Vectors and Tensor Operations in Polar Coordinates

 

 

 
 

Many simple boundary value problems in solid mechanics (such as those that tend to appear in homework assignments or examinations!) are most conveniently solved using spherical or cylindrical-polar coordinate systems.

 

The main drawback of using a polar coordinate system is that there is no convenient way to express the various vector and tensor operations using index notation [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  everything has to be written out in long-hand.  In this section, therefore, we completely abandon index notation [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  vector and tensor components are always expressed as matrices.

 

 

Spherical-polar coordinates

 

1.1 Specifying points in spherical-polar coordinates

 

To specify points in space using spherical-polar coordinates, we first choose two convenient, mutually perpendicular reference directions (i and k in the picture).  For example, to specify position on the Earth’s surface, we might choose k to point from the center of the earth towards the North Pole, and choose i to point from the center of the earth towards the intersection of the equator (which has zero degrees latitude) and the Greenwich Meridian (which has zero degrees longitude, by definition).

 

Then, each point P in space is identified by three numbers, R,θ,ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacaGGSaGaeqiUdeNaaiilaiabew [email protected]@  shown in the picture above.  These are not components of a vector.

 

In words:

R is the distance of P from the origin

θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is the angle between the k direction and OP

ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is the angle between the i direction and the projection of OP onto a plane through O normal to k

 

By convention, we choose  R0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ 0θ 180 o [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcqaH4oqCcqGHKjYOca [email protected]@  and 0ϕ 360 o [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcqaHvpGzcqGHKjYOca [email protected]@

 

 

1.2 Converting between Cartesian and Spherical-Polar representations of points

 

When we use a Cartesian basis, we identify points in space by specifying the components of their position vector relative to the origin (x,y,z), such that r=xi+yj+zk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWG4bGaaCyAaiabgU [email protected]@   When we use a spherical-polar coordinate system, we locate points by specifying their spherical-polar coordinates R,θ,ϕ Ma[email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacaGGSaGaeqiUdeNaaiilaiabew [email protected]@

 

The formulas below relate the two representations.  They are derived using basic trigonometry

x=Rsinθcosϕ y=Rsinθsinϕ z=Rcosθ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4b Gaeyypa0JaamOuaiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGG VbGaai4Caiabew9aMjaaykW7aeaacaWG5bGaeyypa0JaamOuaiGaco hacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMbqa aiaadQhacqGH9aqpcaWGsbGaci4yaiaac+gacaGGZbGaeqiUdehaaa [email protected]@                     R= x 2 + y 2 + z 2 θ= cos 1 z/R ϕ= tan 1 y/x [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGsb Gaeyypa0ZaaOaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadQhadaahaa WcbeqaaiaaikdaaaaabeaakiaaykW7aeaacqaH4oqCcqGH9aqpciGG JbGaai4BaiaacohadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWG6b Gaai4laiaadkfaaeaacqaHvpGzcqGH9aqpciGG0bGaaiyyaiaac6ga daahaaWcbeqaaiabgkHiTiaaigdaaaGccaWG5bGaai4laiaadIhaaa [email protected]@

1.3 Spherical-Polar representation of vectors

 

When we work with vectors in spherical-polar coordinates, we abandon the {i,j,k} basis.  Instead, we specify vectors as components in the { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  basis shown in the figure.  For example, an arbitrary vector a is written as a= a R e R + a θ e θ + a ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaeqy1dygabeaakiaahwga [email protected]@ , where ( a R , a θ , a ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGsb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa [email protected]@  denote the components of a.

 

The basis is different for each point P.  In words

e R [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  points along OP

e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa [email protected]@  is tangent to a line of constant longitude through P

e ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqy1dygabeaaaa [email protected]@  is tangent to a line of constant latitude through P.

 

For example if polar-coordinates are used to specify points on the Earth’s surface,  you can visualize the basis vectors like this.  Suppose you stand at a point P on the Earths surface.  Relative to you: e R [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  points vertically upwards; e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa [email protected]@  points due South; and e ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqy1dygabeaaaa [email protected]@  points due East. Notice that the basis vectors depend on where you are standing.

 

You can also visualize the directions as follows.  To see the direction of e R [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , keep θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  fixed, and increase R. P is moving parallel to e R [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ .  To see the direction of e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa [email protected]@ , keep R and ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  fixed, and increase θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ . P now moves parallel to  e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa [email protected]@ .  To see the direction of e ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqy1dygabeaaaa [email protected]@ , keep R and θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  fixed, and increase ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ .  P now moves parallel to e ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabew9aMbqaba [email protected]@ .  Mathematically, this concept can be expressed as follows.  Let r be the position vector of P.  Then

e R = 1 | r R | r R e θ = 1 | r θ | r θ e ϕ = 1 | r ϕ | r ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGsbaabeaakiabg2da9maalaaabaGaaGymaaqaamaaemaa baWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOuaaaaaiaawE a7caGLiWoaaaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOu aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGH9a qpdaWcaaqaaiaaigdaaeaadaabdaqaamaalaaabaGaeyOaIyRaaCOC aaqaaiabgkGi2kabeI7aXbaaaiaawEa7caGLiWoaaaWaaSaaaeaacq GHciITcaWHYbaabaGaeyOaIyRaeqiUdehaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahwga daWgaaWcbaGaeqy1dygabeaakiabg2da9maalaaabaGaaGymaaqaam aaemaabaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaeqy1dyga aaGaay5bSlaawIa7aaaadaWcaaqaaiabgkGi2kaahkhaaeaacqGHci [email protected]@

By definition, the `natural basis’ for a coordinate system is the derivative of the position vector with respect to the three scalar coordinates that are used to characterize position in space (see Chapter 10 for a more detailed discussion).  The basis vectors for a polar coordinate system are parallel to the natural basis vectors, but are normalized to have unit length.  In addition, the natural basis for a polar coordinate system happens to be orthogonal. Consequently, { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  is an orthonormal basis (basis vectors have unit length, are mutually perpendicular and form a right handed triad)

 

 

1.4 Converting vectors between Cartesian and Spherical-Polar bases

 

Let a= a R e R + a θ e θ + a ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaeqy1dygabeaakiaahwga [email protected]@  be a vector, with components ( a R , a θ , a ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGsb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa [email protected]@  in the spherical-polar basis { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@ .  Let a x , a y , a z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamiEaaqabaGcca GGSaGaamyyamaaBaaaleaacaWG5baabeaakiaacYcacaWGHbWaaSba [email protected]@  denote the components of a in the basis {i,j,k}.

 

The two sets of components are related by

[ a x a y a z ]=[ sinθcosϕ cosθcosϕ sinϕ sinθsinϕ cosθsinϕ cosϕ cosθ sinθ 0 ][ a R a θ a ϕ ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWG4baabeaaaOqaaiaadgga daWgaaWcbaGaamyEaaqabaaakeaacaWGHbWaaSbaaSqaaiaadQhaae qaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbaeqabmWaaaqa aiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGGVbGaai4Caiabew 9aMbqaaiGacogacaGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4C aiabew9aMbqaaiabgkHiTiGacohacaGGPbGaaiOBaiabew9aMbqaai GacohacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9a MbqaaiGacogacaGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBai abew9aMbqaaiGacogacaGGVbGaai4Caiabew9aMbqaaiGacogacaGG VbGaai4CaiabeI7aXbqaaiabgkHiTiGacohacaGGPbGaaiOBaiabeI 7aXbqaaiaaicdaaaaacaGLBbGaayzxaaWaamWaaeaafaqabeWabaaa baGaamyyamaaBaaaleaacaWGsbaabeaaaOqaaiaadggadaWgaaWcba GaeqiUdehabeaaaOqaaiaadggadaWgaaWcbaGaeqy1dygabeaaaaaa [email protected]@

while the inverse relationship is

[ a R a θ a ϕ ]=[ sinθcosϕ sinθsinϕ cosθ cosθcosϕ cosθsinϕ sinθ sinϕ cosϕ 0 ][ a x a y a z ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWGsbaabeaaaOqaaiaadgga daWgaaWcbaGaeqiUdehabeaaaOqaaiaadggadaWgaaWcbaGaeqy1dy gabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqadmaa aeaaciGGZbGaaiyAaiaac6gacqaH4oqCciGGJbGaai4Baiaacohacq aHvpGzaeaaciGGZbGaaiyAaiaac6gacqaH4oqCciGGZbGaaiyAaiaa c6gacqaHvpGzaeaaciGGJbGaai4BaiaacohacqaH4oqCaeaaciGGJb Gaai4BaiaacohacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzaeaa ciGGJbGaai4BaiaacohacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvp GzaeaacqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCaeaacqGHsisl ciGGZbGaaiyAaiaac6gacqaHvpGzaeaaciGGJbGaai4Baiaacohacq aHvpGzaeaacaaIWaaaaaGaay5waiaaw2faamaadmaabaqbaeqabmqa aaqaaiaadggadaWgaaWcbaGaamiEaaqabaaakeaacaWGHbWaaSbaaS qaaiaadMhaaeqaaaGcbaGaamyyamaaBaaaleaacaWG6baabeaaaaaa [email protected]@

 

Observe that the two 3x3 matrices involved in this transformation are transposes (and inverses) of one another.   The transformation matrix is therefore orthogonal, satisfying [Q] [Q] T =[I] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyuaiaac2facaGGBbGaam yuaiaac2fadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcaGGBbGaamys [email protected]@ , where [I] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  denotes the 3x3 identity matrix.

 

Derivation: It is easiest to do the transformation by expressing each basis vector { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  as components in {i,j,k}, and then substituting.  To do this, recall that r=xi+yj+zk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWG4bGaaCyAaiabgU [email protected]@ , recall also the conversion

x=Rsinθcosϕy=Rsinθsinϕz=Rcosθ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadkfaciGGZbGaaiyAaiaac6gacqaH4oqCciGGJbGaai4Baiaa cohacqaHvpGzcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamyEaiabg2da9iaadkfaciGGZbGaaiyAaiaac6gacqaH4oqC ciGGZbGaaiyAaiaac6gacqaHvpGzcaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 [email protected]@

and finally recall that by definition

e R = 1 | r R | r R e θ = 1 | r θ | r θ e ϕ = 1 | r ϕ | r ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOuaaqabaGccq GH9aqpdaWcaaqaaiaaigdaaeaadaabdaqaamaalaaabaGaeyOaIyRa aCOCaaqaaiabgkGi2kaadkfaaaaacaGLhWUaayjcSdaaamaalaaaba GaeyOaIyRaaCOCaaqaaiabgkGi2kaadkfaaaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHLbWaaSba aSqaaiabeI7aXbqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaadaabda qaamaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kabeI7aXbaaaiaa wEa7caGLiWoaaaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaeq iUdehaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaahwgadaWgaaWcbaGaeqy1dygabeaakiab g2da9maalaaabaGaaGymaaqaamaaemaabaWaaSaaaeaacqGHciITca WHYbaabaGaeyOaIyRaeqy1dygaaaGaay5bSlaawIa7aaaadaWcaaqa [email protected]@

Hence, substituting for x,y,z and differentiating

r=Rsinθcosϕi+Rsinθsinϕj+Rcosθk r R =sinθcosϕi+sinθsinϕj+cosθk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHYb Gaeyypa0JaamOuaiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGG VbGaai4Caiabew9aMjaahMgacqGHRaWkcaWGsbGaci4CaiaacMgaca GGUbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dyMaaCOAaiabgUca RiaadkfaciGGJbGaai4BaiaacohacqaH4oqCcaWHRbGaaGPaVdqaai abgkDiEpaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kaadkfaaaGa eyypa0Jaci4CaiaacMgacaGGUbGaeqiUdeNaci4yaiaac+gacaGGZb Gaeqy1dyMaaCyAaiabgUcaRiGacohacaGGPbGaaiOBaiabeI7aXjGa cohacaGGPbGaaiOBaiabew9aMjaahQgacqGHRaWkciGGJbGaai4Bai [email protected]@

Conveniently we find that | r R |=1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaWaaSaaaeaacqGHciITcaWHYb aabaGaeyOaIyRaamOuaaaaaiaawEa7caGLiWoacqGH9aqpcaaIXaaa [email protected]@ . Therefore

e R =sinθcosϕi+sinθsinϕj+cosθk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGsbaabeaakiabg2da9iGacohacaGGPbGaaiOBaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMjaahMgacqGHRaWkciGGZbGaai yAaiaac6gacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzcaWHQbGa [email protected]@

Similarly

r θ =Rcosθcosϕi+RcosθsinϕjRsinθk r ϕ =Rsinθsinϕi+Rsinθcosϕj [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkGi2kaahkhaaeaacqGHciITcqaH4oqCaaGaeyypa0JaamOu aiGacogacaGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4Caiabew 9aMjaahMgacqGHRaWkcaWGsbGaci4yaiaac+gacaGGZbGaeqiUdeNa ci4CaiaacMgacaGGUbGaeqy1dyMaaCOAaiabgkHiTiaadkfaciGGZb GaaiyAaiaac6gacqaH4oqCcaWHRbGaaGPaVdqaamaalaaabaGaeyOa IyRaaCOCaaqaaiabgkGi2kabew9aMbaacqGH9aqpcqGHsislcaWGsb Gaci4CaiaacMgacaGGUbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1 dyMaaCyAaiabgUcaRiaadkfaciGGZbGaaiyAaiaac6gacqaH4oqCci [email protected]@

while | r θ |=R [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaWaaSaaaeaacqGHciITcaWHYb aabaGaeyOaIyRaeqiUdehaaaGaay5bSlaawIa7aiabg2da9iaadkfa [email protected]@ , | r ϕ |=Rsinθ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaamaalaaabaGaeyOaIyRaaC OCaaqaaiabgkGi2kabew9aMbaaaiaawEa7caGLiWoacqGH9aqpcaWG [email protected]@  so that

e θ =cosθcosϕi+cosθsinϕjsinθk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaH4oqCaeqaaOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiU deNaci4yaiaac+gacaGGZbGaeqy1dyMaaCyAaiabgUcaRiGacogaca GGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaahQga [email protected]@      e ϕ =sinϕi+cosϕj [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaHvpGzaeqaaOGaeyypa0JaeyOeI0Iaci4CaiaacMgacaGG UbGaeqy1dyMaaCyAaiabgUcaRiGacogacaGGVbGaai4Caiabew9aMj [email protected]@

Finally, substituting

a= a R [sinθcosϕi+sinθsinϕj+cosθk] + a θ [cosθcosϕi+cosθsinϕjsinθk] + a ϕ [sinϕi+cosϕj] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHHb Gaeyypa0JaamyyamaaBaaaleaacaWGsbaabeaakiaacUfaciGGZbGa aiyAaiaac6gacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzcaWHPb Gaey4kaSIaci4CaiaacMgacaGGUbGaeqiUdeNaci4CaiaacMgacaGG UbGaeqy1dyMaaCOAaiabgUcaRiGacogacaGGVbGaai4CaiabeI7aXj aahUgacaGGDbGaaGPaVdqaaiaaygW7caaMb8UaaGzaVlaaygW7caaM c8UaaGPaVlaaykW7caaMc8Uaey4kaSIaamyyamaaBaaaleaacqaH4o qCaeqaaOGaai4waiGacogacaGGVbGaai4CaiabeI7aXjGacogacaGG VbGaai4Caiabew9aMjaahMgacqGHRaWkciGGJbGaai4Baiaacohacq aH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzcaWHQbGaeyOeI0Iaci4C aiaacMgacaGGUbGaeqiUdeNaaC4Aaiaac2faaeaacaaMb8UaaGzaVl aaygW7caaMb8UaaGzaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWk caWGHbWaaSbaaSqaaiabew9aMbqabaGccaGGBbGaeyOeI0Iaci4Cai aacMgacaGGUbGaeqy1dyMaaCyAaiabgUcaRiGacogacaGGVbGaai4C [email protected]@

Collecting terms in i, j and k, we see that

a x =sinθcosϕ a R +cosθcosϕ a θ sinϕ a ϕ a y =sinθsinϕ a R +cosθsinϕ a θ +cosϕ a ϕ a z =cosθ a R sinθ a θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb WaaSbaaSqaaiaadIhaaeqaaOGaeyypa0Jaci4CaiaacMgacaGGUbGa eqiUdeNaci4yaiaac+gacaGGZbGaeqy1dyMaamyyamaaBaaaleaaca WGsbaabeaakiabgUcaRiGacogacaGGVbGaai4CaiabeI7aXjGacoga caGGVbGaai4Caiabew9aMjaadggadaWgaaWcbaGaeqiUdehabeaaki abgkHiTiGacohacaGGPbGaaiOBaiabew9aMjaadggadaWgaaWcbaGa eqy1dygabeaaaOqaaiaadggadaWgaaWcbaGaamyEaaqabaGccqGH9a qpciGGZbGaaiyAaiaac6gacqaH4oqCciGGZbGaaiyAaiaac6gacqaH vpGzcaWGHbWaaSbaaSqaaiaadkfaaeqaaOGaey4kaSIaci4yaiaac+ gacaGGZbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dyMaamyyamaa BaaaleaacqaH4oqCaeqaaOGaey4kaSIaci4yaiaac+gacaGGZbGaeq y1dyMaamyyamaaBaaaleaacqaHvpGzaeqaaaGcbaGaamyyamaaBaaa leaacaWG6baabeaakiabg2da9iGacogacaGGVbGaai4CaiabeI7aXj aadggadaWgaaWcbaGaamOuaaqabaGccqGHsislciGGZbGaaiyAaiaa c6gacqaH4oqCcaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaaMc8oaaa [email protected]@

This is the result stated.

 

To show the inverse result, start by noting that

a= a R e R + a θ e θ + a ϕ e ϕ = a x i+ a y j+ a z k a e R = a R = a x i e R + a y j e R + a z k e R [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHHb Gaeyypa0JaamyyamaaBaaaleaacaWGsbaabeaakiaahwgadaWgaaWc baGaamOuaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiabeI7aXbqaba GccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGHRaWkcaWGHbWaaSba aSqaaiabew9aMbqabaGccaWHLbWaaSbaaSqaaiabew9aMbqabaGccq GH9aqpcaWGHbWaaSbaaSqaaiaadIhaaeqaaOGaaCyAaiabgUcaRiaa dggadaWgaaWcbaGaamyEaaqabaGccaWHQbGaey4kaSIaamyyamaaBa aaleaacaWG6baabeaakiaahUgacaaMc8oabaGaeyO0H4TaaCyyaiab gwSixlaahwgadaWgaaWcbaGaamOuaaqabaGccqGH9aqpcaWGHbWaaS baaSqaaiaadkfaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaWG4baa beaakiaahMgacqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqaaOGaey 4kaSIaamyyamaaBaaaleaacaWG5baabeaakiaahQgacqGHflY1caWH LbWaaSbaaSqaaiaadkfaaeqaaOGaey4kaSIaamyyamaaBaaaleaaca WG6baabeaakiaahUgacqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqa [email protected]@

(where we have used e θ e R = e ϕ e R =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeI7aXbqaba GccqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqaaOGaeyypa0JaaCyz amaaBaaaleaacqaHvpGzaeqaaOGaeyyXICTaaCyzamaaBaaaleaaca [email protected]@  ).  Recall that

e R =sinθcosϕi+sinθsinϕj+cosθk i e R =sinθcosϕj e R =sinθsinϕk e R =cosθ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHLb WaaSbaaSqaaiaadkfaaeqaaOGaeyypa0Jaci4CaiaacMgacaGGUbGa eqiUdeNaci4yaiaac+gacaGGZbGaeqy1dyMaaCyAaiabgUcaRiGaco hacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaa hQgacqGHRaWkciGGJbGaai4BaiaacohacqaH4oqCcaWHRbGaaGPaVd qaaiabgkDiElaahMgacqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqa aOGaeyypa0Jaci4CaiaacMgacaGGUbGaeqiUdeNaci4yaiaac+gaca GGZbGaeqy1dyMaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWHQbGaeyyXICTaaCyzamaaBaaaleaacaWGsbaabe aakiabg2da9iGacohacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGa aiOBaiabew9aMjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaC4Aai abgwSixlaahwgadaWgaaWcbaGaamOuaaqabaGccqGH9aqpciGGJbGa [email protected]@

Substituting, we get

a R =sinθcosϕ a x +sinθsinϕ a y +cosθ a z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGsbaabeaakiabg2da9iGacohacaGGPbGaaiOBaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMjaadggadaWgaaWcbaGaamiEaa qabaGccqGHRaWkciGGZbGaaiyAaiaac6gacqaH4oqCciGGZbGaaiyA aiaac6gacqaHvpGzcaWGHbWaaSbaaSqaaiaadMhaaeqaaOGaey4kaS Iaci4yaiaac+gacaGGZbGaeqiUdeNaamyyamaaBaaaleaacaWG6baa [email protected]@

Proceeding in exactly the same way for the other two components gives the remaining expressions

a θ =cosθcosϕ a x +cosθsinϕ a y sinθ a z a ϕ =sinϕ a x +cosϕ a y [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb WaaSbaaSqaaiabeI7aXbqabaGccqGH9aqpciGGJbGaai4Baiaacoha cqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzcaWGHbWaaSbaaSqaai aadIhaaeqaaOGaey4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaci4C aiaacMgacaGGUbGaeqy1dyMaamyyamaaBaaaleaacaWG5baabeaaki abgkHiTiGacohacaGGPbGaaiOBaiabeI7aXjaadggadaWgaaWcbaGa amOEaaqabaGccaaMc8oabaGaamyyamaaBaaaleaacqaHvpGzaeqaaO Gaeyypa0JaeyOeI0Iaci4CaiaacMgacaGGUbGaeqy1dyMaamyyamaa BaaaleaacaWG4baabeaakiabgUcaRiGacogacaGGVbGaai4Caiabew [email protected]@

Re-writing the last three equations in matrix form gives the result stated.

 

 

 

1.5 Spherical-Polar representation of tensors

The triad of vectors { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  is an orthonormal basis (i.e. the three basis vectors have unit length, and are mutually perpendicular).  Consequently, tensors can be represented as components in this basis in exactly the same way as for a fixed Cartesian basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz [email protected]@ .  In particular, a general second order tensor S can be represented as a 3x3 matrix

S[ S RR S Rθ S Rϕ S θR S θθ S θϕ S ϕR S ϕθ S ϕϕ ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaWGsbGaamOuaaqabaaakeaacaWG tbWaaSbaaSqaaiaadkfacqaH4oqCaeqaaaGcbaGaam4uamaaBaaale aacaWGsbGaeqy1dygabeaaaOqaaiaadofadaWgaaWcbaGaeqiUdeNa amOuaaqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjabeI7aXbqaba aakeaacaWGtbWaaSbaaSqaaiabeI7aXjabew9aMbqabaaakeaacaWG tbWaaSbaaSqaaiabew9aMjaadkfaaeqaaaGcbaGaam4uamaaBaaale aacqaHvpGzcqaH4oqCaeqaaaGcbaGaam4uamaaBaaaleaacqaHvpGz [email protected]@

You can think of S RR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkfacaWGsb [email protected]@  as being equivalent to S 11 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIXa [email protected]@ , S Rθ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkfacqaH4o [email protected]@  as S 12 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa [email protected]@ , and so on. All tensor operations such as addition, multiplication by a vector, tensor products, etc can be expressed in terms of the corresponding operations on this matrix, as discussed in Section B2 of Appendix B.

 

 

The component representation of a tensor can also be expressed in dyadic form as

S= S RR e R e R + S Rθ e R e θ + S Rϕ e R e ϕ + S θR e θ e R + S θθ e θ e θ + S θϕ e θ e ϕ + S ϕR e ϕ e R + S ϕθ e ϕ e θ + S ϕϕ e ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaC4uaiabg2da9iaaykW7caaMc8 UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadkfacaWGsbaabeaakiaa hwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaai aadkfaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaWGsbGaeqiUdeha beaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWHLbWaaS baaSqaaiabeI7aXbqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaadkfa cqaHvpGzaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgEPiel aahwgadaWgaaWcbaGaeqy1dygabeaaaOqaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlabgUcaRiaaykW7caWGtbWaaSbaaSqaai abeI7aXjaadkfaaeqaaOGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGa ey4LIqSaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaadofada WgaaWcbaGaeqiUdeNaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiU dehabeaakiabgEPielaahwgadaWgaaWcbaGaeqiUdehabeaakiabgU caRiaadofadaWgaaWcbaGaeqiUdeNaeqy1dygabeaakiaahwgadaWg aaWcbaGaeqiUdehabeaakiabgEPielaahwgadaWgaaWcbaGaeqy1dy gabeaaaOqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab gUcaRiaadofadaWgaaWcbaGaeqy1dyMaamOuaaqabaGccaWHLbWaaS baaSqaaiabew9aMbqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiaadkfa aeqaaOGaey4kaSIaam4uamaaBaaaleaacqaHvpGzcqaH4oqCaeqaaO GaaCyzamaaBaaaleaacqaHvpGzaeqaaOGaey4LIqSaaCyzamaaBaaa leaacqaH4oqCaeqaaOGaey4kaSIaam4uamaaBaaaleaacqaHvpGzcq aHvpGzaeqaaOGaaCyzamaaBaaaleaacqaHvpGzaeqaaOGaey4LIqSa [email protected]@

 

Furthermore, the physical significance of the components can be interpreted in exactly the same way as for tensor components in a Cartesian basis.  For example, the spherical-polar coordinate representation for the Cauchy stress tensor has the form

σ[ σ RR σ Rθ σ Rϕ σ θR σ θθ σ θϕ σ ϕR σ ϕθ σ ϕϕ ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdpGaeyyyIO7aamWaaeaafaqabe WadaaabaGaeq4Wdm3aaSbaaSqaaiaadkfacaWGsbaabeaaaOqaaiab eo8aZnaaBaaaleaacaWGsbGaeqiUdehabeaaaOqaaiabeo8aZnaaBa aaleaacaWGsbGaeqy1dygabeaaaOqaaiabeo8aZnaaBaaaleaacqaH 4oqCcaWGsbaabeaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiabeI7aXjabew9aMbqabaaa keaacqaHdpWCdaWgaaWcbaGaeqy1dyMaamOuaaqabaaakeaacqaHdp WCdaWgaaWcbaGaeqy1dyMaeqiUdehabeaaaOqaaiabeo8aZnaaBaaa [email protected]@

The component σ θR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaam [email protected]@  represents the traction component in direction e R [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadkfaaeqaaa [email protected]@  acting on an internal material plane with normal e θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeI7aXbqaba [email protected]@ , and so on.  Of course, the Cauchy stress tensor is symmetric, with σ θR = σ Rθ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaam OuaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamOuaiabeI7aXbqa [email protected]@

 

 

 

1.6 Constitutive equations in spherical-polar coordinates

 

The constitutive equations listed in Chapter 3 all relate some measure of stress in the solid (expressed as a tensor) to some measure of local internal deformation (deformation gradient, Eulerian strain, rate of deformation tensor, etc), also expressed as a tensor.  The constitutive equations can be used without modification in spherical-polar coordinates, as long as the matrices of Cartesian components of the various tensors are replaced by their equivalent matrices in spherical-polar coordinates.

 

For example, the stress-strain relations for an isotropic, linear elastic material in spherical-polar coordinates read

[ ε RR ε θθ ε ϕϕ 2 ε θϕ 2 ε Rϕ 2 ε Rθ ]= 1 E [ 1 ν ν 0 0 0 ν 1 ν 0 0 0 ν ν 1 0 0 0 0 0 0 2( 1+ν ) 0 0 0 0 0 0 2( 1+ν ) 0 0 0 0 0 0 2( 1+ν ) ][ σ RR σ θθ σ ϕϕ σ θϕ σ Rϕ σ Rθ ]+αΔT[ 1 1 1 0 0 0 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabyqaaaaabaGaeqyTdu 2aaSbaaSqaaiaadkfacaWGsbaabeaaaOqaaiabew7aLnaaBaaaleaa cqaH4oqCcqaH4oqCaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiabew9aMj abew9aMbqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaaiabeI7aXjab ew9aMbqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaaiaadkfacqaHvp GzaeqaaaGcbaGaaGOmaiabew7aLnaaBaaaleaacaWGsbGaeqiUdeha beaaaaaakiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaaigdaaeaaca WGfbaaamaadmaabaqbaeqabyGbaaaaaeaacaaIXaaabaGaeyOeI0Ia eqyVd4gabaGaeyOeI0IaeqyVd4gabaGaaGimaaqaaiaaicdaaeaaca aIWaaabaGaeyOeI0IaeqyVd4gabaGaaGymaaqaaiabgkHiTiabe27a UbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiabgkHiTiabe27aUb qaaiabgkHiTiabe27aUbqaaiaaigdaaeaacaaIWaaabaGaaGimaaqa aiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYaWaae WaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaaabaGaaGim aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWa aabaGaaGOmamaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaa wMcaaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiaaikdadaqadaqaaiaaigdacqGHRaWkcqaH 9oGBaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaWaamWaaeaafaqabe GbbaaaaeaacqaHdpWCdaWgaaWcbaGaamOuaiaadkfaaeqaaaGcbaGa eq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqabaaakeaacqaHdpWCda WgaaWcbaGaeqy1dyMaeqy1dygabeaaaOqaaiabeo8aZnaaBaaaleaa cqaH4oqCcqaHvpGzaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkfacq aHvpGzaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkfacqaH4oqCaeqa aaaaaOGaay5waiaaw2faaiabgUcaRiabeg7aHjabfs5aejaadsfada WadaqaauaabeqageaaaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqa [email protected]@

 

HEALTH WARNING: If you are solving a problem involving anisotropic materials using spherical-polar coordinates, it is important to remember that the orientation of the basis vectors { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  vary with position.   For example, for an anisotropic, linear elastic solid you could write the constitutive equation as

σ=C(εαΔT) σ=[ σ RR σ θθ σ ϕϕ σ θϕ σ Rϕ σ Rθ ]C=[ c 11 c 12 c 13 c 14 c 15 c 16 c 12 c 22 c 23 c 24 c 25 c 26 c 13 c 23 c 33 c 34 c 35 c 36 c 14 c 24 c 34 c 44 c 45 c 46 c 15 c 25 c 35 c 45 c 55 c 56 c 16 c 26 c 36 c 46 c 56 c 66 ]ε=[ ε RR ε θθ ε ϕϕ 2 ε θϕ 2 ε Rϕ 2 ε Rθ ]α=[ α RR α θθ α ϕϕ 2 α θϕ 2 α Rϕ 2 α Rθ ] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaho8acqGH9aqpcaWHdbGaai ikaiaahw7acqGHsislcaWHXoGaeuiLdqKaamivaiaacMcaaeaacaWH dpGaeyypa0JaaGPaVpaadmaabaqbaeqabyqaaaaabaGaeq4Wdm3aaS baaSqaaiaadkfacaWGsbaabeaaaOqaaiabeo8aZnaaBaaaleaacqaH 4oqCcqaH4oqCaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiabew9aMjabew 9aMbqabaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeqy1dygabeaa aOqaaiabeo8aZnaaBaaaleaacaWGsbGaeqy1dygabeaaaOqaaiabeo 8aZnaaBaaaleaacaWGsbGaeqiUdehabeaaaaaakiaawUfacaGLDbaa caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahoeacqGH9aqp daWadaqaauaabeqagyaaaaaabaGaam4yamaaBaaaleaacaaIXaGaaG ymaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqa aiaadogadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaam4yamaaBa aaleaacaaIXaGaaGinaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigda caaI1aaabeaaaOqaaiaadogadaWgaaWcbaGaaGymaiaaiAdaaeqaaa GcbaGaam4yamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGJbWa aSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiaadogadaWgaaWcbaGaaG OmaiaaiodaaeqaaaGcbaGaam4yamaaBaaaleaacaaIYaGaaGinaaqa baaakeaacaWGJbWaaSbaaSqaaiaaikdacaaI1aaabeaaaOqaaiaado gadaWgaaWcbaGaaGOmaiaaiAdaaeqaaaGcbaGaam4yamaaBaaaleaa caaIXaGaaG4maaqabaaakeaacaWGJbWaaSbaaSqaaiaaikdacaaIZa aabeaaaOqaaiaadogadaWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGa am4yamaaBaaaleaacaaIZaGaaGinaaqabaaakeaacaWGJbWaaSbaaS qaaiaaiodacaaI1aaabeaaaOqaaiaadogadaWgaaWcbaGaaG4maiaa iAdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIXaGaaGinaaqabaaake aacaWGJbWaaSbaaSqaaiaaikdacaaI0aaabeaaaOqaaiaadogadaWg aaWcbaGaaG4maiaaisdaaeqaaaGcbaGaam4yamaaBaaaleaacaaI0a GaaGinaaqabaaakeaacaWGJbWaaSbaaSqaaiaaisdacaaI1aaabeaa aOqaaiaadogadaWgaaWcbaGaaGinaiaaiAdaaeqaaaGcbaGaam4yam aaBaaaleaacaaIXaGaaGynaaqabaaakeaacaWGJbWaaSbaaSqaaiaa ikdacaaI1aaabeaaaOqaaiaadogadaWgaaWcbaGaaG4maiaaiwdaae qaaaGcbaGaam4yamaaBaaaleaacaaI0aGaaGynaaqabaaakeaacaWG JbWaaSbaaSqaaiaaiwdacaaI1aaabeaaaOqaaiaadogadaWgaaWcba GaaGynaiaaiAdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIXaGaaGOn aaqabaaakeaacaWGJbWaaSbaaSqaaiaaikdacaaI2aaabeaaaOqaai aadogadaWgaaWcbaGaaG4maiaaiAdaaeqaaaGcbaGaam4yamaaBaaa leaacaaI0aGaaGOnaaqabaaakeaacaWGJbWaaSbaaSqaaiaaiwdaca aI2aaabeaaaOqaaiaadogadaWgaaWcbaGaaGOnaiaaiAdaaeqaaaaa aOGaay5waiaaw2faaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyTdiabg2da9maa dmaabaqbaeqabyqaaaaabaGaeqyTdu2aaSbaaSqaaiaadkfacaWGsb aabeaaaOqaaiabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaaGc baGaeqyTdu2aaSbaaSqaaiabew9aMjabew9aMbqabaaakeaacaaIYa GaeqyTdu2aaSbaaSqaaiabeI7aXjabew9aMbqabaaakeaacaaIYaGa eqyTdu2aaSbaaSqaaiaadkfacqaHvpGzaeqaaaGcbaGaaGOmaiabew 7aLnaaBaaaleaacaWGsbGaeqiUdehabeaaaaaakiaawUfacaGLDbaa caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aahg7acqGH9aqpdaWadaqaauaabeqageaaaaqaaiabeg7aHnaaBaaa leaacaWGsbGaamOuaaqabaaakeaacqaHXoqydaWgaaWcbaGaeqiUde NaeqiUdehabeaaaOqaaiabeg7aHnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaaGcbaGaaGOmaiabeg7aHnaaBaaaleaacqaH4oqCcqaHvpGzae qaaaGcbaGaaGOmaiabeg7aHnaaBaaaleaacaWGsbGaeqy1dygabeaa aOqaaiaaikdacqaHXoqydaWgaaWcbaGaamOuaiabeI7aXbqabaaaaa [email protected]@

however, the elastic constants c 11 , c 12 ,... [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaakiaacYcacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaa [email protected]@  would need to be represent the material properties in the basis { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@ , and would therefore be functions of position (you would have to calculate them using the lengthy basis change formulas listed in Section 3.2.11).  In practice the results are so complicated that there would be very little advantage in working with a spherical-polar coordinate system in this situation.

 

 

1.7 Converting tensors between Cartesian and Spherical-Polar bases

 

Let S be a tensor, with components

S[ S RR S Rθ S Rϕ S θR S θθ S θϕ S ϕR S ϕθ S ϕϕ ][ S xx S xy S xz S yx S yy S yz S zx S xy S zz ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaWGsbGaamOuaaqabaaakeaacaWG tbWaaSbaaSqaaiaadkfacqaH4oqCaeqaaaGcbaGaam4uamaaBaaale aacaWGsbGaeqy1dygabeaaaOqaaiaadofadaWgaaWcbaGaeqiUdeNa amOuaaqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjabeI7aXbqaba aakeaacaWGtbWaaSbaaSqaaiabeI7aXjabew9aMbqabaaakeaacaWG tbWaaSbaaSqaaiabew9aMjaadkfaaeqaaaGcbaGaam4uamaaBaaale aacqaHvpGzcqaH4oqCaeqaaaGcbaGaam4uamaaBaaaleaacqaHvpGz cqaHvpGzaeqaaaaaaOGaay5waiaaw2faaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+a amWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaacaWG4bGaamiEaa qabaaakeaacaWGtbWaaSbaaSqaaiaadIhacaWG5baabeaaaOqaaiaa dofadaWgaaWcbaGaamiEaiaadQhaaeqaaaGcbaGaam4uamaaBaaale aacaWG5bGaamiEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadMhacaWG 5baabeaaaOqaaiaadofadaWgaaWcbaGaamyEaiaadQhaaeqaaaGcba Gaam4uamaaBaaaleaacaWG6bGaamiEaaqabaaakeaacaWGtbWaaSba aSqaaiaadIhacaWG5baabeaaaOqaaiaadofadaWgaaWcbaGaamOEai [email protected]@

in the spherical-polar basis { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  and the Cartesian basis {i,j,k}, respectively.  The two sets of components are related by

[ S xx S xy S xz S yx S yy S yz S zx S xy S zz ]= [ sinθcosϕ cosθcosϕ sinϕ sinθsinϕ cosθsinϕ cosϕ cosθ sinθ 0 ][ S RR S Rθ S Rϕ S θR S θθ S θϕ S ϕR S ϕθ S ϕϕ ][ sinθcosϕ sinθsinϕ cosθ cosθcosϕ cosθsinϕ sinθ sinϕ cosϕ 0 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaadmaabaqbaeqabmWaaaqaai aadofadaWgaaWcbaGaamiEaiaadIhaaeqaaaGcbaGaam4uamaaBaaa leaacaWG4bGaamyEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadIhaca WG6baabeaaaOqaaiaadofadaWgaaWcbaGaamyEaiaadIhaaeqaaaGc baGaam4uamaaBaaaleaacaWG5bGaamyEaaqabaaakeaacaWGtbWaaS baaSqaaiaadMhacaWG6baabeaaaOqaaiaadofadaWgaaWcbaGaamOE aiaadIhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG4bGaamyEaaqaba aakeaacaWGtbWaaSbaaSqaaiaadQhacaWG6baabeaaaaaakiaawUfa caGLDbaacqGH9aqpaeaadaWadaqaauaabeqadmaaaeaaciGGZbGaai yAaiaac6gacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzaeaaciGG JbGaai4BaiaacohacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzae aacqGHsislciGGZbGaaiyAaiaac6gacqaHvpGzaeaaciGGZbGaaiyA aiaac6gacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzaeaaciGGJb Gaai4BaiaacohacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzaeaa ciGGJbGaai4BaiaacohacqaHvpGzaeaaciGGJbGaai4Baiaacohacq aH4oqCaeaacqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCaeaacaaI WaaaaaGaay5waiaaw2faamaadmaabaqbaeqabmWaaaqaaiaadofada WgaaWcbaGaamOuaiaadkfaaeqaaaGcbaGaam4uamaaBaaaleaacaWG sbGaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaamOuaiabew9aMb qabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjaadkfaaeqaaaGcbaGa am4uamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaaGcbaGaam4uamaaBa aaleaacqaH4oqCcqaHvpGzaeqaaaGcbaGaam4uamaaBaaaleaacqaH vpGzcaWGsbaabeaaaOqaaiaadofadaWgaaWcbaGaeqy1dyMaeqiUde habeaaaOqaaiaadofadaWgaaWcbaGaeqy1dyMaeqy1dygabeaaaaaa kiaawUfacaGLDbaacaaMc8+aamWaaeaafaqabeWadaaabaGaci4Cai aacMgacaGGUbGaeqiUdeNaci4yaiaac+gacaGGZbGaeqy1dygabaGa ci4CaiaacMgacaGGUbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dy gabaGaci4yaiaac+gacaGGZbGaeqiUdehabaGaci4yaiaac+gacaGG ZbGaeqiUdeNaci4yaiaac+gacaGGZbGaeqy1dygabaGaci4yaiaac+ gacaGGZbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dygabaGaeyOe I0Iaci4CaiaacMgacaGGUbGaeqiUdehabaGaeyOeI0Iaci4CaiaacM gacaGGUbGaeqy1dygabaGaci4yaiaac+gacaGGZbGaeqy1dygabaGa [email protected]@

[ S RR S Rθ S Rϕ S θR S θθ S θϕ S ϕR S ϕθ S ϕϕ ]= [ sinθcosϕ sinθsinϕ cosθ cosθcosϕ cosθsinϕ sinθ sinϕ cosϕ 0 ][ S xx S xy S xz S yx S yy S yz S zx S xy S zz ][ sinθcosϕ cosθcosϕ sinϕ sinθsinϕ cosθsinϕ cosϕ cosθ sinθ 0 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaadmaabaqbaeqabmWaaaqaai aadofadaWgaaWcbaGaamOuaiaadkfaaeqaaaGcbaGaam4uamaaBaaa leaacaWGsbGaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaamOuai abew9aMbqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjaadkfaaeqa aaGcbaGaam4uamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaaGcbaGaam 4uamaaBaaaleaacqaH4oqCcqaHvpGzaeqaaaGcbaGaam4uamaaBaaa leaacqaHvpGzcaWGsbaabeaaaOqaaiaadofadaWgaaWcbaGaeqy1dy MaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaeqy1dyMaeqy1dyga beaaaaaakiaawUfacaGLDbaacqGH9aqpaeaacaaMc8+aamWaaeaafa qabeWadaaabaGaci4CaiaacMgacaGGUbGaeqiUdeNaci4yaiaac+ga caGGZbGaeqy1dygabaGaci4CaiaacMgacaGGUbGaeqiUdeNaci4Cai aacMgacaGGUbGaeqy1dygabaGaci4yaiaac+gacaGGZbGaeqiUdeha baGaci4yaiaac+gacaGGZbGaeqiUdeNaci4yaiaac+gacaGGZbGaeq y1dygabaGaci4yaiaac+gacaGGZbGaeqiUdeNaci4CaiaacMgacaGG UbGaeqy1dygabaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeqiUdehaba GaeyOeI0Iaci4CaiaacMgacaGGUbGaeqy1dygabaGaci4yaiaac+ga caGGZbGaeqy1dygabaGaaGimaaaaaiaawUfacaGLDbaadaWadaqaau aabeqadmaaaeaacaWGtbWaaSbaaSqaaiaadIhacaWG4baabeaaaOqa aiaadofadaWgaaWcbaGaamiEaiaadMhaaeqaaaGcbaGaam4uamaaBa aaleaacaWG4bGaamOEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadMha caWG4baabeaaaOqaaiaadofadaWgaaWcbaGaamyEaiaadMhaaeqaaa GcbaGaam4uamaaBaaaleaacaWG5bGaamOEaaqabaaakeaacaWGtbWa aSbaaSqaaiaadQhacaWG4baabeaaaOqaaiaadofadaWgaaWcbaGaam iEaiaadMhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG6bGaamOEaaqa baaaaaGccaGLBbGaayzxaaWaamWaaeaafaqabeWadaaabaGaci4Cai aacMgacaGGUbGaeqiUdeNaci4yaiaac+gacaGGZbGaeqy1dygabaGa ci4yaiaac+gacaGGZbGaeqiUdeNaci4yaiaac+gacaGGZbGaeqy1dy gabaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeqy1dygabaGaci4Caiaa cMgacaGGUbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dygabaGaci 4yaiaac+gacaGGZbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dyga baGaci4yaiaac+gacaGGZbGaeqy1dygabaGaci4yaiaac+gacaGGZb GaeqiUdehabaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeqiUdehabaGa [email protected]@

 

These results follow immediately from the general basis change formulas for tensors .

 

 

1.8 Vector Calculus using Spherical-Polar Coordinates

 

Calculating derivatives of scalar, vector and tensor functions of position in spherical-polar coordinates is complicated by the fact that the basis vectors are functions of position.  The results can be expressed in a compact form by defining the gradient operator, which, in spherical-polar coordinates, has the representation

( e R R + e θ 1 R θ + e ϕ 1 Rsinθ ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHHjIUdaqadaqaaiaahw gadaWgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi2cqaaiabgkGi 2kaadkfaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaOWaaS aaaeaacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi2cqaaiabgkGi 2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiabew9aMbqabaGcda WcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdeha amaalaaabaGaeyOaIylabaGaeyOaIyRaeqy1dygaaaGaayjkaiaawM [email protected]@

In addition, the derivatives of the basis vectors are

e R R = e θ R = e ϕ R =0 e R θ = e θ e θ θ = e R e ϕ θ =0 e R ϕ =sinθ e ϕ e θ ϕ =cosθ e ϕ e ϕ ϕ =sinθ e R cosθ e θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqGHciITcaWHLbWaaS baaSqaaiaadkfaaeqaaaGcbaGaeyOaIyRaamOuaaaacqGH9aqpdaWc aaqaaiabgkGi2kaahwgadaWgaaWcbaGaeqiUdehabeaaaOqaaiabgk Gi2kaadkfaaaGaeyypa0ZaaSaaaeaacqGHciITcaWHLbWaaSbaaSqa aiabew9aMbqabaaakeaacqGHciITcaWGsbaaaiabg2da9iaaicdaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaa laaabaGaeyOaIyRaaCyzamaaBaaaleaacaWGsbaabeaaaOqaaiabgk Gi2kabeI7aXbaacqGH9aqpcaWHLbWaaSbaaSqaaiabeI7aXbqabaGc caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVp aalaaabaGaeyOaIyRaaCyzamaaBaaaleaacqaH4oqCaeqaaaGcbaGa eyOaIyRaeqiUdehaaiabg2da9iabgkHiTiaahwgadaWgaaWcbaGaam OuaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVpaalaaabaGaeyOaIyRaaCyzamaaBaaaleaacqaHvpGzae qaaaGcbaGaeyOaIyRaeqiUdehaaiabg2da9iaaicdaaeaadaWcaaqa aiabgkGi2kaahwgadaWgaaWcbaGaamOuaaqabaaakeaacqGHciITcq aHvpGzaaGaeyypa0Jaci4CaiaacMgacaGGUbGaeqiUdeNaaCyzamaa BaaaleaacqaHvpGzaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiab gkGi2kaahwgadaWgaaWcbaGaeqiUdehabeaaaOqaaiabgkGi2kabew 9aMbaacqGH9aqpciGGJbGaai4BaiaacohacqaH4oqCcaWHLbWaaSba aSqaaiabew9aMbqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiabgkGi2kaahwgadaWg aaWcbaGaeqy1dygabeaaaOqaaiabgkGi2kabew9aMbaacqGH9aqpcq GHsislciGGZbGaaiyAaiaac6gacqaH4oqCcaWHLbWaaSbaaSqaaiaa dkfaaeqaaOGaeyOeI0Iaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzam [email protected]@

You can derive these formulas by differentiating the expressions for the basis vectors in terms of {i,j,k}

e R =sinθcosϕi+sinθsinϕj+cosθk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGsbaabeaakiabg2da9iGacohacaGGPbGaaiOBaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMjaahMgacqGHRaWkciGGZbGaai yAaiaac6gacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzcaWHQbGa [email protected]@    e θ =cosθcosϕi+cosθsinϕjsinθk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaH4oqCaeqaaOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiU deNaci4yaiaac+gacaGGZbGaeqy1dyMaaCyAaiabgUcaRiGacogaca GGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaahQga [email protected]@       e ϕ =sinϕi+cosϕj [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaHvpGzaeqaaOGaeyypa0JaeyOeI0Iaci4CaiaacMgacaGG UbGaeqy1dyMaaCyAaiabgUcaRiGacogacaGGVbGaai4Caiabew9aMj [email protected]@

and evaluating the various derivatives. When differentiating, note that {i,j,k} are fixed, so their derivatives are zero.  The details are left as an exercise.

 

The various derivatives of scalars, vectors and tensors can be expressed using operator notation as follows. 

 

Gradient of a scalar function: Let f(R,θ,ϕ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaamOuaiaacYcacqaH4o [email protected]@  denote a scalar function of position.  The gradient of f is denoted by

f=f( e R R + e θ 1 R θ + e ϕ 1 Rsinθ ϕ )= e R f R + e θ 1 R f θ + e ϕ 1 Rsinθ f ϕ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpcaWGMbWaae WaaeaacaWHLbWaaSbaaSqaaiaadkfaaeqaaOWaaSaaaeaacqGHciIT aeaacqGHciITcaWGsbaaaiabgUcaRiaahwgadaWgaaWcbaGaeqiUde habeaakmaalaaabaGaaGymaaqaaiaadkfaaaWaaSaaaeaacqGHciIT aeaacqGHciITcqaH4oqCaaGaey4kaSIaaCyzamaaBaaaleaacqaHvp GzaeqaaOWaaSaaaeaacaaIXaaabaGaamOuaiGacohacaGGPbGaaiOB aiabeI7aXbaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabew9aMbaaai aawIcacaGLPaaacqGH9aqpcaWHLbWaaSbaaSqaaiaadkfaaeqaaOWa aSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRaamOuaaaacqGHRaWkca WHLbWaaSbaaSqaaiabeI7aXbqabaGcdaWcaaqaaiaaigdaaeaacaWG sbaaamaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kabeI7aXbaacq GHRaWkcaWHLbWaaSbaaSqaaiabew9aMbqabaGcdaWcaaqaaiaaigda aeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdehaamaalaaabaGaey [email protected]@

Alternatively, in matrix form

f= [ f R , 1 R f θ , 1 Rsinθ f ϕ ] T [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpdaWadaqaam aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadkfaaaGaaiilamaa laaabaGaaGymaaqaaiaadkfaaaWaaSaaaeaacqGHciITcaWGMbaaba GaeyOaIyRaeqiUdehaaiaacYcadaWcaaqaaiaaigdaaeaacaWGsbGa ci4CaiaacMgacaGGUbGaeqiUdehaamaalaaabaGaeyOaIyRaamOzaa qaaiabgkGi2kabew9aMbaaaiaawUfacaGLDbaadaahaaWcbeqaaiaa [email protected]@

 

Gradient of a vector function Let v= v R e R + v θ e θ + v ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaeqy1dygabeaakiaahwga [email protected]@  be a vector function of position. The gradient of v is a tensor, which can be represented as a dyadic product of the vector with the gradient operator as

v=( v R e R + v θ e θ + v ϕ e ϕ )( e R R + e θ 1 R θ + e ϕ 1 Rsinθ ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGHxkcXcqGHhis0cqGH9aqpda qadaqaaiaadAhadaWgaaWcbaGaamOuaaqabaGccaWHLbWaaSbaaSqa aiaadkfaaeqaaOGaey4kaSIaamODamaaBaaaleaacqaH4oqCaeqaaO GaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaamODamaaBaaa leaacqaHvpGzaeqaaOGaaCyzamaaBaaaleaacqaHvpGzaeqaaaGcca GLOaGaayzkaaGaey4LIq8aaeWaaeaacaWHLbWaaSbaaSqaaiaadkfa aeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWGsbaaaiabgUcaRi aahwgadaWgaaWcbaGaeqiUdehabeaakmaalaaabaGaaGymaaqaaiaa dkfaaaWaaSaaaeaacqGHciITaeaacqGHciITcqaH4oqCaaGaey4kaS IaaCyzamaaBaaaleaacqaHvpGzaeqaaOWaaSaaaeaacaaIXaaabaGa amOuaiGacohacaGGPbGaaiOBaiabeI7aXbaadaWcaaqaaiabgkGi2c [email protected]@

The dyadic product can be expanded [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  but when evaluating the derivatives it is important to recall that the basis vectors are functions of the coordinates (R,θ,ϕ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGsbGaaiilaiabeI7aXjaacY [email protected]@  and consequently their derivatives do not vanish.  For example

1 R θ ( v R e R ) e θ = 1 R v R θ e R e θ + v R R e R θ e θ = 1 R v R θ e R e θ + v R R e θ e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGymaaqaaiaadkfaaaWaaS aaaeaacqGHciITaeaacqGHciITcqaH4oqCaaWaaeWaaeaacaWG2bWa aSbaaSqaaiaadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaaaO GaayjkaiaawMcaaiabgEPielaahwgadaWgaaWcbaGaeqiUdehabeaa kiabg2da9maalaaabaGaaGymaaqaaiaadkfaaaWaaSaaaeaacqGHci ITcaWG2bWaaSbaaSqaaiaadkfaaeqaaaGcbaGaeyOaIyRaeqiUdeha aiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWHLbWaaSbaaS qaaiabeI7aXbqabaGccqGHRaWkdaWcaaqaaiaadAhadaWgaaWcbaGa amOuaaqabaaakeaacaWGsbaaamaalaaabaGaeyOaIyRaaCyzamaaBa aaleaacaWGsbaabeaaaOqaaiabgkGi2kabeI7aXbaacqGHxkcXcaWH LbWaaSbaaSqaaiabeI7aXbqabaGccqGH9aqpdaWcaaqaaiaaigdaae aacaWGsbaaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGsbaa beaaaOqaaiabgkGi2kabeI7aXbaacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSYa aSaaaeaacaWG2bWaaSbaaSqaaiaadkfaaeqaaaGcbaGaamOuaaaaca WHLbWaaSbaaSqaaiabeI7aXbqabaGccqGHxkcXcaWHLbWaaSbaaSqa [email protected]@

Verify for yourself that the matrix representing the components of the gradient of a vector is

v[ v R R 1 R v R θ v θ R 1 Rsinθ v R ϕ v ϕ R v θ R 1 R v θ θ + v R R 1 Rsinθ v θ ϕ cotθ v ϕ R v ϕ R 1 R v ϕ θ 1 Rsinθ v ϕ ϕ +cotθ v θ R + v R R ] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGHxkcXcqGHhis0cqGHHjIUda WadaqaauaabeqadmaaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWc baGaamOuaaqabaaakeaacqGHciITcaWGsbaaaaqaamaalaaabaGaaG ymaaqaaiaadkfaaaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaa dkfaaeqaaaGcbaGaeyOaIyRaeqiUdehaaiabgkHiTmaalaaabaGaam ODamaaBaaaleaacqaH4oqCaeqaaaGcbaGaamOuaaaaaeaadaWcaaqa aiaaigdaaeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdehaamaala aabaGaeyOaIyRaamODamaaBaaaleaacaWGsbaabeaaaOqaaiabgkGi 2kabew9aMbaacqGHsisldaWcaaqaaiaadAhadaWgaaWcbaGaeqy1dy gabeaaaOqaaiaadkfaaaaabaWaaSaaaeaacqGHciITcaWG2bWaaSba aSqaaiabeI7aXbqabaaakeaacqGHciITcaWGsbaaaaqaamaalaaaba GaaGymaaqaaiaadkfaaaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiabeI7aXbqabaaakeaacqGHciITcqaH4oqCaaGaey4kaSYaaSaaae aacaWG2bWaaSbaaSqaaiaadkfaaeqaaaGcbaGaamOuaaaaaeaadaWc aaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdehaam aalaaabaGaeyOaIyRaamODamaaBaaaleaacqaH4oqCaeqaaaGcbaGa eyOaIyRaeqy1dygaaiabgkHiTiGacogacaGGVbGaaiiDaiabeI7aXn aalaaabaGaamODamaaBaaaleaacqaHvpGzaeqaaaGcbaGaamOuaaaa aeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaeqy1dygabeaaaO qaaiabgkGi2kaadkfaaaaabaWaaSaaaeaacaaIXaaabaGaamOuaaaa daWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaeqy1dygabeaaaOqaai abgkGi2kabeI7aXbaaaeaadaWcaaqaaiaaigdaaeaacaWGsbGaci4C aiaacMgacaGGUbGaeqiUdehaamaalaaabaGaeyOaIyRaamODamaaBa aaleaacqaHvpGzaeqaaaGcbaGaeyOaIyRaeqy1dygaaiabgUcaRiGa cogacaGGVbGaaiiDaiabeI7aXnaalaaabaGaamODamaaBaaaleaacq aH4oqCaeqaaaGcbaGaamOuaaaacqGHRaWkdaWcaaqaaiaadAhadaWg aaWcbaGaamOuaaqabaaakeaacaWGsbaaaaaaaiaawUfacaGLDbaaaa [email protected]@

 

Divergence of a vector function Let v= v R e R + v θ e θ + v ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaeqy1dygabeaakiaahwga [email protected]@  be a vector function of position. The divergence of v is a scalar, which can be represented as a dot product of the vector with the gradient operator as

v=( e R R + e θ 1 R θ + e ϕ 1 Rsinθ ϕ )( v R e R + v θ e θ + v ϕ e ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkfaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiabew 9aMbqabaGcdaWcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGG UbGaeqiUdehaamaalaaabaGaeyOaIylabaGaeyOaIyRaeqy1dygaaa GaayjkaiaawMcaaiabgwSixpaabmaabaGaamODamaaBaaaleaacaWG sbaabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHRaWkcaWG2b WaaSbaaSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqa baGccqGHRaWkcaWG2bWaaSbaaSqaaiabew9aMbqabaGccaWHLbWaaS [email protected]@

Again, when expanding the dot product, it is important to remember to differentiate the basis vectors. Alternatively, the divergence can be expressed as trace(v) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaabshacaqGYbGaaeyyaiaabogacaqGLb [email protected]@ , which immediately gives

v v R R +2 v R R + 1 R v θ θ + 1 Rsinθ v ϕ ϕ +cotθ v θ R [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGHHjIUda WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamOuaaqabaaakeaacqGH ciITcaWGsbaaaiabgUcaRiaaikdadaWcaaqaaiaadAhadaWgaaWcba GaamOuaaqabaaakeaacaWGsbaaaiabgUcaRmaalaaabaGaaGymaaqa aiaadkfaaaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiabeI7aXb qabaaakeaacqGHciITcqaH4oqCaaGaey4kaSYaaSaaaeaacaaIXaaa baGaamOuaiGacohacaGGPbGaaiOBaiabeI7aXbaadaWcaaqaaiabgk Gi2kaadAhadaWgaaWcbaGaeqy1dygabeaaaOqaaiabgkGi2kabew9a MbaacqGHRaWkciGGJbGaai4BaiaacshacqaH4oqCdaWcaaqaaiaadA [email protected]@

 

Curl of a vector function Let v= v R e R + v θ e θ + v ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaeqy1dygabeaakiaahwga [email protected]@  be a vector function of position. The curl of v is a vector, which can be represented as a cross product of the vector with the gradient operator as

×v=( e R R + e θ 1 R θ + e ϕ 1 Rsinθ ϕ )×( v R e R + v θ e θ + v ϕ e ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgEna0kaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkfaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiabew 9aMbqabaGcdaWcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGG UbGaeqiUdehaamaalaaabaGaeyOaIylabaGaeyOaIyRaeqy1dygaaa GaayjkaiaawMcaaiabgEna0oaabmaabaGaamODamaaBaaaleaacaWG sbaabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHRaWkcaWG2b WaaSbaaSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqa baGccqGHRaWkcaWG2bWaaSbaaSqaaiabew9aMbqabaGccaWHLbWaaS [email protected]@

The curl rarely appears in solid mechanics so the components will not be expanded in full

 

 

Divergence of a tensor function.   Let S [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  be a tensor, with dyadic representation

S= S RR e R e R + S Rθ e R e θ + S Rϕ e R e ϕ + S θR e θ e R + S θθ e θ e θ + S θϕ e θ e ϕ + S ϕR e ϕ e R + S ϕθ e ϕ e θ + S ϕϕ e ϕ e ϕ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaC4uaiabg2da9iaaykW7caaMc8 UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadkfacaWGsbaabeaakiaa hwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaai aadkfaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaWGsbGaeqiUdeha beaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWHLbWaaS baaSqaaiabeI7aXbqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaadkfa cqaHvpGzaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgEPiel aahwgadaWgaaWcbaGaeqy1dygabeaaaOqaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlabgUcaRiaaykW7caWGtbWaaSbaaSqaai abeI7aXjaadkfaaeqaaOGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGa ey4LIqSaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaadofada WgaaWcbaGaeqiUdeNaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiU dehabeaakiabgEPielaahwgadaWgaaWcbaGaeqiUdehabeaakiabgU caRiaadofadaWgaaWcbaGaeqiUdeNaeqy1dygabeaakiaahwgadaWg aaWcbaGaeqiUdehabeaakiabgEPielaahwgadaWgaaWcbaGaeqy1dy gabeaaaOqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab gUcaRiaadofadaWgaaWcbaGaeqy1dyMaamOuaaqabaGccaWHLbWaaS baaSqaaiabew9aMbqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiaadkfa aeqaaOGaey4kaSIaam4uamaaBaaaleaacqaHvpGzcqaH4oqCaeqaaO GaaCyzamaaBaaaleaacqaHvpGzaeqaaOGaey4LIqSaaCyzamaaBaaa leaacqaH4oqCaeqaaOGaey4kaSIaam4uamaaBaaaleaacqaHvpGzcq aHvpGzaeqaaOGaaCyzamaaBaaaleaacqaHvpGzaeqaaOGaey4LIqSa [email protected]@

The divergence of S is a vector, which can be represented as

S=( e R R + e θ 1 R θ + e ϕ 1 Rsinθ ϕ )( S RR e R e R + S Rθ e R e θ + S Rϕ e R e ϕ + S θR e θ e R + S θθ e θ e θ + S θϕ e θ e ϕ + S ϕR e ϕ e R + S ϕθ e ϕ e θ + S ϕϕ e ϕ e ϕ ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahofacqGH9aqpca aMc8UaaGPaVlaaykW7daqadaqaaiaahwgadaWgaaWcbaGaamOuaaqa baGcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkfaaaGaey4kaSIaaC yzamaaBaaaleaacqaH4oqCaeqaaOWaaSaaaeaacaaIXaaabaGaamOu aaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaacqGHRaWkca WHLbWaaSbaaSqaaiabew9aMbqabaGcdaWcaaqaaiaaigdaaeaacaWG sbGaci4CaiaacMgacaGGUbGaeqiUdehaamaalaaabaGaeyOaIylaba GaeyOaIyRaeqy1dygaaaGaayjkaiaawMcaaiabgwSixlaaykW7daqa daqaauaabeqadeaaaeaacaWGtbWaaSbaaSqaaiaadkfacaWGsbaabe aakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWHLbWaaSba aSqaaiaadkfaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaWGsbGaeq iUdehabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWH LbWaaSbaaSqaaiabeI7aXbqabaGccqGHRaWkcaWGtbWaaSbaaSqaai aadkfacqaHvpGzaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiab gEPielaahwgadaWgaaWcbaGaeqy1dygabeaaaOqaaiabgUcaRiaayk W7caWGtbWaaSbaaSqaaiabeI7aXjaadkfaaeqaaOGaaCyzamaaBaaa leaacqaH4oqCaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWGsbaabe aakiabgUcaRiaadofadaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiaa hwgadaWgaaWcbaGaeqiUdehabeaakiabgEPielaahwgadaWgaaWcba GaeqiUdehabeaakiabgUcaRiaadofadaWgaaWcbaGaeqiUdeNaeqy1 dygabeaakiaahwgadaWgaaWcbaGaeqiUdehabeaakiabgEPielaahw gadaWgaaWcbaGaeqy1dygabeaaaOqaaiabgUcaRiaadofadaWgaaWc baGaeqy1dyMaamOuaaqabaGccaWHLbWaaSbaaSqaaiabew9aMbqaba GccqGHxkcXcaWHLbWaaSbaaSqaaiaadkfaaeqaaOGaey4kaSIaam4u amaaBaaaleaacqaHvpGzcqaH4oqCaeqaaOGaaCyzamaaBaaaleaacq aHvpGzaeqaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCaeqaaOGa ey4kaSIaam4uamaaBaaaleaacqaHvpGzcqaHvpGzaeqaaOGaaCyzam aaBaaaleaacqaHvpGzaeqaaOGaey4LIqSaaCyzamaaBaaaleaacqaH [email protected]@

Evaluating the components of the divergence is an extremely tedious operation, because each of the basis vectors in the dyadic representation of S must be differentiated, in addition to the components themselves.  The final result (expressed as a column vector) is

S[ S RR R +2 S RR R + 1 R S θR θ +cotθ S θR R + 1 Rsinθ S ϕR ϕ 1 R ( S θθ + S ϕϕ ) S Rθ R +2 S Rθ R + 1 R S θθ θ +cotθ S θθ R + 1 Rsinθ S ϕθ ϕ + S θR R cotθ S ϕϕ R S Rϕ R +2 S Rϕ R + sinθ R S θϕ θ +cosθ S θϕ R + 1 Rsinθ S ϕϕ ϕ + 1 R ( S ϕR + S ϕθ ) ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahofacqGHHjIUda WadaqaauaabeqadeaaaeaadaWcaaqaaiabgkGi2kaadofadaWgaaWc baGaamOuaiaadkfaaeqaaaGcbaGaeyOaIyRaamOuaaaacqGHRaWkca aIYaWaaSaaaeaacaWGtbWaaSbaaSqaaiaadkfacaWGsbaabeaaaOqa aiaadkfaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOuaaaadaWcaa qaaiabgkGi2kaadofadaWgaaWcbaGaeqiUdeNaamOuaaqabaaakeaa cqGHciITcqaH4oqCaaGaey4kaSIaci4yaiaac+gacaGG0bGaeqiUde 3aaSaaaeaacaWGtbWaaSbaaSqaaiabeI7aXjaadkfaaeqaaaGcbaGa amOuaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacM gacaGGUbGaeqiUdehaamaalaaabaGaeyOaIyRaam4uamaaBaaaleaa cqaHvpGzcaWGsbaabeaaaOqaaiabgkGi2kabew9aMbaacqGHsislda WcaaqaaiaaigdaaeaacaWGsbaaamaabmaabaGaam4uamaaBaaaleaa cqaH4oqCcqaH4oqCaeqaaOGaey4kaSIaam4uamaaBaaaleaacqaHvp GzcqaHvpGzaeqaaaGccaGLOaGaayzkaaaabaWaaSaaaeaacqGHciIT caWGtbWaaSbaaSqaaiaadkfacqaH4oqCaeqaaaGcbaGaeyOaIyRaam OuaaaacqGHRaWkcaaIYaWaaSaaaeaacaWGtbWaaSbaaSqaaiaadkfa cqaH4oqCaeqaaaGcbaGaamOuaaaacqGHRaWkdaWcaaqaaiaaigdaae aacaWGsbaaamaalaaabaGaeyOaIyRaam4uamaaBaaaleaacqaH4oqC cqaH4oqCaeqaaaGcbaGaeyOaIyRaeqiUdehaaiabgUcaRiGacogaca GGVbGaaiiDaiabeI7aXnaalaaabaGaam4uamaaBaaaleaacqaH4oqC cqaH4oqCaeqaaaGcbaGaamOuaaaacqGHRaWkdaWcaaqaaiaaigdaae aacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdehaamaalaaabaGaeyOa IyRaam4uamaaBaaaleaacqaHvpGzcqaH4oqCaeqaaaGcbaGaeyOaIy Raeqy1dygaaiabgUcaRmaalaaabaGaam4uamaaBaaaleaacqaH4oqC caWGsbaabeaaaOqaaiaadkfaaaGaeyOeI0Iaci4yaiaac+gacaGG0b GaeqiUde3aaSaaaeaacaWGtbWaaSbaaSqaaiabew9aMjabew9aMbqa baaakeaacaWGsbaaaaqaamaalaaabaGaeyOaIyRaam4uamaaBaaale aacaWGsbGaeqy1dygabeaaaOqaaiabgkGi2kaadkfaaaGaey4kaSIa aGOmamaalaaabaGaam4uamaaBaaaleaacaWGsbGaeqy1dygabeaaaO qaaiaadkfaaaGaey4kaSYaaSaaaeaaciGGZbGaaiyAaiaac6gacqaH 4oqCaeaacaWGsbaaamaalaaabaGaeyOaIyRaam4uamaaBaaaleaacq aH4oqCcqaHvpGzaeqaaaGcbaGaeyOaIyRaeqiUdehaaiabgUcaRiGa cogacaGGVbGaai4CaiabeI7aXnaalaaabaGaam4uamaaBaaaleaacq aH4oqCcqaHvpGzaeqaaaGcbaGaamOuaaaacqGHRaWkdaWcaaqaaiaa igdaaeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdehaamaalaaaba GaeyOaIyRaam4uamaaBaaaleaacqaHvpGzcqaHvpGzaeqaaaGcbaGa eyOaIyRaeqy1dygaaiabgUcaRmaalaaabaGaaGymaaqaaiaadkfaaa WaaeWaaeaacaWGtbWaaSbaaSqaaiabew9aMjaadkfaaeqaaOGaey4k aSIaam4uamaaBaaaleaacqaHvpGzcqaH4oqCaeqaaaGccaGLOaGaay [email protected]@

 

 

 

2: Cylindrical-polar coordinates

 

 

2.1 Specifying points in space using in cylindrical-polar coordinates

To specify the location of a point in cylindrical-polar coordinates, we choose an origin at some point on the axis of the cylinder, select a unit vector k to be parallel to the axis of the cylinder, and choose a convenient direction for the basis vector i, as shown in the picture.  We then use the three numbers r,θ,z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacaGGSaGaeqiUdeNaaiilaiaadQ [email protected]@  to locate a point inside the cylinder, as shown in the picture.  These are not components of a vector.

 

In words

r is the radial distance of P from the axis of the cylinder

ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr [email protected]@  is the angle between the i direction and the  projection of OP onto the i,j plane

z is the length of the projection of OP on the axis of the cylinder.

By convention r>0 and 0θ 360 o [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcqaH4oqCcqGHKjYOca [email protected]@

 

2.2 Converting between cylindrical polar and rectangular cartesian coordinates

 

When we use a Cartesian basis, we identify points in space by specifying the components of their position vector relative to the origin (x,y,z), such that r=xi+yj+zk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWG4bGaaCyAaiabgU [email protected]@   When we use a spherical-polar coordinate system, we locate points by specifying their spherical-polar coordinates r,θ,z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacaGGSaGaeqiUdeNaaiilaiaadQ [email protected]@

 

The formulas below relate the two representations.  They are derived using basic trigonometry

x=rcosθ y=rsinθ z=z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4b Gaeyypa0JaamOCaiGacogacaGGVbGaai4CaiabeI7aXjaaykW7aeaa caWG5bGaeyypa0JaamOCaiGacohacaGGPbGaaiOBaiabeI7aXbqaai [email protected]@                    r= x 2 + y 2 θ= tan 1 y/x z=z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGYb Gaeyypa0ZaaOaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamyEamaaCaaaleqabaGaaGOmaaaaaeqaaOGaaGPaVdqaaiabeI 7aXjabg2da9iGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0Ia aGymaaaakiaadMhacaGGVaGaamiEaaqaaiaadQhacqGH9aqpcaWG6b [email protected]@

 

 

2.3 Cylindrical-polar representation of vectors

 

When we work with vectors in spherical-polar coordinates, we specify vectors as components in the { e r , e θ , e z } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  basis shown in the figure.  For example, an arbitrary vector a is written as a= a r e r + a θ e θ + a z e z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaamOEaaqabaGccaWHLbWa [email protected]@ , where ( a r , a θ , a z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGYb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa [email protected]@  denote the components of a.

 

The basis vectors are selected as follows

e r [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is a unit vector normal to the cylinder at P

e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa [email protected]@  is a unit vector circumferential to the cylinder at P, chosen to make { e r , e θ , e z } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  a right handed triad

e z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is parallel to the k vector.

 

You will see that the position vector of point P would be expressed as

r=r e r +z e z =rcosθi+rsinθj+zk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCaiabg2 da9iaadkhacaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaamOE aiaahwgadaWgaaWcbaGaamOEaaqabaGccqGH9aqpcaWGYbGaci4yai aac+gacaGGZbGaeqiUdeNaaCyAaiabgUcaRiaadkhaciGGZbGaaiyA [email protected]@

 

Note also that the basis vectors are intentionally chosen to satisfy

e r = 1 | r r | r r e ϕ = 1 | r θ | r θ e z = 1 | r z | r z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGYbaabeaakiabg2da9maalaaabaGaaGymaaqaamaaemaa baWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOCaaaaaiaawE a7caGLiWoaaaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOC aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWHLbWaaSbaaSqaaiabew9aMbqabaGccqGH9a qpdaWcaaqaaiaaigdaaeaadaabdaqaamaalaaabaGaeyOaIyRaaCOC aaqaaiabgkGi2kabeI7aXbaaaiaawEa7caGLiWoaaaWaaSaaaeaacq GHciITcaWHYbaabaGaeyOaIyRaeqiUdehaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahwga daWgaaWcbaGaamOEaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaada abdaqaamaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kaadQhaaaaa caGLhWUaayjcSdaaamaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2k [email protected]@

The basis vectors have unit length, are mutually perpendicular, and form a right handed triad and therefore { e r , e θ , e z } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  is an orthonormal basis.  The basis vectors are parallel to (but not equivalent to) the natural basis vectors for a cylindrical polar coordinate system (see Chapter 10 for a more detailed discussion).

 

 

2.4 Converting vectors between Cylindrical and Cartesian bases

Let a= a r e r + a θ e θ + a z e z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaamOEaaqabaGccaWHLbWa [email protected]@  be a vector, with components ( a r , a θ , a z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGYb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa [email protected]@  in the spherical-polar basis { e R , e θ , e ϕ } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@ .  Let a x , a y , a z [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamiEaaqabaGcca GGSaGaamyyamaaBaaaleaacaWG5baabeaakiaacYcacaWGHbWaaSba [email protected]@  denote the components of a in the basis {i,j,k}.

 

The two sets of components are related by

[ a x a y a z ]=[ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ][ a r a θ a z ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWG4baabeaaaOqaaiaadgga daWgaaWcbaGaamyEaaqabaaakeaacaWGHbWaaSbaaSqaaiaadQhaae qaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbaeqabmWaaaqa aiGacogacaGGVbGaai4CaiabeI7aXbqaaiabgkHiTiGacohacaGGPb GaaiOBaiabeI7aXbqaaiaaicdaaeaaciGGZbGaaiyAaiaac6gacqaH 4oqCaeaaciGGJbGaai4BaiaacohacqaH4oqCaeaacaaIWaaabaGaaG imaaqaaiaaicdaaeaacaaIXaaaaaGaay5waiaaw2faamaadmaabaqb aeqabmqaaaqaaiaadggadaWgaaWcbaGaamOCaaqabaaakeaacaWGHb WaaSbaaSqaaiabeI7aXbqabaaakeaacaWGHbWaaSbaaSqaaiaadQha [email protected]@      [ a r a θ a z ]=[ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ][ a x a y a z ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWGYbaabeaaaOqaaiaadgga daWgaaWcbaGaeqiUdehabeaaaOqaaiaadggadaWgaaWcbaGaamOEaa qabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqabeWadaaa baGaci4yaiaac+gacaGGZbGaeqiUdehabaGaci4CaiaacMgacaGGUb GaeqiUdehabaGaaGimaaqaaiabgkHiTiGacohacaGGPbGaaiOBaiab eI7aXbqaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiaaigdaaaaacaGLBbGaayzxaaWaamWaaeaa faqabeWabaaabaGaamyyamaaBaaaleaacaWG4baabeaaaOqaaiaadg gadaWgaaWcbaGaamyEaaqabaaakeaacaWGHbWaaSbaaSqaaiaadQha [email protected]@

 

Observe that the two 3x3 matrices involved in this transformation are transposes (and inverses) of one another.   The transformation matrix is therefore orthogonal, satisfying [Q] [Q] T =[I] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyuaiaac2facaGGBbGaam yuaiaac2fadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcaGGBbGaamys [email protected]@ , where [I] [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  denotes the 3x3 identity matrix.

 

The derivation of these results follows the procedure outlined in E.1.4 exactly, and is left as an exercise.
 

 

2.5 Cylindrical-Polar representation of tensors

 

The triad of vectors { e r , e θ , e z } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  is an orthonormal basis (i.e. the three basis vectors have unit length, and are mutually perpendicular).  Consequently, tensors can be represented as components in this basis in exactly the same way as for a fixed Cartesian basis { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz [email protected]@ .  In particular, a general second order tensor S can be represented as a 3x3 matrix

S[ S rr S rθ S rz S θr S θθ S θz S zr S zθ S zz ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaWGYbGaamOCaaqabaaakeaacaWG tbWaaSbaaSqaaiaadkhacqaH4oqCaeqaaaGcbaGaam4uamaaBaaale aacaWGYbGaamOEaaqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjaa dkhaaeqaaaGcbaGaam4uamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaa GcbaGaam4uamaaBaaaleaacqaH4oqCcaWG6baabeaaaOqaaiaadofa daWgaaWcbaGaamOEaiaadkhaaeqaaaGcbaGaam4uamaaBaaaleaaca WG6bGaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaamOEaiaadQha [email protected]@

You can think of S rr [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkhacaWGYb [email protected]@  as being equivalent to S 11 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIXa [email protected]@ , S rθ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkhacqaH4o [email protected]@  as S 12 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa [email protected]@ , and so on. All tensor operations such as addition, multiplication by a vector, tensor products, etc can be expressed in terms of the corresponding operations on this matrix, as discussed in Section B2 of Appendix B.

 

The component representation of a tensor can also be expressed in dyadic form as

S= S rr e r e r + S rθ e r e θ + S rz e r e z + S θr e θ e r + S θθ e θ e θ + S θz e θ e z + S zr e z e r + S zθ e z e θ + S zz e z e z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaC4uaiabg2da9iaaykW7caaMc8 UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadkhacaWGYbaabeaakiaa hwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaai aadkhaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaWGYbGaeqiUdeha beaakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXcaWHLbWaaS baaSqaaiabeI7aXbqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaadkha caWG6baabeaakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXca WHLbWaaSbaaSqaaiaadQhaaeqaaaGcbaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaey4kaSIaaGPaVlaadofadaWgaaWcbaGaeq iUdeNaamOCaaqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGH xkcXcaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaam4uamaaBa aaleaacqaH4oqCcqaH4oqCaeqaaOGaaCyzamaaBaaaleaacqaH4oqC aeqaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaS Iaam4uamaaBaaaleaacqaH4oqCcaWG6baabeaakiaahwgadaWgaaWc baGaeqiUdehabeaakiabgEPielaahwgadaWgaaWcbaGaamOEaaqaba aakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWk caWGtbWaaSbaaSqaaiaadQhacaWGYbaabeaakiaahwgadaWgaaWcba GaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiaadkhaaeqaaOGa ey4kaSIaam4uamaaBaaaleaacaWG6bGaeqiUdehabeaakiaahwgada WgaaWcbaGaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiabeI7a XbqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaadQhacaWG6baabeaaki aahwgadaWgaaWcbaGaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqa [email protected]@

The remarks in Section E.1.5 regarding the physical significance of tensor components also applies to tensor components in cylindrical-polar coordinates.

 

 

2.6 Constitutive equations in cylindrical-polar coordinates

 

The constitutive equations listed in Chapter 3 all relate some measure of stress in the solid (expressed as a tensor) to some measure of local internal deformation (deformation gradient, Eulerian strain, rate of deformation tensor, etc), also expressed as a tensor.  The constitutive equations can be used without modification in cylindrical-polar coordinates, as long as the matrices of Cartesian components of the various tensors are replaced by their equivalent matrices in spherical-polar coordinates.

 

For example, the stress-strain relations for an isotropic, linear elastic material in cylindrical-polar coordinates read

[ ε rr ε θθ ε zz 2 ε θz 2 ε rz 2 ε rθ ]= 1 E [ 1 ν ν 0 0 0 ν 1 ν 0 0 0 ν ν 1 0 0 0 0 0 0 2( 1+ν ) 0 0 0 0 0 0 2( 1+ν ) 0 0 0 0 0 0 2( 1+ν ) ][ σ rr σ θθ σ zz σ θz σ rz σ rθ ]+αΔT[ 1 1 1 0 0 0 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabyqaaaaabaGaeqyTdu 2aaSbaaSqaaiaadkhacaWGYbaabeaaaOqaaiabew7aLnaaBaaaleaa cqaH4oqCcqaH4oqCaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaadQhaca WG6baabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGaeqiUdeNaamOE aaqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaaiaadkhacaWG6baabe aaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGaamOCaiabeI7aXbqabaaa aaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamyraa aadaWadaqaauaabeqagyaaaaaabaGaaGymaaqaaiabgkHiTiabe27a UbqaaiabgkHiTiabe27aUbqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiabgkHiTiabe27aUbqaaiaaigdaaeaacqGHsislcqaH9oGBaeaa caaIWaaabaGaaGimaaqaaiaaicdaaeaacqGHsislcqaH9oGBaeaacq GHsislcqaH9oGBaeaacaaIXaaabaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGOmamaabmaaba GaaGymaiabgUcaRiabe27aUbGaayjkaiaawMcaaaqaaiaaicdaaeaa caaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaai aaikdadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaa aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaeaacaaIYaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4ga caGLOaGaayzkaaaaaaGaay5waiaaw2faamaadmaabaqbaeqabyqaaa aabaGaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaaaOqaaiabeo8a ZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaaGcbaGaeq4Wdm3aaSbaaS qaaiaadQhacaWG6baabeaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqC caWG6baabeaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOEaaqaba aakeaacqaHdpWCdaWgaaWcbaGaamOCaiabeI7aXbqabaaaaaGccaGL BbGaayzxaaGaey4kaSIaeqySdeMaeuiLdqKaamivamaadmaabaqbae qabyqaaaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaqa [email protected]@

The cautionary remarks regarding anisotropic materials in E.1.6 also applies to cylindrical-polar coordinate systems.

 

 

2.7 Converting tensors between Cartesian and Spherical-Polar bases

 

Let S be a tensor, with components

S[ S rr S rθ S rz S θr S θθ S θz S zr S zθ S zz ][ S xx S xy S xz S yx S yy S yz S zx S xy S zz ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaWGYbGaamOCaaqabaaakeaacaWG tbWaaSbaaSqaaiaadkhacqaH4oqCaeqaaaGcbaGaam4uamaaBaaale aacaWGYbGaamOEaaqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjaa dkhaaeqaaaGcbaGaam4uamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaa GcbaGaam4uamaaBaaaleaacqaH4oqCcaWG6baabeaaaOqaaiaadofa daWgaaWcbaGaamOEaiaadkhaaeqaaaGcbaGaam4uamaaBaaaleaaca WG6bGaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaamOEaiaadQha aeqaaaaaaOGaay5waiaaw2faaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaeyyyIORaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+a amWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaacaWG4bGaamiEaa qabaaakeaacaWGtbWaaSbaaSqaaiaadIhacaWG5baabeaaaOqaaiaa dofadaWgaaWcbaGaamiEaiaadQhaaeqaaaGcbaGaam4uamaaBaaale aacaWG5bGaamiEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadMhacaWG 5baabeaaaOqaaiaadofadaWgaaWcbaGaamyEaiaadQhaaeqaaaGcba Gaam4uamaaBaaaleaacaWG6bGaamiEaaqabaaakeaacaWGtbWaaSba aSqaaiaadIhacaWG5baabeaaaOqaaiaadofadaWgaaWcbaGaamOEai [email protected]@

in the cylindrical-polar basis { e r , e θ , e z } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH [email protected]@  and the Cartesian basis {i,j,k}, respectively.  The two sets of components are related by

[ S xx S xy S xz S yx S yy S yz S zx S xy S zz ]=[ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ][ S rr S rθ S rz S θr S θθ S θz S zr S zθ S zz ][ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaadIhacaWG4baabeaaaOqaaiaadofadaWgaaWcbaGa amiEaiaadMhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG4bGaamOEaa qabaaakeaacaWGtbWaaSbaaSqaaiaadMhacaWG4baabeaaaOqaaiaa dofadaWgaaWcbaGaamyEaiaadMhaaeqaaaGcbaGaam4uamaaBaaale aacaWG5bGaamOEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadQhacaWG 4baabeaaaOqaaiaadofadaWgaaWcbaGaamiEaiaadMhaaeqaaaGcba Gaam4uamaaBaaaleaacaWG6bGaamOEaaqabaaaaaGccaGLBbGaayzx aaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaci4yaiaac+gacaGGZb GaeqiUdehabaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeqiUdehabaGa aGimaaqaaiGacohacaGGPbGaaiOBaiabeI7aXbqaaiGacogacaGGVb Gaai4CaiabeI7aXbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaa igdaaaaacaGLBbGaayzxaaWaamWaaeaafaqabeWadaaabaGaam4uam aaBaaaleaacaWGYbGaamOCaaqabaaakeaacaWGtbWaaSbaaSqaaiaa dkhacqaH4oqCaeqaaaGcbaGaam4uamaaBaaaleaacaWGYbGaamOEaa qabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjaadkhaaeqaaaGcbaGa am4uamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaaGcbaGaam4uamaaBa aaleaacqaH4oqCcaWG6baabeaaaOqaaiaadofadaWgaaWcbaGaamOE aiaadkhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG6bGaeqiUdehabe aaaOqaaiaadofadaWgaaWcbaGaamOEaiaadQhaaeqaaaaaaOGaay5w aiaaw2faaiaaykW7daWadaqaauaabeqadmaaaeaaciGGJbGaai4Bai aacohacqaH4oqCaeaaciGGZbGaaiyAaiaac6gacqaH4oqCaeaacaaI WaaabaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeqiUdehabaGaci4yai aac+gacaGGZbGaeqiUdehabaGaaGimaaqaaiaaicdaaeaacaaIWaaa [email protected]@

[ S rr S rθ S rz S θr S θθ S θz S zr S zθ S zz ]=[ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ][ S xx S xy S xz S yx S yy S yz S zx S xy S zz ][ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaadkhacaWGYbaabeaaaOqaaiaadofadaWgaaWcbaGa amOCaiabeI7aXbqabaaakeaacaWGtbWaaSbaaSqaaiaadkhacaWG6b aabeaaaOqaaiaadofadaWgaaWcbaGaeqiUdeNaamOCaaqabaaakeaa caWGtbWaaSbaaSqaaiabeI7aXjabeI7aXbqabaaakeaacaWGtbWaaS baaSqaaiabeI7aXjaadQhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG 6bGaamOCaaqabaaakeaacaWGtbWaaSbaaSqaaiaadQhacqaH4oqCae qaaaGcbaGaam4uamaaBaaaleaacaWG6bGaamOEaaqabaaaaaGccaGL BbGaayzxaaGaeyypa0JaaGPaVpaadmaabaqbaeqabmWaaaqaaiGaco gacaGGVbGaai4CaiabeI7aXbqaaiGacohacaGGPbGaaiOBaiabeI7a XbqaaiaaicdaaeaacqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCae aaciGGJbGaai4BaiaacohacqaH4oqCaeaacaaIWaaabaGaaGimaaqa aiaaicdaaeaacaaIXaaaaaGaay5waiaaw2faamaadmaabaqbaeqabm WaaaqaaiaadofadaWgaaWcbaGaamiEaiaadIhaaeqaaaGcbaGaam4u amaaBaaaleaacaWG4bGaamyEaaqabaaakeaacaWGtbWaaSbaaSqaai aadIhacaWG6baabeaaaOqaaiaadofadaWgaaWcbaGaamyEaiaadIha aeqaaaGcbaGaam4uamaaBaaaleaacaWG5bGaamyEaaqabaaakeaaca WGtbWaaSbaaSqaaiaadMhacaWG6baabeaaaOqaaiaadofadaWgaaWc baGaamOEaiaadIhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG4bGaam yEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadQhacaWG6baabeaaaaaa kiaawUfacaGLDbaadaWadaqaauaabeqadmaaaeaaciGGJbGaai4Bai aacohacqaH4oqCaeaacqGHsislciGGZbGaaiyAaiaac6gacqaH4oqC aeaacaaIWaaabaGaci4CaiaacMgacaGGUbGaeqiUdehabaGaci4yai aac+gacaGGZbGaeqiUdehabaGaaGimaaqaaiaaicdaaeaacaaIWaaa [email protected]@

 

 

 

 

 

 

2.8 Vector Calculus using Cylindrical-Polar Coordinates

 

Calculating derivatives of scalar, vector and tensor functions of position in cylindrical-polar coordinates is complicated by the fact that the basis vectors are functions of position.  The results can be expressed in a compact form by defining the gradient operator, which, in spherical-polar coordinates, has the representation

( e r r + e θ 1 r θ + e z z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHHjIUdaqadaqaaiaahw gadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi2cqaaiabgkGi 2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaOWaaS aaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi2cqaaiabgkGi 2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQhaaeqaaOWaaS aaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjkaiaawMcaaaaa @[email protected]

In addition, the nonzero derivatives of the basis vectors are

e r θ = e θ e θ θ = e r (all other derivatives are zero) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaaCyzamaaBaaale aacaWGYbaabeaaaOqaaiabgkGi2kabeI7aXbaacqGH9aqpcaWHLbWa aSbaaSqaaiabeI7aXbqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVpaalaaabaGaeyOaIyRaaCyzamaaBaaa leaacqaH4oqCaeqaaaGcbaGaeyOaIyRaeqiUdehaaiabg2da9iabgk HiTiaahwgadaWgaaWcbaGaamOCaaqabaGccaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caGGOaGaaeyyai aabYgacaqGSbGaaeiiaiaab+gacaqG0bGaaeiAaiaabwgacaqGYbGa aeiiaiaabsgacaqGLbGaaeOCaiaabMgacaqG2bGaaeyyaiaabshaca qGPbGaaeODaiaabwgacaqGZbGaaeiiaiaabggacaqGYbGaaeyzaiaa [email protected]@

 

The various derivatives of scalars, vectors and tensors can be expressed using operator notation as follows. 

 

Gradient of a scalar function: Let f(r,θ,z) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaamOCaiaacYcacqaH4o [email protected]@  denote a scalar function of position.  The gradient of f is denoted by

f=( e r r + e θ 1 r θ + e z z )f= e r f r + e θ 1 r f θ + e z f z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpdaqadaqaai aahwgadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi2cqaaiab gkGi2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaO WaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi2cqaaiab gkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQhaaeqaaO WaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjkaiaawMca aiaadAgacqGH9aqpcaWHLbWaaSbaaSqaaiaadkhaaeqaaOWaaSaaae aacqGHciITcaWGMbaabaGaeyOaIyRaamOCaaaacqGHRaWkcaWHLbWa aSbaaSqaaiabeI7aXbqabaGcdaWcaaqaaiaaigdaaeaacaWGYbaaam aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kabeI7aXbaacqGHRaWk caWHLbWaaSbaaSqaaiaadQhaaeqaaOWaaSaaaeaacqGHciITcaWGMb [email protected]@

Alternatively, in matrix form

f= [ f r , 1 r f θ , f z ] T [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpdaWadaqaam aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadkhaaaGaaiilamaa laaabaGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHciITcaWGMbaaba GaeyOaIyRaeqiUdehaaiaacYcadaWcaaqaaiabgkGi2kaadAgaaeaa cqGHciITcaWG6baaaaGaay5waiaaw2faamaaCaaaleqabaGaamivaa [email protected]@

 

Gradient of a vector function Let v= v r e r + v θ e θ + v z e z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaamOEaaqabaGccaWHLbWa [email protected]@  be a vector function of position. The gradient of v is a tensor, which can be represented as a dyadic product of the vector with the gradient operator as

v=( v r e r + v θ e θ + v z e z )( e r r + e θ 1 r θ + e z z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGHxkcXcqGHhis0cqGH9aqpda qadaqaaiaadAhadaWgaaWcbaGaamOCaaqabaGccaWHLbWaaSbaaSqa aiaadkhaaeqaaOGaey4kaSIaamODamaaBaaaleaacqaH4oqCaeqaaO GaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaamODamaaBaaa leaacaWG6baabeaakiaahwgadaWgaaWcbaGaamOEaaqabaaakiaawI cacaGLPaaacqGHxkcXdaqadaqaaiaahwgadaWgaaWcbaGaamOCaaqa baGcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaC yzamaaBaaaleaacqaH4oqCaeqaaOWaaSaaaeaacaaIXaaabaGaamOC aaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaacqGHRaWkca WHLbWaaSbaaSqaaiaadQhaaeqaaOWaaSaaaeaacqGHciITaeaacqGH [email protected]@

The dyadic product can be expanded [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  but when evaluating the derivatives it is important to recall that the basis vectors are functions of the coordinate θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqab[email protected]@  and consequently their derivatives may not vanish.  For example

1 r θ ( v r e r ) e θ = 1 r v r θ e r e θ + v r r e r θ e θ = 1 r v r θ e r e θ + v r r e θ e θ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGymaaqaaiaadkhaaaWaaS aaaeaacqGHciITaeaacqGHciITcqaH4oqCaaWaaeWaaeaacaWG2bWa aSbaaSqaaiaadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaaaO GaayjkaiaawMcaaiabgEPielaahwgadaWgaaWcbaGaeqiUdehabeaa kiabg2da9maalaaabaGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHci ITcaWG2bWaaSbaaSqaaiaadkhaaeqaaaGcbaGaeyOaIyRaeqiUdeha aiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXcaWHLbWaaSbaaS qaaiabeI7aXbqabaGccqGHRaWkdaWcaaqaaiaadAhadaWgaaWcbaGa amOCaaqabaaakeaacaWGYbaaamaalaaabaGaeyOaIyRaaCyzamaaBa aaleaacaWGYbaabeaaaOqaaiabgkGi2kabeI7aXbaacqGHxkcXcaWH LbWaaSbaaSqaaiabeI7aXbqabaGccqGH9aqpdaWcaaqaaiaaigdaae aacaWGYbaaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGYbaa beaaaOqaaiabgkGi2kabeI7aXbaacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSYa aSaaaeaacaWG2bWaaSbaaSqaaiaadkhaaeqaaaGcbaGaamOCaaaaca WHLbWaaSbaaSqaaiabeI7aXbqabaGccqGHxkcXcaWHLbWaaSbaaSqa [email protected]@

Verify for yourself that the matrix representing the components of the gradient of a vector is

v[ v r r 1 r v r θ v θ r v r z v θ r 1 r v θ θ + v r r v θ z v z r 1 r v z θ v z z ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGHxkcXcqGHhis0cqGHHjIUda WadaqaauaabeqadmaaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWc baGaamOCaaqabaaakeaacqGHciITcaWGYbaaaaqaamaalaaabaGaaG ymaaqaaiaadkhaaaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaa dkhaaeqaaaGcbaGaeyOaIyRaeqiUdehaaiabgkHiTmaalaaabaGaam ODamaaBaaaleaacqaH4oqCaeqaaaGcbaGaamOCaaaaaeaadaWcaaqa aiabgkGi2kaadAhadaWgaaWcbaGaamOCaaqabaaakeaacqGHciITca WG6baaaaqaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacqaH4oqC aeqaaaGcbaGaeyOaIyRaamOCaaaaaeaadaWcaaqaaiaaigdaaeaaca WGYbaaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacqaH4oqCaeqa aaGcbaGaeyOaIyRaeqiUdehaaiabgUcaRmaalaaabaGaamODamaaBa aaleaacaWGYbaabeaaaOqaaiaadkhaaaaabaWaaSaaaeaacqGHciIT caWG2bWaaSbaaSqaaiabeI7aXbqabaaakeaacqGHciITcaWG6baaaa qaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWG6baabeaaaOqa aiabgkGi2kaadkhaaaaabaWaaSaaaeaacaaIXaaabaGaamOCaaaada WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamOEaaqabaaakeaacqGH ciITcqaH4oqCaaaabaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaai aadQhaaeqaaaGcbaGaeyOaIyRaamOEaaaaaaaacaGLBbGaayzxaaaa [email protected]@

 

Divergence of a vector function Let v= v r e r + v θ e θ + v z e z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaamOEaaqabaGccaWHLbWa [email protected]@  be a vector function of position. The divergence of v is a scalar, which can be represented as a dot product of the vector with the gradient operator as

v=( e r r + e θ 1 r θ + e z z )( v r e r + v θ e θ + v z e z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQ haaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjk aiaawMcaaiabgwSixpaabmaabaGaamODamaaBaaaleaacaWGYbaabe aakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWkcaWG2bWaaSba aSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccq GHRaWkcaWG2bWaaSbaaSqaaiaadQhaaeqaaOGaaCyzamaaBaaaleaa [email protected]@

Again, when expanding the dot product, it is important to remember to differentiate the basis vectors. Alternatively, the divergence can be expressed as trace(v) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaabshacaqGYbGaaeyyaiaabogacaqGLb [email protected]@ , which immediately gives

v v r r + v r r + 1 r v θ θ + v z z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGHHjIUda WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamOCaaqabaaakeaacqGH ciITcaWGYbaaaiabgUcaRmaalaaabaGaamODamaaBaaaleaacaWGYb aabeaaaOqaaiaadkhaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOC aaaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaeqiUdehabeaaaO qaaiabgkGi2kabeI7aXbaacqGHRaWkdaWcaaqaaiabgkGi2kaadAha [email protected]@

 

Curl of a vector function Let v= v R e R + v θ e θ + v z e z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaamOEaaqabaGccaWHLbWa [email protected]@  be a vector function of position. The curl of v is a vector, which can be represented as a cross product of the vector with the gradient operator as

×v=( e r r + e θ 1 r θ + e z z )×( v r e r + v θ e θ + v z e z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgEna0kaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQ haaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjk aiaawMcaaiabgEna0oaabmaabaGaamODamaaBaaaleaacaWGYbaabe aakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWkcaWG2bWaaSba aSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccq GHRaWkcaWG2bWaaSbaaSqaaiaadQhaaeqaaOGaaCyzamaaBaaaleaa [email protected]@

The curl rarely appears in solid mechanics so the components will not be expanded in full

 

 

Divergence of a tensor function.   Let S [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  be a tensor, with dyadic representation

S= S rr e r e r + S rθ e r e θ + S rz e r e z + S θr e θ e r + S θθ e θ e θ + S θz e θ e z + S zr e z e r + S zθ e z e θ + S zz e z e z [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaC4uaiabg2da9iaaykW7caaMc8 UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadkhacaWGYbaabeaakiaa hwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaai aadkhaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaWGYbGaeqiUdeha beaakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXcaWHLbWaaS baaSqaaiabeI7aXbqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaadkha caWG6baabeaakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHxkcXca WHLbWaaSbaaSqaaiaadQhaaeqaaaGcbaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaey4kaSIaaGPaVlaadofadaWgaaWcbaGaeq iUdeNaamOCaaqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGH xkcXcaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaam4uamaaBa aaleaacqaH4oqCcqaH4oqCaeqaaOGaaCyzamaaBaaaleaacqaH4oqC aeqaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaS Iaam4uamaaBaaaleaacqaH4oqCcaWG6baabeaakiaahwgadaWgaaWc baGaeqiUdehabeaakiabgEPielaahwgadaWgaaWcbaGaamOEaaqaba aakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWk caWGtbWaaSbaaSqaaiaadQhacaWGYbaabeaakiaahwgadaWgaaWcba GaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiaadkhaaeqaaOGa ey4kaSIaam4uamaaBaaaleaacaWG6bGaeqiUdehabeaakiaahwgada WgaaWcbaGaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiabeI7a XbqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaadQhacaWG6baabeaaki aahwgadaWgaaWcbaGaamOEaaqabaGccqGHxkcXcaWHLbWaaSbaaSqa [email protected]@

The divergence of S is a vector, which can be represented as

S=( e r r + e θ 1 r θ + e z z )( S rr e r e r + S rθ e r e θ + S rz e r e z + S θr e θ e r + S θθ e θ e θ + S θz e θ e z + S zr e z e r + S zθ e z e θ + S zz e z e z ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahofacqGH9aqpca aMc8UaaGPaVlaaykW7daqadaqaaiaahwgadaWgaaWcbaGaamOCaaqa baGcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaC yzamaaBaaaleaacqaH4oqCaeqaaOWaaSaaaeaacaaIXaaabaGaamOC aaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaacqGHRaWkca WHLbWaaSbaaSqaaiaadQhaaeqaaOWaaSaaaeaacqGHciITaeaacqGH ciITcaWG6baaaaGaayjkaiaawMcaaiabgwSixlaaykW7daqadaabae qabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8Uaam4uamaaBaaaleaacaWGYbGaamOCaaqabaGccaWHLb WaaSbaaSqaaiaadkhaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWG YbaabeaakiabgUcaRiaadofadaWgaaWcbaGaamOCaiabeI7aXbqaba GccaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4LIqSaaCyzamaaBaaa leaacqaH4oqCaeqaaOGaey4kaSIaam4uamaaBaaaleaacaWGYbGaam OEaaqabaGccaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4LIqSaaCyz amaaBaaaleaacaWG6baabeaaaOqaaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlabgUcaRiaaykW7caWGtbWaaSbaaSqaaiabeI7a XjaadkhaaeqaaOGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4LIq SaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaadofadaWgaaWc baGaeqiUdeNaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUdehabe aakiabgEPielaahwgadaWgaaWcbaGaeqiUdehabeaakiabgUcaRiaa dofadaWgaaWcbaGaeqiUdeNaamOEaaqabaGccaWHLbWaaSbaaSqaai abeI7aXbqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiaadQhaaeqaaaGc baGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaSIaam 4uamaaBaaaleaacaWG6bGaamOCaaqabaGccaWHLbWaaSbaaSqaaiaa dQhaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWGYbaabeaakiabgU caRiaadofadaWgaaWcbaGaamOEaiabeI7aXbqabaGccaWHLbWaaSba aSqaaiaadQhaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacqaH4oqCae qaaOGaey4kaSIaam4uamaaBaaaleaacaWG6bGaamOEaaqabaGccaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaey4LIqSaaCyzamaaBaaaleaaca [email protected]@

Evaluating the components of the divergence is an extremely tedious operation, because each of the basis vectors in the dyadic representation of S must be differentiated, in addition to the components themselves.  The final result (expressed as a column vector) is

S[ S rr r + S rr r + 1 r S θr θ + S zR z S θθ r 1 r S θθ θ + S rθ r + S rθ r + S θr r + S zθ z S zz z + S rz r + S rz r + 1 r S θz θ ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahofacqGHHjIUda WadaqaauaabeqadeaaaeaadaWcaaqaaiabgkGi2kaadofadaWgaaWc baGaamOCaiaadkhaaeqaaaGcbaGaeyOaIyRaamOCaaaacqGHRaWkda WcaaqaaiaadofadaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGaamOC aaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGYbaaamaalaaabaGaey OaIyRaam4uamaaBaaaleaacqaH4oqCcaWGYbaabeaaaOqaaiabgkGi 2kabeI7aXbaacqGHRaWkdaWcaaqaaiabgkGi2kaadofadaWgaaWcba GaamOEaiaadkfaaeqaaaGcbaGaeyOaIyRaamOEaaaacqGHsisldaWc aaqaaiaadofadaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiaadk haaaaabaWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi 2kaadofadaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiabgkGi2k abeI7aXbaacqGHRaWkdaWcaaqaaiabgkGi2kaadofadaWgaaWcbaGa amOCaiabeI7aXbqabaaakeaacqGHciITcaWGYbaaaiabgUcaRmaala aabaGaam4uamaaBaaaleaacaWGYbGaeqiUdehabeaaaOqaaiaadkha aaGaey4kaSYaaSaaaeaacaWGtbWaaSbaaSqaaiabeI7aXjaadkhaae qaaaGcbaGaamOCaaaacqGHRaWkdaWcaaqaaiabgkGi2kaadofadaWg aaWcbaGaamOEaiabeI7aXbqabaaakeaacqGHciITcaWG6baaaaqaam aalaaabaGaeyOaIyRaam4uamaaBaaaleaacaWG6bGaamOEaaqabaaa keaacqGHciITcaWG6baaaiabgUcaRmaalaaabaGaeyOaIyRaam4uam aaBaaaleaacaWGYbGaamOEaaqabaaakeaacqGHciITcaWGYbaaaiab gUcaRmaalaaabaGaam4uamaaBaaaleaacaWGYbGaamOEaaqabaaake aacaWGYbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaadkhaaaWaaSaa aeaacqGHciITcaWGtbWaaSbaaSqaaiabeI7aXjaadQhaaeqaaaGcba [email protected]@