Appendix D

 

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.  This appendix reviews the main ideas and procedures associated with polar coordinate systems.  A more sophisticated discussion of general non-orthogonal coordinate systems is given in Chapter 10.

 

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  everything has to be written out in long-hand.  In this appendix, therefore, we completely abandon index notation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  vectors and tensors components are always expressed as matrices.

 

 

D.1: Spherical-polar coordinates

 

D.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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacaGGSaGaeqiUdeNaaiilaiabew 9aMjaaykW7aaa@37A7@  shown in the picture above.  These are not components of a vector.

 

In words:

R is the distance of P from the origin

θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321D@  is the angle between the k direction and OP

ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@322F@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGHLjYScaaIWaaaaa@33BE@ 0θ 180 o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcqaH4oqCcqGHKjYOca aIXaGaaGioaiaaicdadaahaaWcbeqaaiaad+gaaaaaaa@3999@  and 0ϕ 360 o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcqaHvpGzcqGHKjYOca aIZaGaaGOnaiaaicdadaahaaWcbeqaaiaad+gaaaaaaa@39AB@

 

 

 

D.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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWG4bGaaCyAaiabgU caRiaadMhacaWHQbGaey4kaSIaamOEaiaahUgaaaa@3A00@   When we use a spherical-polar coordinate system, we locate points by specifying their spherical-polar coordinates R,θ,ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacaGGSaGaeqiUdeNaaiilaiabew 9aMbaa@361C@

 

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

x=Rsinθcosϕ y=Rsinθsinϕ z=Rcosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4b Gaeyypa0JaamOuaiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGG VbGaai4Caiabew9aMjaaykW7aeaacaWG5bGaeyypa0JaamOuaiGaco hacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMbqa aiaadQhacqGH9aqpcaWGsbGaci4yaiaac+gacaGGZbGaeqiUdehaaa a@5801@                     R= x 2 + y 2 + z 2 θ= cos 1 z/R ϕ= tan 1 y/x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGsb Gaeyypa0ZaaOaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadQhadaahaa WcbeqaaiaaikdaaaaabeaakiaaykW7aeaacqaH4oqCcqGH9aqpciGG JbGaai4BaiaacohadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWG6b Gaai4laiaadkfaaeaacqaHvpGzcqGH9aqpciGG0bGaaiyyaiaac6ga daahaaWcbeqaaiabgkHiTiaaigdaaaGccaWG5bGaai4laiaadIhaaa aa@5637@

 

D.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 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaeqy1dygabeaakiaahwga daWgaaWcbaGaeqy1dygabeaakiaaykW7aaa@4510@ , where ( a R , a θ , a ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGsb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa amyyamaaBaaaleaacqaHvpGzaeqaaOGaaiykaaaa@3D28@  denote the components of a.

 

The basis is different for each point P.  In words

e R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOuaaqabaaaaa@3258@  points along OP

e θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa a@3337@  is tangent to a line of constant longitude through P

e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqy1dygabeaaaa a@3349@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOuaaqabaaaaa@3258@  points vertically upwards; e θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa a@3337@  points due South; and e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqy1dygabeaaaa a@3349@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOuaaqabaaaaa@3258@ , keep θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321D@  and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@322F@  fixed, and increase R. P is moving parallel to e R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOuaaqabaaaaa@3258@ .  To see the direction of e θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa a@3337@ , keep R and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@322F@  fixed, and increase θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321D@ . P now moves parallel to  e θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa a@3337@ .  To see the direction of e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqy1dygabeaaaa a@3349@ , keep R and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321D@  fixed, and increase ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@322F@ .  P now moves parallel to e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabew9aMbqaba aaaa@35A7@ .  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGsbaabeaakiabg2da9maalaaabaGaaGymaaqaamaaemaa baWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOuaaaaaiaawE a7caGLiWoaaaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOu aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGH9a qpdaWcaaqaaiaaigdaaeaadaabdaqaamaalaaabaGaeyOaIyRaaCOC aaqaaiabgkGi2kabeI7aXbaaaiaawEa7caGLiWoaaaWaaSaaaeaacq GHciITcaWHYbaabaGaeyOaIyRaeqiUdehaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahwga daWgaaWcbaGaeqy1dygabeaakiabg2da9maalaaabaGaaGymaaqaam aaemaabaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaeqy1dyga aaGaay5bSlaawIa7aaaadaWcaaqaaiabgkGi2kaahkhaaeaacqGHci ITcqaHvpGzaaaaaa@A4D7@

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 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@  is an orthonormal basis (basis vectors have unit length, are mutually perpendicular and form a right handed triad)

 

 

D.1.4 Converting vectors between Cartesian and Spherical-Polar bases

 

Let a= a R e R + a θ e θ + a ϕ e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaeqy1dygabeaakiaahwga daWgaaWcbaGaeqy1dygabeaakiaaykW7aaa@4510@  be a vector, with components ( a R , a θ , a ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGsb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa amyyamaaBaaaleaacqaHvpGzaeqaaOGaaiykaaaa@3D28@  in the spherical-polar basis { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@ .  Let a x , a y , a z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamiEaaqabaGcca GGSaGaamyyamaaBaaaleaacaWG5baabeaakiaacYcacaWGHbWaaSba aSqaaiaadQhaaeqaaaaa@380B@  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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWG4baabeaaaOqaaiaadgga daWgaaWcbaGaamyEaaqabaaakeaacaWGHbWaaSbaaSqaaiaadQhaae qaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbaeqabmWaaaqa aiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGGVbGaai4Caiabew 9aMbqaaiGacogacaGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4C aiabew9aMbqaaiabgkHiTiGacohacaGGPbGaaiOBaiabew9aMbqaai GacohacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9a MbqaaiGacogacaGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBai abew9aMbqaaiGacogacaGGVbGaai4Caiabew9aMbqaaiGacogacaGG VbGaai4CaiabeI7aXbqaaiabgkHiTiGacohacaGGPbGaaiOBaiabeI 7aXbqaaiaaicdaaaaacaGLBbGaayzxaaWaamWaaeaafaqabeWabaaa baGaamyyamaaBaaaleaacaWGsbaabeaaaOqaaiaadggadaWgaaWcba GaeqiUdehabeaaaOqaaiaadggadaWgaaWcbaGaeqy1dygabeaaaaaa kiaawUfacaGLDbaacaaMc8oaaa@871A@

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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWGsbaabeaaaOqaaiaadgga daWgaaWcbaGaeqiUdehabeaaaOqaaiaadggadaWgaaWcbaGaeqy1dy gabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqadmaa aeaaciGGZbGaaiyAaiaac6gacqaH4oqCciGGJbGaai4Baiaacohacq aHvpGzaeaaciGGZbGaaiyAaiaac6gacqaH4oqCciGGZbGaaiyAaiaa c6gacqaHvpGzaeaaciGGJbGaai4BaiaacohacqaH4oqCaeaaciGGJb Gaai4BaiaacohacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzaeaa ciGGJbGaai4BaiaacohacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvp GzaeaacqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCaeaacqGHsisl ciGGZbGaaiyAaiaac6gacqaHvpGzaeaaciGGJbGaai4Baiaacohacq aHvpGzaeaacaaIWaaaaaGaay5waiaaw2faamaadmaabaqbaeqabmqa aaqaaiaadggadaWgaaWcbaGaamiEaaqabaaakeaacaWGHbWaaSbaaS qaaiaadMhaaeqaaaGcbaGaamyyamaaBaaaleaacaWG6baabeaaaaaa kiaawUfacaGLDbaacaaMc8oaaa@871A@

 

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] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyuaiaac2facaGGBbGaam yuaiaac2fadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcaGGBbGaamys aiaac2faaaa@3C96@ , where [I] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamysaiaac2faaaa@3554@  denotes the 3x3 identity matrix.

 

Derivation: It is easiest to do the transformation by expressing each basis vector { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@  as components in {i,j,k}, and then substituting.  To do this, recall that r=xi+yj+zk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWG4bGaaCyAaiabgU caRiaadMhacaWHQbGaey4kaSIaamOEaiaahUgaaaa@39FF@ , recall also the conversion

x=Rsinθcosϕy=Rsinθsinϕz=Rcosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadkfaciGGZbGaaiyAaiaac6gacqaH4oqCciGGJbGaai4Baiaa cohacqaHvpGzcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamyEaiabg2da9iaadkfaciGGZbGaaiyAaiaac6gacqaH4oqC ciGGZbGaaiyAaiaac6gacqaHvpGzcaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caWG6bGaeyypa0JaamOuaiGacogacaGGVbGaai4CaiabeI7aXbaa@72B8@

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOuaaqabaGccq GH9aqpdaWcaaqaaiaaigdaaeaadaabdaqaamaalaaabaGaeyOaIyRa aCOCaaqaaiabgkGi2kaadkfaaaaacaGLhWUaayjcSdaaamaalaaaba GaeyOaIyRaaCOCaaqaaiabgkGi2kaadkfaaaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHLbWaaSba aSqaaiabeI7aXbqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaadaabda qaamaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kabeI7aXbaaaiaa wEa7caGLiWoaaaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaeq iUdehaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaahwgadaWgaaWcbaGaeqy1dygabeaakiab g2da9maalaaabaGaaGymaaqaamaaemaabaWaaSaaaeaacqGHciITca WHYbaabaGaeyOaIyRaeqy1dygaaaGaay5bSlaawIa7aaaadaWcaaqa aiabgkGi2kaahkhaaeaacqGHciITcqaHvpGzaaaaaa@9CB6@

Hence, substituting for x,y,z and differentiating

r=Rsinθcosϕi+Rsinθsinϕj+Rcosθk r R =sinθcosϕi+sinθsinϕj+cosθk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHYb Gaeyypa0JaamOuaiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGG VbGaai4Caiabew9aMjaahMgacqGHRaWkcaWGsbGaci4CaiaacMgaca GGUbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dyMaaCOAaiabgUca RiaadkfaciGGJbGaai4BaiaacohacqaH4oqCcaWHRbGaaGPaVdqaai abgkDiEpaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kaadkfaaaGa eyypa0Jaci4CaiaacMgacaGGUbGaeqiUdeNaci4yaiaac+gacaGGZb Gaeqy1dyMaaCyAaiabgUcaRiGacohacaGGPbGaaiOBaiabeI7aXjGa cohacaGGPbGaaiOBaiabew9aMjaahQgacqGHRaWkciGGJbGaai4Bai aacohacqaH4oqCcaWHRbaaaaa@7C20@

Conveniently we find that | r R |=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaWaaSaaaeaacqGHciITcaWHYb aabaGaeyOaIyRaamOuaaaaaiaawEa7caGLiWoacqGH9aqpcaaIXaaa aa@39F8@ . Therefore

e R =sinθcosϕi+sinθsinϕj+cosθk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGsbaabeaakiabg2da9iGacohacaGGPbGaaiOBaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMjaahMgacqGHRaWkciGGZbGaai yAaiaac6gacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzcaWHQbGa ey4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaaC4AaiaaykW7aaa@5706@

Similarly

r θ =Rcosθcosϕi+RcosθsinϕjRsinθk r ϕ =Rsinθsinϕi+Rsinθcosϕj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkGi2kaahkhaaeaacqGHciITcqaH4oqCaaGaeyypa0JaamOu aiGacogacaGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4Caiabew 9aMjaahMgacqGHRaWkcaWGsbGaci4yaiaac+gacaGGZbGaeqiUdeNa ci4CaiaacMgacaGGUbGaeqy1dyMaaCOAaiabgkHiTiaadkfaciGGZb GaaiyAaiaac6gacqaH4oqCcaWHRbGaaGPaVdqaamaalaaabaGaeyOa IyRaaCOCaaqaaiabgkGi2kabew9aMbaacqGH9aqpcqGHsislcaWGsb Gaci4CaiaacMgacaGGUbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1 dyMaaCyAaiabgUcaRiaadkfaciGGZbGaaiyAaiaac6gacqaH4oqCci GGJbGaai4BaiaacohacqaHvpGzcaWHQbGaaGPaVdaaaa@7D13@

while | r θ |=R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaWaaSaaaeaacqGHciITcaWHYb aabaGaeyOaIyRaeqiUdehaaaGaay5bSlaawIa7aiabg2da9iaadkfa aaa@3AF3@ , | r ϕ |=Rsinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaamaalaaabaGaeyOaIyRaaC OCaaqaaiabgkGi2kabew9aMbaaaiaawEa7caGLiWoacqGH9aqpcaWG sbGaci4CaiaacMgacaGGUbGaeqiUdehaaa@41F1@  so that

e θ =cosθcosϕi+cosθsinϕjsinθk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaH4oqCaeqaaOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiU deNaci4yaiaac+gacaGGZbGaeqy1dyMaaCyAaiabgUcaRiGacogaca GGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaahQga cqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCcaWHRbGaaGPaVdaa@57EB@      e ϕ =sinϕi+cosϕj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaHvpGzaeqaaOGaeyypa0JaeyOeI0Iaci4CaiaacMgacaGG UbGaeqy1dyMaaCyAaiabgUcaRiGacogacaGGVbGaai4Caiabew9aMj aahQgacaaMc8oaaa@4969@

Finally, substituting

a= a R [sinθcosϕi+sinθsinϕj+cosθk] + a θ [cosθcosϕi+cosθsinϕjsinθk] + a ϕ [sinϕi+cosϕj] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHHb Gaeyypa0JaamyyamaaBaaaleaacaWGsbaabeaakiaacUfaciGGZbGa aiyAaiaac6gacqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzcaWHPb Gaey4kaSIaci4CaiaacMgacaGGUbGaeqiUdeNaci4CaiaacMgacaGG UbGaeqy1dyMaaCOAaiabgUcaRiGacogacaGGVbGaai4CaiabeI7aXj aahUgacaGGDbGaaGPaVdqaaiaaygW7caaMb8UaaGzaVlaaygW7caaM c8UaaGPaVlaaykW7caaMc8Uaey4kaSIaamyyamaaBaaaleaacqaH4o qCaeqaaOGaai4waiGacogacaGGVbGaai4CaiabeI7aXjGacogacaGG VbGaai4Caiabew9aMjaahMgacqGHRaWkciGGJbGaai4Baiaacohacq aH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzcaWHQbGaeyOeI0Iaci4C aiaacMgacaGGUbGaeqiUdeNaaC4Aaiaac2faaeaacaaMb8UaaGzaVl aaygW7caaMb8UaaGzaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWk caWGHbWaaSbaaSqaaiabew9aMbqabaGccaGGBbGaeyOeI0Iaci4Cai aacMgacaGGUbGaeqy1dyMaaCyAaiabgUcaRiGacogacaGGVbGaai4C aiabew9aMjaahQgacaGGDbaaaaa@A74E@

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 MathType@MTEF@5@5@+= 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 a@917D@

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHHb Gaeyypa0JaamyyamaaBaaaleaacaWGsbaabeaakiaahwgadaWgaaWc baGaamOuaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiabeI7aXbqaba GccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGHRaWkcaWGHbWaaSba aSqaaiabew9aMbqabaGccaWHLbWaaSbaaSqaaiabew9aMbqabaGccq GH9aqpcaWGHbWaaSbaaSqaaiaadIhaaeqaaOGaaCyAaiabgUcaRiaa dggadaWgaaWcbaGaamyEaaqabaGccaWHQbGaey4kaSIaamyyamaaBa aaleaacaWG6baabeaakiaahUgacaaMc8oabaGaeyO0H4TaaCyyaiab gwSixlaahwgadaWgaaWcbaGaamOuaaqabaGccqGH9aqpcaWGHbWaaS baaSqaaiaadkfaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaWG4baa beaakiaahMgacqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqaaOGaey 4kaSIaamyyamaaBaaaleaacaWG5baabeaakiaahQgacqGHflY1caWH LbWaaSbaaSqaaiaadkfaaeqaaOGaey4kaSIaamyyamaaBaaaleaaca WG6baabeaakiaahUgacqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqa aaaaaa@7AD9@

(where we have used e θ e R = e ϕ e R =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeI7aXbqaba GccqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqaaOGaeyypa0JaaCyz amaaBaaaleaacqaHvpGzaeqaaOGaeyyXICTaaCyzamaaBaaaleaaca WGsbaabeaakiabg2da9iaaicdaaaa@43DB@  ).  Recall that

e R =sinθcosϕi+sinθsinϕj+cosθk i e R =sinθcosϕj e R =sinθsinϕk e R =cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWHLb WaaSbaaSqaaiaadkfaaeqaaOGaeyypa0Jaci4CaiaacMgacaGGUbGa eqiUdeNaci4yaiaac+gacaGGZbGaeqy1dyMaaCyAaiabgUcaRiGaco hacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaa hQgacqGHRaWkciGGJbGaai4BaiaacohacqaH4oqCcaWHRbGaaGPaVd qaaiabgkDiElaahMgacqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqa aOGaeyypa0Jaci4CaiaacMgacaGGUbGaeqiUdeNaci4yaiaac+gaca GGZbGaeqy1dyMaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWHQbGaeyyXICTaaCyzamaaBaaaleaacaWGsbaabe aakiabg2da9iGacohacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGa aiOBaiabew9aMjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaC4Aai abgwSixlaahwgadaWgaaWcbaGaamOuaaqabaGccqGH9aqpciGGJbGa ai4BaiaacohacqaH4oqCaaaa@9713@

Substituting, we get

a R =sinθcosϕ a x +sinθsinϕ a y +cosθ a z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGsbaabeaakiabg2da9iGacohacaGGPbGaaiOBaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMjaadggadaWgaaWcbaGaamiEaa qabaGccqGHRaWkciGGZbGaaiyAaiaac6gacqaH4oqCciGGZbGaaiyA aiaac6gacqaHvpGzcaWGHbWaaSbaaSqaaiaadMhaaeqaaOGaey4kaS Iaci4yaiaac+gacaGGZbGaeqiUdeNaamyyamaaBaaaleaacaWG6baa beaakiaaykW7aaa@5A73@

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb WaaSbaaSqaaiabeI7aXbqabaGccqGH9aqpciGGJbGaai4Baiaacoha cqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzcaWGHbWaaSbaaSqaai aadIhaaeqaaOGaey4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaci4C aiaacMgacaGGUbGaeqy1dyMaamyyamaaBaaaleaacaWG5baabeaaki abgkHiTiGacohacaGGPbGaaiOBaiabeI7aXjaadggadaWgaaWcbaGa amOEaaqabaGccaaMc8oabaGaamyyamaaBaaaleaacqaHvpGzaeqaaO Gaeyypa0JaeyOeI0Iaci4CaiaacMgacaGGUbGaeqy1dyMaamyyamaa BaaaleaacaWG4baabeaakiabgUcaRiGacogacaGGVbGaai4Caiabew 9aMjaadggadaWgaaWcbaGaamyEaaqabaaaaaa@6E7C@

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

 

 

 

D.1.5 Spherical-Polar representation of tensors

 

The triad of vectors { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@  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 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3967@ .  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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaWGsbGaamOuaaqabaaakeaacaWG tbWaaSbaaSqaaiaadkfacqaH4oqCaeqaaaGcbaGaam4uamaaBaaale aacaWGsbGaeqy1dygabeaaaOqaaiaadofadaWgaaWcbaGaeqiUdeNa amOuaaqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjabeI7aXbqaba aakeaacaWGtbWaaSbaaSqaaiabeI7aXjabew9aMbqabaaakeaacaWG tbWaaSbaaSqaaiabew9aMjaadkfaaeqaaaGcbaGaam4uamaaBaaale aacqaHvpGzcqaH4oqCaeqaaaGcbaGaam4uamaaBaaaleaacqaHvpGz cqaHvpGzaeqaaaaaaOGaay5waiaaw2faaaaa@5AEF@

You can think of S RR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkfacaWGsb aabeaaaaa@3578@  as being equivalent to S 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIXa aabeaaaaa@3540@ , S Rθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkfacqaH4o qCaeqaaaaa@3657@  as S 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa aabeaaaaa@3541@ , 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 MathType@MTEF@5@5@+= 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 aCyzamaaBaaaleaacqaHvpGzaeqaaaaaaa@B7E6@

 

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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdpGaeyyyIO7aamWaaeaafaqabe WadaaabaGaeq4Wdm3aaSbaaSqaaiaadkfacaWGsbaabeaaaOqaaiab eo8aZnaaBaaaleaacaWGsbGaeqiUdehabeaaaOqaaiabeo8aZnaaBa aaleaacaWGsbGaeqy1dygabeaaaOqaaiabeo8aZnaaBaaaleaacqaH 4oqCcaWGsbaabeaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiabeI7aXjabew9aMbqabaaa keaacqaHdpWCdaWgaaWcbaGaeqy1dyMaamOuaaqabaaakeaacqaHdp WCdaWgaaWcbaGaeqy1dyMaeqiUdehabeaaaOqaaiabeo8aZnaaBaaa leaacqaHvpGzcqaHvpGzaeqaaaaaaOGaay5waiaaw2faaaaa@63A5@

The component σ θR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaam Ouaaqabaaaaa@3742@  represents the traction component in direction e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadkfaaeqaaa aa@34B7@  acting on an internal material plane with normal e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeI7aXbqaba aaaa@3596@ , and so on.  Of course, the Cauchy stress tensor is symmetric, with σ θR = σ Rθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaam OuaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamOuaiabeI7aXbqa baaaaa@3CCE@

 

 

 

D.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 ] MathType@MTEF@5@5@+= 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 aiaaicdaaeaacaaIWaaabaGaaGimaaaaaiaawUfacaGLDbaaaaa@B48D@

 

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 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@  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 ] MathType@MTEF@5@5@+= 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 GccaGLBbGaayzxaaaaaaa@3A41@

however, the elastic constants c 11 , c 12 ,... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaakiaacYcacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaa cYcacaGGUaGaaiOlaiaac6caaaa@3B65@  would need to be represent the material properties in the basis { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@ , 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.

 

 

 

D.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 ] MathType@MTEF@5@5@+= 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 aadQhaaeqaaaaaaOGaay5waiaaw2faaaaa@8949@

in the spherical-polar basis { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@  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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaadIhacaWG4baabeaaaOqaaiaadofadaWgaaWcbaGa amiEaiaadMhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG4bGaamOEaa qabaaakeaacaWGtbWaaSbaaSqaaiaadMhacaWG4baabeaaaOqaaiaa dofadaWgaaWcbaGaamyEaiaadMhaaeqaaaGcbaGaam4uamaaBaaale aacaWG5bGaamOEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadQhacaWG 4baabeaaaOqaaiaadofadaWgaaWcbaGaamiEaiaadMhaaeqaaaGcba Gaam4uamaaBaaaleaacaWG6bGaamOEaaqabaaaaaGccaGLBbGaayzx aaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaci4CaiaacMgacaGGUb GaeqiUdeNaci4yaiaac+gacaGGZbGaeqy1dygabaGaci4yaiaac+ga caGGZbGaeqiUdeNaci4yaiaac+gacaGGZbGaeqy1dygabaGaeyOeI0 Iaci4CaiaacMgacaGGUbGaeqy1dygabaGaci4CaiaacMgacaGGUbGa eqiUdeNaci4CaiaacMgacaGGUbGaeqy1dygabaGaci4yaiaac+gaca GGZbGaeqiUdeNaci4CaiaacMgacaGGUbGaeqy1dygabaGaci4yaiaa c+gacaGGZbGaeqy1dygabaGaci4yaiaac+gacaGGZbGaeqiUdehaba GaeyOeI0Iaci4CaiaacMgacaGGUbGaeqiUdehabaGaaGimaaaaaiaa wUfacaGLDbaadaWadaqaauaabeqadmaaaeaacaWGtbWaaSbaaSqaai aadkfacaWGsbaabeaaaOqaaiaadofadaWgaaWcbaGaamOuaiabeI7a XbqabaaakeaacaWGtbWaaSbaaSqaaiaadkfacqaHvpGzaeqaaaGcba Gaam4uamaaBaaaleaacqaH4oqCcaWGsbaabeaaaOqaaiaadofadaWg aaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaeq iUdeNaeqy1dygabeaaaOqaaiaadofadaWgaaWcbaGaeqy1dyMaamOu aaqabaaakeaacaWGtbWaaSbaaSqaaiabew9aMjabeI7aXbqabaaake aacaWGtbWaaSbaaSqaaiabew9aMjabew9aMbqabaaaaaGccaGLBbGa ayzxaaGaaGPaVpaadmaabaqbaeqabmWaaaqaaiGacohacaGGPbGaai OBaiabeI7aXjGacogacaGGVbGaai4Caiabew9aMbqaaiGacohacaGG PbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMbqaaiGaco gacaGGVbGaai4CaiabeI7aXbqaaiGacogacaGGVbGaai4CaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMbqaaiGacogacaGGVbGaai4Cai abeI7aXjGacohacaGGPbGaaiOBaiabew9aMbqaaiabgkHiTiGacoha caGGPbGaaiOBaiabeI7aXbqaaiabgkHiTiGacohacaGGPbGaaiOBai abew9aMbqaaiGacogacaGGVbGaai4Caiabew9aMbqaaiaaicdaaaaa caGLBbGaayzxaaaaaa@EF62@

[ 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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaadkfacaWGsbaabeaaaOqaaiaadofadaWgaaWcbaGa amOuaiabeI7aXbqabaaakeaacaWGtbWaaSbaaSqaaiaadkfacqaHvp GzaeqaaaGcbaGaam4uamaaBaaaleaacqaH4oqCcaWGsbaabeaaaOqa aiaadofadaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiaadofada WgaaWcbaGaeqiUdeNaeqy1dygabeaaaOqaaiaadofadaWgaaWcbaGa eqy1dyMaamOuaaqabaaakeaacaWGtbWaaSbaaSqaaiabew9aMjabeI 7aXbqabaaakeaacaWGtbWaaSbaaSqaaiabew9aMjabew9aMbqabaaa aaGccaGLBbGaayzxaaGaeyypa0JaaGPaVpaadmaabaqbaeqabmWaaa qaaiGacohacaGGPbGaaiOBaiabeI7aXjGacogacaGGVbGaai4Caiab ew9aMbqaaiGacohacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaai OBaiabew9aMbqaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiGacoga caGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4Caiabew9aMbqaai GacogacaGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9a MbqaaiabgkHiTiGacohacaGGPbGaaiOBaiabeI7aXbqaaiabgkHiTi GacohacaGGPbGaaiOBaiabew9aMbqaaiGacogacaGGVbGaai4Caiab ew9aMbqaaiaaicdaaaaacaGLBbGaayzxaaWaamWaaeaafaqabeWada aabaGaam4uamaaBaaaleaacaWG4bGaamiEaaqabaaakeaacaWGtbWa aSbaaSqaaiaadIhacaWG5baabeaaaOqaaiaadofadaWgaaWcbaGaam iEaiaadQhaaeqaaaGcbaGaam4uamaaBaaaleaacaWG5bGaamiEaaqa baaakeaacaWGtbWaaSbaaSqaaiaadMhacaWG5baabeaaaOqaaiaado fadaWgaaWcbaGaamyEaiaadQhaaeqaaaGcbaGaam4uamaaBaaaleaa caWG6bGaamiEaaqabaaakeaacaWGtbWaaSbaaSqaaiaadIhacaWG5b aabeaaaOqaaiaadofadaWgaaWcbaGaamOEaiaadQhaaeqaaaaaaOGa ay5waiaaw2faamaadmaabaqbaeqabmWaaaqaaiGacohacaGGPbGaai OBaiabeI7aXjGacogacaGGVbGaai4Caiabew9aMbqaaiGacogacaGG VbGaai4CaiabeI7aXjGacogacaGGVbGaai4Caiabew9aMbqaaiabgk HiTiGacohacaGGPbGaaiOBaiabew9aMbqaaiGacohacaGGPbGaaiOB aiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMbqaaiGacogacaGGVb Gaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMbqaaiGacoga caGGVbGaai4Caiabew9aMbqaaiGacogacaGGVbGaai4CaiabeI7aXb qaaiabgkHiTiGacohacaGGPbGaaiOBaiabeI7aXbqaaiaaicdaaaaa caGLBbGaayzxaaaaaa@EF62@

 

These results follow immediately from the general basis change formulas for tensors given in Appendix B.

 

 

 

D.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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHHjIUdaqadaqaaiaahw gadaWgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi2cqaaiabgkGi 2kaadkfaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaOWaaS aaaeaacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi2cqaaiabgkGi 2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiabew9aMbqabaGcda WcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdeha amaalaaabaGaeyOaIylabaGaeyOaIyRaeqy1dygaaaGaayjkaiaawM caaaaa@55DE@

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 MathType@MTEF@5@5@+= 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 aaBaaaleaacqaH4oqCaeqaaaaaaa@E168@

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

e R =sinθcosϕi+sinθsinϕj+cosθk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGsbaabeaakiabg2da9iGacohacaGGPbGaaiOBaiabeI7a XjGacogacaGGVbGaai4Caiabew9aMjaahMgacqGHRaWkciGGZbGaai yAaiaac6gacqaH4oqCciGGZbGaaiyAaiaac6gacqaHvpGzcaWHQbGa ey4kaSIaci4yaiaac+gacaGGZbGaeqiUdeNaaC4AaiaaykW7aaa@5706@    e θ =cosθcosϕi+cosθsinϕjsinθk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaH4oqCaeqaaOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiU deNaci4yaiaac+gacaGGZbGaeqy1dyMaaCyAaiabgUcaRiGacogaca GGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjaahQga cqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCcaWHRbGaaGPaVdaa@57EB@       e ϕ =sinϕi+cosϕj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacqaHvpGzaeqaaOGaeyypa0JaeyOeI0Iaci4CaiaacMgacaGG UbGaeqy1dyMaaCyAaiabgUcaRiGacogacaGGVbGaai4Caiabew9aMj aahQgacaaMc8oaaa@4969@

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) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaamOuaiaacYcacqaH4o qCcaGGSaGaeqy1dyMaaiykaaaa@3861@  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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpcaWGMbWaae WaaeaacaWHLbWaaSbaaSqaaiaadkfaaeqaaOWaaSaaaeaacqGHciIT aeaacqGHciITcaWGsbaaaiabgUcaRiaahwgadaWgaaWcbaGaeqiUde habeaakmaalaaabaGaaGymaaqaaiaadkfaaaWaaSaaaeaacqGHciIT aeaacqGHciITcqaH4oqCaaGaey4kaSIaaCyzamaaBaaaleaacqaHvp GzaeqaaOWaaSaaaeaacaaIXaaabaGaamOuaiGacohacaGGPbGaaiOB aiabeI7aXbaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabew9aMbaaai aawIcacaGLPaaacqGH9aqpcaWHLbWaaSbaaSqaaiaadkfaaeqaaOWa aSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRaamOuaaaacqGHRaWkca WHLbWaaSbaaSqaaiabeI7aXbqabaGcdaWcaaqaaiaaigdaaeaacaWG sbaaamaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kabeI7aXbaacq GHRaWkcaWHLbWaaSbaaSqaaiabew9aMbqabaGcdaWcaaqaaiaaigda aeaacaWGsbGaci4CaiaacMgacaGGUbGaeqiUdehaamaalaaabaGaey OaIyRaamOzaaqaaiabgkGi2kabew9aMbaaaaa@769A@

Alternatively, in matrix form

f= [ f R , 1 R f θ , 1 Rsinθ f ϕ ] T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpdaWadaqaam aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadkfaaaGaaiilamaa laaabaGaaGymaaqaaiaadkfaaaWaaSaaaeaacqGHciITcaWGMbaaba GaeyOaIyRaeqiUdehaaiaacYcadaWcaaqaaiaaigdaaeaacaWGsbGa ci4CaiaacMgacaGGUbGaeqiUdehaamaalaaabaGaeyOaIyRaamOzaa qaaiabgkGi2kabew9aMbaaaiaawUfacaGLDbaadaahaaWcbeqaaiaa dsfaaaaaaa@4FB3@

 

Gradient of a vector function Let v= v R e R + v θ e θ + v ϕ e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaeqy1dygabeaakiaahwga daWgaaWcbaGaeqy1dygabeaaaaa@43D0@  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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGHxkcXcqGHhis0cqGH9aqpda qadaqaaiaadAhadaWgaaWcbaGaamOuaaqabaGccaWHLbWaaSbaaSqa aiaadkfaaeqaaOGaey4kaSIaamODamaaBaaaleaacqaH4oqCaeqaaO GaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaamODamaaBaaa leaacqaHvpGzaeqaaOGaaCyzamaaBaaaleaacqaHvpGzaeqaaaGcca GLOaGaayzkaaGaey4LIq8aaeWaaeaacaWHLbWaaSbaaSqaaiaadkfa aeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWGsbaaaiabgUcaRi aahwgadaWgaaWcbaGaeqiUdehabeaakmaalaaabaGaaGymaaqaaiaa dkfaaaWaaSaaaeaacqGHciITaeaacqGHciITcqaH4oqCaaGaey4kaS IaaCyzamaaBaaaleaacqaHvpGzaeqaaOWaaSaaaeaacaaIXaaabaGa amOuaiGacohacaGGPbGaaiOBaiabeI7aXbaadaWcaaqaaiabgkGi2c qaaiabgkGi2kabew9aMbaaaiaawIcacaGLPaaaaaa@6AC4@

The dyadic product can be expanded MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but when evaluating the derivatives it is important to recall that the basis vectors are functions of the coordinates (R,θ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGsbGaaiilaiabeI7aXjaacY cacqaHvpGzcaGGPaaaaa@3776@  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 MathType@MTEF@5@5@+= 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 aiabeI7aXbqabaaaaa@7C0F@

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 ] MathType@MTEF@5@5@+= 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 a@B26C@

 

Divergence of a vector function Let v= v R e R + v θ e θ + v ϕ e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaeqy1dygabeaakiaahwga daWgaaWcbaGaeqy1dygabeaaaaa@43D0@  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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkfaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiabew 9aMbqabaGcdaWcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGG UbGaeqiUdehaamaalaaabaGaeyOaIylabaGaeyOaIyRaeqy1dygaaa GaayjkaiaawMcaaiabgwSixpaabmaabaGaamODamaaBaaaleaacaWG sbaabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHRaWkcaWG2b WaaSbaaSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqa baGccqGHRaWkcaWG2bWaaSbaaSqaaiabew9aMbqabaGccaWHLbWaaS baaSqaaiabew9aMbqabaaakiaawIcacaGLPaaaaaa@6B46@

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) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaabshacaqGYbGaaeyyaiaabogacaqGLb GaaiikaiaahAhacqGHxkcXcqGHhis0caGGPaaaaa@3AED@ , 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGHHjIUda WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamOuaaqabaaakeaacqGH ciITcaWGsbaaaiabgUcaRiaaikdadaWcaaqaaiaadAhadaWgaaWcba GaamOuaaqabaaakeaacaWGsbaaaiabgUcaRmaalaaabaGaaGymaaqa aiaadkfaaaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiabeI7aXb qabaaakeaacqGHciITcqaH4oqCaaGaey4kaSYaaSaaaeaacaaIXaaa baGaamOuaiGacohacaGGPbGaaiOBaiabeI7aXbaadaWcaaqaaiabgk Gi2kaadAhadaWgaaWcbaGaeqy1dygabeaaaOqaaiabgkGi2kabew9a MbaacqGHRaWkciGGJbGaai4BaiaacshacqaH4oqCdaWcaaqaaiaadA hadaWgaaWcbaGaeqiUdehabeaaaOqaaiaadkfaaaaaaa@632E@

 

Curl of a vector function Let v= v R e R + v θ e θ + v ϕ e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaeqy1dygabeaakiaahwga daWgaaWcbaGaeqy1dygabeaaaaa@43D0@  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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgEna0kaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkfaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiabew 9aMbqabaGcdaWcaaqaaiaaigdaaeaacaWGsbGaci4CaiaacMgacaGG UbGaeqiUdehaamaalaaabaGaeyOaIylabaGaeyOaIyRaeqy1dygaaa GaayjkaiaawMcaaiabgEna0oaabmaabaGaamODamaaBaaaleaacaWG sbaabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHRaWkcaWG2b WaaSbaaSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqa baGccqGHRaWkcaWG2bWaaSbaaSqaaiabew9aMbqabaGccaWHLbWaaS baaSqaaiabew9aMbqabaaakiaawIcacaGLPaaaaaa@6AE0@

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

 

 

Divergence of a tensor function.   Let S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahofaaaa@3144@  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 MathType@MTEF@5@5@+= 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 aCyzamaaBaaaleaacqaHvpGzaeqaaaaaaa@B7E6@

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 ) MathType@MTEF@5@5@+= 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 vpGzaeqaaaaaaOGaayjkaiaawMcaaiaaykW7aaa@CE6A@

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 ) ] MathType@MTEF@5@5@+= 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 zkaaaaaaGaay5waiaaw2faaaaa@006F@

 

 

 

 

 

D.2: Cylindrical-polar coordinates

 

 

D.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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacaGGSaGaeqiUdeNaaiilaiaadQ haaaa@3573@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcqaH4oqCcqGHKjYOca aIZaGaaGOnaiaaicdadaahaaWcbeqaaiaad+gaaaaaaa@3999@

 

D.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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWG4bGaaCyAaiabgU caRiaadMhacaWHQbGaey4kaSIaamOEaiaahUgaaaa@3A00@   When we use a spherical-polar coordinate system, we locate points by specifying their spherical-polar coordinates r,θ,z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacaGGSaGaeqiUdeNaaiilaiaadQ haaaa@3573@

 

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

x=rcosθ y=rsinθ z=z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4b Gaeyypa0JaamOCaiGacogacaGGVbGaai4CaiabeI7aXjaaykW7aeaa caWG5bGaeyypa0JaamOCaiGacohacaGGPbGaaiOBaiabeI7aXbqaai aadQhacqGH9aqpcaWG6baaaaa@4989@                   r= x 2 + y 2 θ= tan 1 y/x z=z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGYb Gaeyypa0ZaaOaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamyEamaaCaaaleqabaGaaGOmaaaaaeqaaOGaaGPaVdqaaiabeI 7aXjabg2da9iGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0Ia aGymaaaakiaadMhacaGGVaGaamiEaaqaaiaadQhacqGH9aqpcaWG6b aaaaa@4B67@

 

 

D.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 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaaiyFaaaa@3ADF@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaamOEaaqabaGccaWHLbWa aSbaaSqaaiaadQhaaeqaaOGaaGPaVdaa@43BE@ , where ( a r , a θ , a z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGYb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa amyyamaaBaaaleaacaWG6baabeaakiaacMcaaaa@3C7F@  denote the components of a.

 

The basis vectors are selected as follows

e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOCaaqabaaaaa@3278@  is a unit vector normal to the cylinder at P

e θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaeqiUdehabeaaaa a@3337@  is a unit vector circumferential to the cylinder at P, chosen to make { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaaiyFaaaa@3ADF@  a right handed triad

e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaamOEaaqabaaaaa@3280@  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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCaiabg2 da9iaadkhacaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaamOE aiaahwgadaWgaaWcbaGaamOEaaqabaGccqGH9aqpcaWGYbGaci4yai aac+gacaGGZbGaeqiUdeNaaCyAaiabgUcaRiaadkhaciGGZbGaaiyA aiaac6gacqaH4oqCcaWHQbGaey4kaSIaamOEaiaahUgacaaMc8oaaa@5346@

 

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyzamaaBa aaleaacaWGYbaabeaakiabg2da9maalaaabaGaaGymaaqaamaaemaa baWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOCaaaaaiaawE a7caGLiWoaaaWaaSaaaeaacqGHciITcaWHYbaabaGaeyOaIyRaamOC aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWHLbWaaSbaaSqaaiabew9aMbqabaGccqGH9a qpdaWcaaqaaiaaigdaaeaadaabdaqaamaalaaabaGaeyOaIyRaaCOC aaqaaiabgkGi2kabeI7aXbaaaiaawEa7caGLiWoaaaWaaSaaaeaacq GHciITcaWHYbaabaGaeyOaIyRaeqiUdehaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahwga daWgaaWcbaGaamOEaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaada abdaqaamaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2kaadQhaaaaa caGLhWUaayjcSdaaamaalaaabaGaeyOaIyRaaCOCaaqaaiabgkGi2k aadQhaaaGaaGPaVdaa@A479@

The basis vectors have unit length, are mutually perpendicular, and form a right handed triad and therefore { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaaiyFaaaa@3ADF@  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).

 

 

D.2.4 Converting vectors between Cylindrical and Cartesian bases

 

Let a= a r e r + a θ e θ + a z e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGHbWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dggadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadggadaWgaaWcbaGaamOEaaqabaGccaWHLbWa aSbaaSqaaiaadQhaaeqaaOGaaGPaVdaa@43BE@  be a vector, with components ( a r , a θ , a z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaWGYb aabeaakiaacYcacaWGHbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa amyyamaaBaaaleaacaWG6baabeaakiaacMcaaaa@3C7F@  in the spherical-polar basis { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkfaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiabew9aMbqabaGccaGG9baaaa@3B88@ .  Let a x , a y , a z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamiEaaqabaGcca GGSaGaamyyamaaBaaaleaacaWG5baabeaakiaacYcacaWGHbWaaSba aSqaaiaadQhaaeqaaaaa@380B@  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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWG4baabeaaaOqaaiaadgga daWgaaWcbaGaamyEaaqabaaakeaacaWGHbWaaSbaaSqaaiaadQhaae qaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbaeqabmWaaaqa aiGacogacaGGVbGaai4CaiabeI7aXbqaaiabgkHiTiGacohacaGGPb GaaiOBaiabeI7aXbqaaiaaicdaaeaaciGGZbGaaiyAaiaac6gacqaH 4oqCaeaaciGGJbGaai4BaiaacohacqaH4oqCaeaacaaIWaaabaGaaG imaaqaaiaaicdaaeaacaaIXaaaaaGaay5waiaaw2faamaadmaabaqb aeqabmqaaaqaaiaadggadaWgaaWcbaGaamOCaaqabaaakeaacaWGHb WaaSbaaSqaaiabeI7aXbqabaaakeaacaWGHbWaaSbaaSqaaiaadQha aeqaaaaaaOGaay5waiaaw2faaiaaykW7aaa@63A5@      [ a r a θ a z ]=[ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ][ a x a y a z ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaamyyamaaBaaaleaacaWGYbaabeaaaOqaaiaadgga daWgaaWcbaGaeqiUdehabeaaaOqaaiaadggadaWgaaWcbaGaamOEaa qabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqabeWadaaa baGaci4yaiaac+gacaGGZbGaeqiUdehabaGaci4CaiaacMgacaGGUb GaeqiUdehabaGaaGimaaqaaiabgkHiTiGacohacaGGPbGaaiOBaiab eI7aXbqaaiGacogacaGGVbGaai4CaiabeI7aXbqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiaaigdaaaaacaGLBbGaayzxaaWaamWaaeaa faqabeWabaaabaGaamyyamaaBaaaleaacaWG4baabeaaaOqaaiaadg gadaWgaaWcbaGaamyEaaqabaaakeaacaWGHbWaaSbaaSqaaiaadQha aeqaaaaaaOGaay5waiaaw2faaiaaykW7aaa@63A5@

 

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] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyuaiaac2facaGGBbGaam yuaiaac2fadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcaGGBbGaamys aiaac2faaaa@3C96@ , where [I] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamysaiaac2faaaa@3554@  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.
 

 

D.2.5 Cylindrical-Polar representation of tensors

 

The triad of vectors { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaaiyFaaaa@3ADF@  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 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3967@ .  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 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaWGYbGaamOCaaqabaaakeaacaWG tbWaaSbaaSqaaiaadkhacqaH4oqCaeqaaaGcbaGaam4uamaaBaaale aacaWGYbGaamOEaaqabaaakeaacaWGtbWaaSbaaSqaaiabeI7aXjaa dkhaaeqaaaGcbaGaam4uamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaa GcbaGaam4uamaaBaaaleaacqaH4oqCcaWG6baabeaaaOqaaiaadofa daWgaaWcbaGaamOEaiaadkhaaeqaaaGcbaGaam4uamaaBaaaleaaca WG6bGaeqiUdehabeaaaOqaaiaadofadaWgaaWcbaGaamOEaiaadQha aeqaaaaaaOGaay5waiaaw2faaaaa@56F9@

You can think of S rr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkhacaWGYb aabeaaaaa@35B8@  as being equivalent to S 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIXa aabeaaaaa@3540@ , S rθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadkhacqaH4o qCaeqaaaaa@3677@  as S 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa aabeaaaaa@3541@ , 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 MathType@MTEF@5@5@+= 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 aiaadQhaaeqaaaaaaa@AFFA@

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

 

 

D.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 ] MathType@MTEF@5@5@+= 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 aiaaicdaaeaacaaIWaaaaaGaay5waiaaw2faaaaa@AF45@

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

 

 

D.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 ] MathType@MTEF@5@5@+= 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 aadQhaaeqaaaaaaOGaay5waiaaw2faaaaa@871C@

in the cylindrical-polar basis { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaaiilaiaahwgadaWgaaWcbaGaeqiUdehabeaakiaacYcacaWH LbWaaSbaaSqaaiaadQhaaeqaaOGaaiyFaaaa@3ADF@  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 ] MathType@MTEF@5@5@+= 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 baGaaGymaaaaaiaawUfacaGLDbaaaaa@A5D4@

[ 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 ] MathType@MTEF@5@5@+= 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 baGaaGymaaaaaiaawUfacaGLDbaaaaa@A5D4@

 

 

 

 

 

D.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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0cqGHHjIUdaqadaqaaiaahw gadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi2cqaaiabgkGi 2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaOWaaS aaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi2cqaaiabgkGi 2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQhaaeqaaOWaaS aaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjkaiaawMcaaaaa @4E7C@

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) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaaCyzamaaBaaale aacaWGYbaabeaaaOqaaiabgkGi2kabeI7aXbaacqGH9aqpcaWHLbWa aSbaaSqaaiabeI7aXbqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVpaalaaabaGaeyOaIyRaaCyzamaaBaaa leaacqaH4oqCaeqaaaGcbaGaeyOaIyRaeqiUdehaaiabg2da9iabgk HiTiaahwgadaWgaaWcbaGaamOCaaqabaGccaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caGGOaGaaeyyai aabYgacaqGSbGaaeiiaiaab+gacaqG0bGaaeiAaiaabwgacaqGYbGa aeiiaiaabsgacaqGLbGaaeOCaiaabMgacaqG2bGaaeyyaiaabshaca qGPbGaaeODaiaabwgacaqGZbGaaeiiaiaabggacaqGYbGaaeyzaiaa bccacaqG6bGaaeyzaiaabkhacaqGVbGaaeykaaaa@7CD5@

 

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) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaamOCaiaacYcacqaH4o qCcaGGSaGaamOEaiaacMcaaaa@37B8@  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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpdaqadaqaai aahwgadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi2cqaaiab gkGi2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4oqCaeqaaO WaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi2cqaaiab gkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQhaaeqaaO WaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjkaiaawMca aiaadAgacqGH9aqpcaWHLbWaaSbaaSqaaiaadkhaaeqaaOWaaSaaae aacqGHciITcaWGMbaabaGaeyOaIyRaamOCaaaacqGHRaWkcaWHLbWa aSbaaSqaaiabeI7aXbqabaGcdaWcaaqaaiaaigdaaeaacaWGYbaaam aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kabeI7aXbaacqGHRaWk caWHLbWaaSbaaSqaaiaadQhaaeqaaOWaaSaaaeaacqGHciITcaWGMb aabaGaeyOaIyRaamOEaaaaaaa@67D6@

Alternatively, in matrix form

f= [ f r , 1 r f θ , f z ] T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlaadAgacqGH9aqpdaWadaqaam aalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadkhaaaGaaiilamaa laaabaGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHciITcaWGMbaaba GaeyOaIyRaeqiUdehaaiaacYcadaWcaaqaaiabgkGi2kaadAgaaeaa cqGHciITcaWG6baaaaGaay5waiaaw2faamaaCaaaleqabaGaamivaa aaaaa@48FA@

 

Gradient of a vector function Let v= v r e r + v θ e θ + v z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaamOEaaqabaGccaWHLbWa aSbaaSqaaiaadQhaaeqaaaaa@427E@  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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGHxkcXcqGHhis0cqGH9aqpda qadaqaaiaadAhadaWgaaWcbaGaamOCaaqabaGccaWHLbWaaSbaaSqa aiaadkhaaeqaaOGaey4kaSIaamODamaaBaaaleaacqaH4oqCaeqaaO GaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaamODamaaBaaa leaacaWG6baabeaakiaahwgadaWgaaWcbaGaamOEaaqabaaakiaawI cacaGLPaaacqGHxkcXdaqadaqaaiaahwgadaWgaaWcbaGaamOCaaqa baGcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaC yzamaaBaaaleaacqaH4oqCaeqaaOWaaSaaaeaacaaIXaaabaGaamOC aaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabeI7aXbaacqGHRaWkca WHLbWaaSbaaSqaaiaadQhaaeqaaOWaaSaaaeaacqGHciITaeaacqGH ciITcaWG6baaaaGaayjkaiaawMcaaaaa@6210@

The dyadic product can be expanded MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but when evaluating the derivatives it is important to recall that the basis vectors are functions of the coordinate θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321E@  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 MathType@MTEF@5@5@+= 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 aiabeI7aXbqabaaaaa@7DCF@

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 ] MathType@MTEF@5@5@+= 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 aa@8176@

 

Divergence of a vector function Let v= v r e r + v θ e θ + v z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkhaaeqaaOGaaCyzamaaBaaaleaacaWGYbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaamOEaaqabaGccaWHLbWa aSbaaSqaaiaadQhaaeqaaaaa@427E@  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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQ haaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjk aiaawMcaaiabgwSixpaabmaabaGaamODamaaBaaaleaacaWGYbaabe aakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWkcaWG2bWaaSba aSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccq GHRaWkcaWG2bWaaSbaaSqaaiaadQhaaeqaaOGaaCyzamaaBaaaleaa caWG6baabeaaaOGaayjkaiaawMcaaaaa@6292@

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) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaabshacaqGYbGaaeyyaiaabogacaqGLb GaaiikaiaahAhacqGHxkcXcqGHhis0caGGPaaaaa@3AED@ , which immediately gives

v v r r + v r r + 1 r v θ θ + v z z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgwSixlaahAhacqGHHjIUda WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamOCaaqabaaakeaacqGH ciITcaWGYbaaaiabgUcaRmaalaaabaGaamODamaaBaaaleaacaWGYb aabeaaaOqaaiaadkhaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOC aaaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaeqiUdehabeaaaO qaaiabgkGi2kabeI7aXbaacqGHRaWkdaWcaaqaaiabgkGi2kaadAha daWgaaWcbaGaamOEaaqabaaakeaacqGHciITcaWG6baaaaaa@5216@

 

Curl of a vector function Let v= v R e R + v θ e θ + v z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWG2bWaaSbaaSqaai aadkfaaeqaaOGaaCyzamaaBaaaleaacaWGsbaabeaakiabgUcaRiaa dAhadaWgaaWcbaGaeqiUdehabeaakiaahwgadaWgaaWcbaGaeqiUde habeaakiabgUcaRiaadAhadaWgaaWcbaGaamOEaaqabaGccaWHLbWa aSbaaSqaaiaadQhaaeqaaaaa@423E@  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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirlabgEna0kaahAhacqGH9aqpda qadaqaaiaahwgadaWgaaWcbaGaamOCaaqabaGcdaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadkhaaaGaey4kaSIaaCyzamaaBaaaleaacqaH4o qCaeqaaOWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQ haaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG6baaaaGaayjk aiaawMcaaiabgEna0oaabmaabaGaamODamaaBaaaleaacaWGYbaabe aakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWkcaWG2bWaaSba aSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI7aXbqabaGccq GHRaWkcaWG2bWaaSbaaSqaaiaadQhaaeqaaOGaaCyzamaaBaaaleaa caWG6baabeaaaOGaayjkaiaawMcaaaaa@622C@

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

 

 

Divergence of a tensor function.   Let S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahofaaaa@3144@  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 MathType@MTEF@5@5@+= 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 aiaadQhaaeqaaaaaaa@AFFA@

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 ) MathType@MTEF@5@5@+= 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 WG6baabeaaaaGccaGLOaGaayzkaaGaaGPaVdaa@DF7C@

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 ] MathType@MTEF@5@5@+= 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 GaeyOaIyRaeqiUdehaaaaaaiaawUfacaGLDbaaaaa@A31D@