Tensors for Physics

138 8 Integration of Fields

∫

̂rμ̂rνd^2 ̂r=

4 π

3

δμν

is obtained. The expressions just computed for the volume and surface integrals are
in accord with the Gauss theorem.
On the other hand, the Gauss theorem can be applied for the determination of
a more complicated surface integral when the evaluation of the pertaining volume
integral is easier, or vice versa. An example: the relation

∫

V

∇μrνd^3 r=δμν

∫

V

d^3 r=δμνV

holds true for a well bounded volumeVwith any shape, not just for a sphere, as
considered above. The Gauss theorem (8.71) now implies the remarkable result

∮

∂V

rνnμd^2 s=δμνV, (8.74)

irrespective of the shape of the surface∂V, as long as the outer normalnis well
defined everywhere on the surface of the volume. The trace part of this relation, viz.:

∮

∂V

r·nd^2 s= 3 V, (8.75)

shows that the volumeVcan also be computed with the help of a surface integral.

8.3.5 Application: Gauss Theorem in Electrodynamics,

Coulomb Force

In electrodynamics, the symbolρis used for the charge density. Then the integral
over the volumeV
∫

V

ρd^3 r=QV, (8.76)

is equal to the electric charge contained in this volume. One of the Maxwell equations
links the divergence of the electric displacement fieldDwith the charge density, viz.:

∇μDμ=ρ. (8.77)