Tensors for Physics

124 8 Integration of Fields

Fig. 8.10Flux through an

areaAand side view of an

effective areaAeff

S=vAeff.

Herevis the magnitude of the homogeneous vector field. The effective area

Aeff=v̂μnμA

is the actual area of the surface, reduced by the cosine of the angle betweenvand

the normal to the plane, see Fig.8.10.

8.2 Exercise: Surface Integrals over a Hemisphere

Consider a hemisphere with radiusR, with the center at the origin. The unit vector

pointing from its center to the North pole is denoted byu.

Compute the surface integralsSμν=

∫

vνdsμ, over the hemisphere and the pertain-

ing fluxS=Sμμfor

(i) the homogeneous vector fieldvν=vv̂ν=const. and
(ii) the radial fieldvν=rν.

Hint: use the symmetry argumentsSμν∼uμv̂ν(case i) andSμν∼uμuν(case ii),

to simplify the calculations. Put the base of the hemisphere onto thex–y-plane, for

the explicit integration.

8.2.6 Generalized Stokes Law

TheStokes lawprovides a relation between surface integrals of a certain type with a

line integral along the closed rim of the surface. Thus the “dimension” of the integral

is reduced from 2 to 1. This applies when the integrand of the surface integral is a

spatial derivative of a functionf=f(r), which then occurs as integrand in the line

integral. To be more specific, thegeneralized Stokes lawreads:

∫

A

ελνμ∇νf(r)dsλ=

∮

C

f(r)drμ. (8.35)