Tensors for Physics

8.2 Surface Integrals, Stokes 129

over the areaA, bounded by the wire and application of the Stokes law (8.36) yields
the equation governing theFaraday induction,viz.

∮

∂A

E·dr=

d

dt

∫

A

B·ds≡

d

dt

ΦA, (8.42)

whereΦA=

∫

AB·dsis the flux ofBthrough the areaA, cf. Sect.8.2.5.Theleft
hand side of (8.42) quantifies the electric tension or the voltageV =

∮

∂AE·dr
generated by the change of the magnetic flux, underlying theFaraday induc-
tion. Notice, the time change of the magnetic fluxΦcan be brought about by
a time change of the fieldBor by a change of the areaA, e.g. by a change
of the surface normal of the area with respect to the direction of the magnetic field.

8.3 Volume Integrals, Gauss

8.3.1 Volume Integrals inR

The integral of a functionf=f(r)over a regionVinR^3 is denoted by

V =

∫

V

f(r)d^3 r, (8.43)

where the scalard^3 ris the volume element. Forf=1, the volume integral yields the
content or the size of the volume, which is also referred to as “volume” and denoted
byV:

V=

∫

V

d^3 r, (8.44)

When the Cartesian componentsx,y,zare used for the integration, the volume
integral is just the threefold integral

V =

∫

V

f(r)dxdydz. (8.45)

Often it is advantageous to express the vectorrin terms of three parameters
p 1 ,p 2 ,p 3 , viz.:

r=r(p 1 ,p 2 ,p 3 ),

and to use those as general coordinates for the integration. Then the prescription for
the evaluation of the volume integral is

V =

∫

V

f(r(p 1 ,p 2 ,p 3 ))

∣

∣J(p 1 ,p 2 ,p 3 )

∣

∣dp 1 dp 2 dp 3. (8.46)