Tensors for Physics

130 8 Integration of Fields

HereJis theJacobi determinant, also calledfunctional determinant, which can be
computed according to

J=εμνλ

∂rμ

∂p 1

∂rν

∂p 2

∂rλ

∂p 3

. (8.47)

Notice thatJdp 1 dp 2 dp 3 can also be written as

Jdp 1 dp 2 dp 3 =dsμ

∂rμ

∂p 1

dp 1 , (8.48)

where dsμis the surface element of a surface parameterized byp 2 ,p 3 , withp 1 =
const., cf. Sect.8.2.3.
Two examples for general coordinates are discussed next.

(i) Cylinder Coordinates

Consider a circular cylinder whose axis coincides with thez-axis. The presentation
r={ρcosφ, ρsinφ,z}is used, whereρ,φ,zare the parameters, see also (8.16).
Now the volume integral is

V =

∫

V

f(r(ρ, φ,z)) ρdρdφdz. (8.49)

When the region in space to be integrated over is a cylinder with radiusRand length
L, and furthermore the integrand is independent of the angleφ, the relation (8.49)
reduces to

V = 2 π

∫R

0

ρdρ

∫L

0

dzf(r(ρ,z)). (8.50)

The factor 2πstems from the integration overφ.Forf=1, one obtains the volume
V=πR^2 Lof the circular cylinder.

(ii) Spherical Coordinates

The standard parametrization with the spherical coordinatesr,θ,φcorresponds to
r={rcosφsinθ,rsinφsinθ,rcosθ}. The volume integral is given by

V =

∫

V

f(r(r,θ,φ))r^2 drsinθdθdφ=

∫

V

f(r(r,θ,φ))r^2 drd^2 ̂r. (8.51)

Here, as in (8.31), the symbol d^2 ̂r=sinθdθdφstands for the scalar surface element
of the unit sphere. Sometimes it is advantageous to useζ =cosθas integration
variable instead ofθ. Then one has

d^2 ̂r=−dζdφ. (8.52)