Tensors for Physics

8.2 Surface Integrals, Stokes 119

Fig. 8.6 Cylinder
coordinates with the vectors
eφ,ezandeρ

with the two parametersφandz. The tangential vectors are

∂r

∂φ

={−ρsinφ, ρcosφ, 0 }=ρeφ,

∂r

∂z

={ 0 , 0 , 1 }=ez. (8.17)

The unit vectorseφandez=eρare orthogonal, cf. Fig.8.6. Their cross product
eφ×ezis parallel to the outer normalnof the cylinder mantle.

(iii) Surface of a Sphere

The surface of a sphere with constant radiusRparameterized with the ansatz

r={Rcosφsinθ,Rsinφsinθ,Rcosθ}. (8.18)

Here the parameters are the polar anglesθandφ. Now the tangential vectors are

∂r

∂φ

={−Rsinφsinθ,Rcosφsinθ, 0 }=Rsinθeφ, (8.19)

and

∂r

∂θ

={Rcosφcosθ,Rsinφcosθ,−Rsinθ}=Reθ, (8.20)

The cross product of these tangential vectors yields

∂r

∂θ

×

∂r

∂φ

=R^2 sinθeθ×eφ=R^2 sin^2 θ̂r, (8.21)

with the unit vector̂rpointing in radial direction. The mesh on the surface consists
of circles around the polar axis with radiusRsinθ,forθ=const., and grand semi-
circles running from the North to the South pole, forφ=const.. The unit vectors
eθ,eφand̂rare mutually orthogonal, see Fig.8.7.