Tensors for Physics

8.2 Surface Integrals, Stokes 121

Fig. 8.8 Surface in real space and in thep–qparameter plane

Often the symbol

∮

is used to indicate that the integral is extended over a closed
surface which is topologically equivalent to the surface of a sphere. Notice that the
area in the parameter space has a well defined rim or border line even when the closed
surface has none in the 3D space. This is obvious for the parameter representation of
the surface of a sphere. There the polar anglesθandφare within the intervals[ 0 ,π]
and[ 0 , 2 π].
Surface integrals are discussed next for the examples of parameter presentations
of surfaces shown in Sect.8.2.2.

8.2.4 Examples for Surface Integrals

(i) Plane

Firstly, consider as areaAover which the surface integral shall be evaluated a
rectangle in thex–y-plane where the variables are within the intervals[x 1 ,x 2 ]and
[y 1 ,y 2 ]. The vector normal to the plane is parallel to thez-direction. The unit vector
in this direction is denoted byez. The surface element is dsμ=ezμdxdy. Thus the
surface integral of a functionf=f(r)withr={x,y, 0 }, located within the plane,
is evaluated according to

Sμ=ezμ

∫

A

f(r(x,y))dxdy=ezμ

∫x 2

x 1

dx

∫y 2

y 1

dyf(r(x,y)). (8.25)

The representation by the planar polar coordinatesρ,φis appropriate for an areaA
whose border lines are parts of two circular arcs and two radial lines, see Fig.8.9.
Here the position vector within the plane is given byr={ρcosφ, ρsinφ, 0 }.
The variables are within the intervals[ρ 1 ,ρ 2 ]and[φ 1 ,φ 2 ]. Now the surface
element is dsμ=ezμρdρdφand the surface integral is to be evaluated according to

Sμ=ezμ

∫

A

f(r(ρ, φ))ρdρdφ=eμz

∫ρ 2

ρ 1

ρdρ

∫φ 2

φ 1

dφf(r(ρ, φ)). (8.26)