiaaikdacaaIYaaabeaakiabgkHiTiaadggadaWgaaWcbaGaaG ymaiaaikdaaeqaaOGaamyyamaaBaaaleaacaaIYaGaaGymaaqabaaa kiaawIcacaGLPaaaaiaawUhacaGL9baadaahaaWcbeqaaiaaigdaca GGVaGaaGOmaaaaaOGaayjkaiaawMcaaiaadIhadaqhaaWcbaGaaGym aaqaaiaacIcacaaIYaGaaiykaaaakiabgUcaRiaadggadaWgaaWcba GaaGymaiaaikdaaeqaaOGaamiEamaaDaaaleaacaaIYaaabaGaaiik aiaaikdacaGGPaaaaOGaeyypa0JaaGimaaaaaa@9E81@

which are satisfied by any vector of the form

x (1) =[ 2 a 12 ( a 11 a 22 )+ { ( a 11 + a 22 ) 2 4( a 11 a 22 a 12 a 21 ) } 1/2 ]p x (2) =[ 2 a 12 ( a 11 a 22 ) { ( a 11 + a 22 ) 2 4( a 11 a 22 a 12 a 21 ) } 1/2 ]q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahIhadaahaaWcbeqaaiaacI cacaaIXaGaaiykaaaakiabg2da9maadmaaeaqabeaacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaikdacaWGHbWaaSbaaSqaaiaaigdacaaIYa aabeaaaOqaamaabmaabaGaamyyamaaBaaaleaacaaIXaGaaGymaaqa baGccqGHsislcaWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOGaay jkaiaawMcaaiabgUcaRmaacmaabaWaaeWaaeaacaWGHbWaaSbaaSqa aiaaigdacaaIXaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGOmai aaikdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa eyOeI0IaaGinamaabmaabaGaamyyamaaBaaaleaacaaIXaGaaGymaa qabaGccaWGHbWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgkHiTiaa dggadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaamyyamaaBaaaleaaca aIYaGaaGymaaqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baadaah aaWcbeqaaiaaigdacaGGVaGaaGOmaaaaaaGccaGLBbGaayzxaaGaam iCaaqaaiaahIhadaahaaWcbeqaaiaacIcacaaIYaGaaiykaaaakiab g2da9maadmaaeaqabeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIYaGaamyyamaaBa aaleaacaaIXaGaaGOmaaqabaaakeaadaqadaqaaiaadggadaWgaaWc baGaaGymaiaaigdaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaaIYa GaaGOmaaqabaaakiaawIcacaGLPaaacqGHsisldaGadaqaamaabmaa baGaamyyamaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcaWGHb WaaSbaaSqaaiaaikdacaaIYaaabeaaaOGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaakiabgkHiTiaaisdadaqadaqaaiaadggadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaamyyamaaBaaaleaacaaIYaGaaGOm aaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaaigdacaaIYaaabeaaki aadggadaWgaaWcbaGaaGOmaiaaigdaaeqaaaGccaGLOaGaayzkaaaa caGL7bGaayzFaaWaaWbaaSqabeaacaaIXaGaai4laiaaikdaaaaaaO Gaay5waiaaw2faaiaadghaaaaa@3071@

where p and q are arbitrary real numbers.

 

It is often convenient to normalize eigenvectors so that they have unit `length’.  For this purpose, choose p and q so that x (i) x (i) =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahIhadaahaa WcbeqaaiaacIcacaWGPbGaaiykaaaakiabgwSixlaahIhadaahaaWc beqaaiaacIcacaWGPbGaaiykaaaakiabg2da9iaaigdaaaa@4142@ .  (For vectors of dimension n, the generalized dot product is defined such that xx= i=1 n x i x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahIhacqGHfl Y1caWH4bGaeyypa0ZaaabmaeaacaWG4bWaaSbaaSqaaiaadMgaaeqa aOGaamiEamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaa0GaeyyeIuoaaaa@455C@  )

 

One may calculate explicit expressions for eigenvalues and eigenvectors for any matrix up to order ( 4×4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG inaiabgEna0kaaisdaaiaawIcacaGLPaaaaaa@3B55@ , but the results are so cumbersome that, except for the ( 2×2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG OmaiabgEna0kaaikdaaiaawIcacaGLPaaaaaa@3B51@  results, they are virtually useless.  In practice, numerical values may be computed using several iterative techniques.  Packages like Mathematica, Maple or Matlab make calculations like this easy.

 

The eigenvalues of a real symmetric matrix are always real, and its eigenvectors are orthogonal, i.e. the ith and jth eigenvectors (with ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMgacqGHGj sUcaWGQbaaaa@39DD@  ) satisfy x (i) x (j) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahIhadaahaa WcbeqaaiaacIcacaWGPbGaaiykaaaakiabgwSixlaahIhadaahaaWc beqaaiaacIcacaWGQbGaaiykaaaakiabg2da9iaaicdaaaa@4142@ .

 

The eigenvalues of a skew symmetric matrix are pure imaginary.

 

 Spectral and singular value decomposition.  Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be a real symmetric  ( n×n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam OBaiabgEna0kaad6gaaiaawIcacaGLPaaaaaa@3BBF@  matrix. Denote the n (real) eigenvalues of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  by λ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@3907@ , and let w (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahEhadaahaa WcbeqaaiaacIcacaWGPbGaaiykaaaaaaa@39AD@  be the corresponding normalized eigenvectors, such that w (i) w (i) =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahEhadaahaa WcbeqaaiaacIcacaWGPbGaaiykaaaakiabgwSixlaahEhadaahaaWc beqaaiaacIcacaWGPbGaaiykaaaakiabg2da9iaaigdaaaa@4140@ .  Then, for any arbitrary vector b,

[ A ]b= i=1 n λ i ( w (i) b ) w (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiaahkgacqGH9aqpdaaeWbqaaiabeU7aSnaa BaaaleaacaWGPbaabeaakmaabmaabaGaaC4DamaaCaaaleqabaGaai ikaiaadMgacaGGPaaaaOGaeyyXICTaaCOyaaGaayjkaiaawMcaaiaa ykW7caaMc8UaaC4DamaaCaaaleqabaGaaiikaiaadMgacaGGPaaaaa qaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa@5259@

Let [ Λ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeu 4MdWeacaGLBbGaayzxaaaaaa@39A0@  be a diagonal matrix which contains the n eigenvalues of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  as elements of the diagonal, and let [ Q ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yuaaGaay5waiaaw2faaaaa@3901@  be a matrix consisting of the n eigenvectors as columns, i.e.

[ Λ ]=[ λ 1 000 0 λ 2 00 000 λ n ][ Q ]=[ w 1 (1) w 1 (2) w 1 (3) w 1 (n) w 2 (1) w 2 (2) w 2 (3) w 2 (n) w n (1) w n (2) w n (3) w n (n) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeu 4MdWeacaGLBbGaayzxaaGaeyypa0JaaGPaVpaadmaaeaqabeaacqaH 7oaBdaWgaaWcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabl+UimjaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaIWaaabaGaaGimaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeU7aSnaa BaaaleaacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabl+Ui mjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWa aabaGaaGPaVlabl6UinjaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabl6Uinj aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabl6UinjaaykW7caaMc8UaaGPaVlaaykW7cqWIXlYtcaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 cqWIUlstaeaacaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeS47IWKaaGPaVlaa ykW7caaMc8UaaGPaVlabeU7aSnaaBaaaleaacaWGUbaabeaaaaGcca GLBbGaayzxaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVpaadmaabaGaamyuaaGaay5waiaaw2 faaiabg2da9iaaykW7daWadaabaeqabaGaam4DamaaDaaaleaacaaI XaaabaGaaiikaiaaigdacaGGPaaaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWG3bWaa0baaSqaaiaaigdaaeaacaGGOaGaaGOmaiaa cMcaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3b Waa0baaSqaaiaaigdaaeaacaGGOaGaaG4maiaacMcaaaGccaaMc8Ua aGPaVlaaykW7caaMc8UaeS47IWKaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWG3bWaa0baaSqaaiaaigdaaeaacaGGOaGaamOBaiaacMca aaGccaaMc8oabaGaam4DamaaDaaaleaacaaIYaaabaGaaiikaiaaig dacaGGPaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3bWa a0baaSqaaiaaikdaaeaacaGGOaGaaGOmaiaacMcaaaGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3bWaa0baaSqaaiaaikda aeaacaGGOaGaaG4maiaacMcaaaGccaaMc8UaaGPaVlaaykW7caaMc8 UaeS47IWKaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3bWaa0ba aSqaaiaaikdaaeaacaGGOaGaamOBaiaacMcaaaaakeaacaaMc8UaaG PaVlaaykW7cqWIUlstcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqWIUlstca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqWIUlstca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7cqWIXlYtcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaeSO7I0eabaGaam4DamaaDaaaleaacaWGUbaabaGaaiikaiaa igdacaGGPaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3b Waa0baaSqaaiaad6gaaeaacaGGOaGaaGOmaiaacMcaaaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3bWaa0baaSqaaiaad6 gaaeaacaGGOaGaaG4maiaacMcaaaGccaaMc8UaaGPaVlaaykW7caaM c8UaeS47IWKaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG3bWaa0 baaSqaaiaad6gaaeaacaGGOaGaamOBaiaacMcaaaaaaOGaay5waiaa w2faaaaa@10ED@

Then

[ A ]=[ Q ][ Λ ] [ Q ] T [ Q ] T [ Q ]=[ Q ] [ Q ] T =[ I ] [ Q ] T [ A ][ Q ]=[ Λ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyuaaGaay5waiaa w2faamaadmaabaGaeu4MdWeacaGLBbGaayzxaaWaamWaaeaacaWGrb aacaGLBbGaayzxaaWaaWbaaSqabeaacaWGubaaaOGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWadaqaaiaadg faaiaawUfacaGLDbaadaahaaWcbeqaaiaadsfaaaGcdaWadaqaaiaa dgfaaiaawUfacaGLDbaacqGH9aqpdaWadaqaaiaadgfaaiaawUfaca GLDbaadaWadaqaaiaadgfaaiaawUfacaGLDbaadaahaaWcbeqaaiaa dsfaaaGccqGH9aqpdaWadaqaaiaadMeaaiaawUfacaGLDbaacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7daWadaqaaiaadgfaaiaawUfacaGLDbaada ahaaWcbeqaaiaadsfaaaGcdaWadaqaaiaadgeaaiaawUfacaGLDbaa daWadaqaaiaadgfaaiaawUfacaGLDbaacqGH9aqpdaWadaqaaiabfU 5ambGaay5waiaaw2faaaaa@82AB@

Note that this gives another (generally quite useless) way to invert [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@

[ A ] 1 =[ Q ] [ Λ ] 1 [ Q ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaakiab g2da9maadmaabaGaamyuaaGaay5waiaaw2faamaadmaabaGaeu4MdW eacaGLBbGaayzxaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaamWa aeaacaWGrbaacaGLBbGaayzxaaWaaWbaaSqabeaacaWGubaaaaaa@47B2@

where [ Λ ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeu 4MdWeacaGLBbGaayzxaaWaaWbaaSqabeaacqGHsislcaaIXaaaaaaa @3B75@  is easy to compute since [ Λ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeu 4MdWeacaGLBbGaayzxaaaaaa@39A0@  is diagonal.

 

 Square root of a matrix.   Let [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  be a real symmetric  ( n×n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam OBaiabgEna0kaad6gaaiaawIcacaGLPaaaaaa@3BBF@  matrix.  Denote the singular value decomposition of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@  by [ A ]=[ Q ][ Λ ] [ Q ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyuaaGaay5waiaa w2faamaadmaabaGaeu4MdWeacaGLBbGaayzxaaWaamWaaeaacaWGrb aacaGLBbGaayzxaaWaaWbaaSqabeaacaWGubaaaOGaaGPaVdaa@4589@  as defined above.  Suppose that [ S ]= [ A ] 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4uaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyqaaGaay5waiaa w2faamaaCaaaleqabaGaaGymaiaac+cacaaIYaaaaaaa@3F18@  denotes the square root of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@ , defined so that

[ S ][ S ]=[ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4uaaGaay5waiaaw2faamaadmaabaGaam4uaaGaay5waiaaw2faaiab g2da9maadmaabaGaamyqaaGaay5waiaaw2faaaaa@3F8B@

One way to compute [ S ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4uaaGaay5waiaaw2faaaaa@3903@  is through the singular value decomposition of [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@

[ S ]=[ Q ] [ Λ ] 1/2 [ Q ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4uaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyuaaGaay5waiaa w2faamaadmaabaGaeu4MdWeacaGLBbGaayzxaaWaaWbaaSqabeaaca aIXaGaai4laiaaikdaaaGcdaWadaqaaiaadgfaaiaawUfacaGLDbaa daahaaWcbeqaaiaadsfaaaaaaa@4667@

where

[ Λ ] 1/2 =[ λ 1 000 0 λ 2 00 000 λ n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaeu 4MdWeacaGLBbGaayzxaaWaaWbaaSqabeaacaaIXaGaai4laiaaikda aaGccqGH9aqpcaaMc8+aamWaaqaabeqaamaakaaabaGaeq4UdW2aaS baaSqaaiaaigdaaeqaaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlabl+UimjaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaIWaaabaGaaGimaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaakaaabaGaeq 4UdW2aaSbaaSqaaiaaikdaaeqaaaqabaGccaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqWIVlctcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGimaaqaaiaaykW7cqWIUlstcaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7cqWIUlstcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqWIUlstcaaMc8UaaGPaVlaaykW7caaMc8Ua eSy8I8KaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaeSO7I0eabaGaaGimaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG imaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabl+ UimjaaykW7caaMc8UaaGPaVlaaykW7daGcaaqaaiabeU7aSnaaBaaa leaacaWGUbaabeaaaeqaaaaakiaawUfacaGLDbaacaaMc8UaaGPaVd aa@1997@