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  or using bold type .  Its magnitude is denoted  or .  There are two special cases of vectors: the unit vector  has ; and the null vector  has .

2. Vector Operations

Let  and  be vectors.  Then  is also a vector.  The vector  may be shown diagramatically by placing arrows representing  and  head to tail, as shown in the figure.

Multiplication

1.      Multiplication by a scalar. Let  be a vector, and  a scalar.  Then  is a vector.  The direction of  is parallel to  and its magnitude is given by .

Note that you can form a unit vector n which is parallel to a by setting .

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 , and is defined by

,

where  is the angle subtended by a and b. Note that , and .  If  and  then  if and only if ; 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 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

Note that  and .

Some useful vector identities

3. Cartesian components of vectors

Let  be three mutually perpendicular unit vectors which form a right handed triad, Fig. 4.  Then  are said to form and orthonormal basis. The vectors satisfy

We may express any vector a as a suitable combination of the unit vectors ,  and .  For example, we may write

where  are scalars, called the components of a in the basis .   The components of a have a simple physical interpretation.  For example, if we evaluate the dot product  we find that

in view of the properties of the three vectors ,  and .  Recall that

Then, noting that , we have

Thus,  represents the projected length of the vector a  in the direction of , as illustrated in the figure.  Similarly,  and  may be shown to represent the projection of  in the directions  and , 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 ,  and , respectively.  Then, it is straightforward to show that

4. Change of basis

Let a be a vector, and let  be a Cartesian basis.  Suppose that the components of a in the basis  are known to be .  Now, suppose that we wish to compute the components of a in a second Cartesian basis, .  This means we wish to find components , such that

To do so, note that

This transformation is conveniently written as a matrix operation

,

where  is a matrix consisting of the components of a in the basis ,  is a matrix consisting of the components of a in the basis , and  is a `rotation matrix’ as follows

Note that the elements of  have a simple physical interpretation.  For example, , where  is the angle between the  and  axes.  Similarly  where  is the angle between the  and  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  by putting the cosines of the known angles in the appropriate places.

Index notation provides another convenient way to write this transformation:

You don’t need to know index notation in detail to understand this  all you need to know is that

The same approach may be used to find an expression for  in terms of .  If you work through the details, you will find that

Comparing this result with the formula for  in terms of , we see that

where the superscript T denotes the transpose (rows and columns interchanged). The transformation matrix  is therefore orthogonal, and satisfies

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

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

Calculating angles

The angle between two vectors a and b is

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

If the surface is specified by a parametric equation of the form , 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

Calculating Volumes

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

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

VECTOR FIELDS AND VECTOR CALCULUS

1. Scalar field.

Let   be a Cartesian basis with origin O in three dimensional space.  Let

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 . The value of  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   be a Cartesian basis with origin O in three dimensional space.  Let

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 .  The vector may also be expressed as components in the basis

The magnitude and direction of  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  be a Cartesian basis with origin O in three dimensional space. Express the position vector of a point relative to O in  as

and let  be a scalar field.

Let  be a second Cartesian basis, with origin P.  Let  denote the position vector of  P relative to O. Express the position vector of a point relative to P in  as

To find , use the following procedure.  First, express  p as components in the basis , using the procedure outlined in Section 1.4:

where

or, using index notation

where the transformation matrix  is defined in Sect 1.4.

Now, express c as components in , and note that

so that

4. Change of basis for vector fields.

Let  be a Cartesian basis with origin O in three dimensional space. Express the position vector of a point relative to O in  as

and let  be a vector  field, with components

Let  be a second Cartesian basis, with origin P.  Let  denote the position vector of  P relative to O. Express the position vector of a point relative to P in  as

To express the vector field as components in  and as a function of the components of p, use the following procedure.  First, express  in terms of  using the procedure outlined for scalar fields in the preceding section

for k=1,2,3.  Now, find the components  of v in  using the procedure outlined in Section 1.4.  Using index notation, the result is

5. Time derivatives of vectors

Let a(t) be a vector whose magnitude and direction vary with time, t.  Suppose that  is a fixed basis, i.e. independent of time.  We may express a(t) in terms of components  in the basis  as

.

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

,

or in component form as

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

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  be a basis which rotates with instantaneous angular velocity .  Then,

7. Gradient of a scalar field.

Let  be a scalar field in three dimensional space.  The gradient of  is a vector field denoted by  or , and is defined so that

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

Let   be a Cartesian basis with origin O in three dimensional space.  Let

denote the position vector of a point in space.  Express  as a function of the components of r .  The gradient of   in this basis is then given by

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  or , and is defined so that

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

Let   be a Cartesian basis with origin O in three dimensional space.  Let

denote the position vector of a point in space.  Express v as a function of the components of r, so that .  The gradient of  v in this basis is then given by

Alternatively, in index notation

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  or .  Formally, it is defined as  (the trace of a tensor is the sum of its diagonal terms).

Let   be a Cartesian basis with origin O in three dimensional space.  Let

denote the position vector of a point in space.  Express v as a function of the components of r: . The divergence of v is then

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  or .  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   be a Cartesian basis with origin O in three dimensional space.  Let

denote the position vector of a point in space.  Express v as a function of the components of r  . The curl of  v in this basis is then given by

Using index notation, this may be expressed as

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

alternatively, expressed in index notation

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

MATRICES

1. Definition

An  matrix  is a set of numbers, arranged in m rows and n columns

A square matrix has equal numbers of rows and columns

A diagonal matrix is a square matrix with elements such that  for

The identity matrix  is a diagonal matrix for which all diagonal elements

A symmetric matrix is a square matrix with elements such that

A skew symmetric matrix is a square matrix with elements such that

2. Matrix operations

Addition  Let   and  be two matrices of order  with elements  and .  Then

Multiplication by a scalar.  Let  be a matrix with elements , and let k be a scalar.  Then

Multiplication by a matrix. Let  be a matrix of order  with elements , and let  be a matrix of order  with elements .  The product  is defined only if n=p, and is an  matrix such that

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

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  matrices.  Let  be an  matrix.  Thus

Now,

i.e.

Transpose. Let  be a matrix of order  with elements .  The transpose of  is denoted .  If  is an  matrix such that , then , i.e.

Note that

Determinant  The determinant is defined only for a square matrix.  Let  be a  matrix with components .  The determinant of   is denoted by  or  and is given by

Now, let  be an  matrix.  Define the minors  of  as the determinant formed by omitting the ith row and jth column of .  For example, the minors  and  for a  matrix are computed as follows.   Let

Then

Define the cofactors  of  as

Then, the determinant of the  matrix  is computed as follows

The result is the same whichever row i is chosen for the expansion.  For the particular case of a  matrix

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

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

Inversion.  Let  be an  matrix.  The inverse of  is denoted by  and is defined such that

The inverse of  exists if and only if .  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  with components  as defined in the preceding section.  Then

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

Note that

For a diagonal matrix, the inverse is

For a  matrix, the inverse is

Eigenvalues and eigenvectors. Let  be an  matrix, with coefficients .  Consider the vector equation

(1)

where x is a vector with n components, and  is a scalar (which may be complex).  The n nonzero vectors x and corresponding scalars  which satisfy this equation are the eigenvectors and eigenvalues of .

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

(2)

This has nontrivial solutions for x only if the determinant of the matrix  vanishes.  The equation

is an nth order polynomial which may be solved for .  In general the polynomial will have n roots, which may be complex.  The eigenvectors may then be computed using equation (2).  For example, a  matrix generally has two eigenvectors, which satisfy

Solve the quadratic equation to see that

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

so that, multiplying out the first row of the matrix (you can use the second row too, if you wish  since we chose  to make the determinant of the matrix vanish, the two equations have the same solutions.  In fact, if , you will need to do this, because the first equation will simply give 0=0 when trying to solve for one of the eigenvectors)

which are satisfied by any vector of the form

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 .  (For vectors of dimension n, the generalized dot product is defined such that  )

One may calculate explicit expressions for eigenvalues and eigenvectors for any matrix up to order , but the results are so cumbersome that, except for the  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  ) satisfy .

The eigenvalues of a skew symmetric matrix are pure imaginary.

Spectral and singular value decomposition.  Let  be a real symmetric   matrix. Denote the n (real) eigenvalues of  by , and let  be the corresponding normalized eigenvectors, such that .  Then, for any arbitrary vector b,

Let  be a diagonal matrix which contains the n eigenvalues of  as elements of the diagonal, and let  be a matrix consisting of the n eigenvectors as columns, i.e.

Then

Note that this gives another (generally quite useless) way to invert

where  is easy to compute since  is diagonal.

Square root of a matrix.   Let  be a real symmetric   matrix.  Denote the singular value decomposition of  by  as defined above.  Suppose that  denotes the square root of , defined so that

One way to compute  is through the singular value decomposition of

where