Tensors for Physics

8.2 Surface Integrals, Stokes 123

The surface integral over a regionAon the surface of the sphere is

Sμ=R^2

∫

A

f(r(θ, φ))̂rμ(θ, φ)sinθdθdφ. (8.30)

In the following, the abbreviation

d^2 ̂r=sinθdθdφ (8.31)

is used for the scalar part of the surface element pertaining to the surface of a sphere
with radiusR=1. This sphere is referred to asunit sphere.
The symbol
∫
d^2 ̂r..., (8.32)

without any indication of a specific area, is used for integrals over the whole unit
sphere.

8.2.5 Flux of a Vector Field.

The surface integralSμ=

∫

Af(r)dsμ, defined in (8.22) with (8.23), is a tensor of
rank+1 whenfstands for the components of a tensor of rank. In particular, for
f=vνwherev=v(r)is a vector field, the corresponding integral over a surface
Ais the second rank tensor

Sμν=

∫

A

vνdsμ. (8.33)

The isotropic part of this tensor, cf. Chap. 6 , involves its traceS=Sμμ. This scalar
quantity, viz.

S=

∫

A

vμdsμ=

∫

A

v·ds (8.34)

is referred to as theflux of the vector fieldvthrough the surface A.
A simple example demonstrates the meaning of the term “flux”. LetAbe a plane
surface with a fixed normal vectornandv=const. a homogeneous vector field.
With dsμ=nμdsone obtains

S=vμnμ

∫

ds=vμnμA,

whereA stands for the area of the surface. Withvμ =vv̂μ, this result can be
written as