Tensors for Physics

122 8 Integration of Fields

Fig. 8.9 Planar polar
coordinates for a segment
of a planar ring

For the simple casef =1 and an integration over the full disc with the constant
radiusR, the integration overφyields 2π, that overρgives( 1 / 2 )R^2. Then (8.26)
leads toSμ=ezμR^2 π. As expected, in this case the surface integral gives the area

R^2 πof the circular disc.
Inbothexamplesconsideredsofar,theunitvectornormaltothesurfaceisconstant.
As a consequence, it could be put outside the integral, just as a factor. This is no longer
the case when the integration is to be taken over curved surfaces.

(ii) Cylinder Mantle

For a circular cylinder with constant radiusρ=R, its mantle surface is described
byr ={Rcosφ,Rsinφ,z}. The two parameters areφandz. Here the surface
element is

dsμ=Rnμ(φ)dφdz, (8.27)

where vector normal to the cylinder mantle is given byn={cosφ,sinφ, 0 }.The
surface integral over a regionAlocated on the cylinder mantle is

Sμ=R

∫

A

f(r(φ,z))nμ(φ)dφdz. (8.28)

(iii) Surface of a Sphere

The surface of a sphere with constant radiusRis described by

r={Rcosφsinθ,Rsinφsinθ,Rcosθ},

with the polar anglesθandφas parameters. The unit vector normal to the surface is
the radial unit vector̂r=R−^1 r. Here the surface element is

dsμ=R^2 ̂rμ(θ, φ)sinθdθdφ. (8.29)