Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS


ware constant, and soPQhas components (∂x/∂u)du,(∂y/∂u)duand (∂z/∂u)du


in the directions of thex-,y-andz- axes respectively. The components of the


line elementsPSandSTare found by replacingubyvandwrespectively.


The expression for the volume of a parallelepiped in terms of the components

of its edges with respect to thex-,y-andz-axes is given in chapter 7. Using this,


we find that the element of volume inu, v, wcoordinates is given by


dVuvw=





∂(x, y, z)
∂(u, v, w)




∣du dv dw,

where the Jacobian ofx, y, zwith respect tou, v, wis a short-hand for a 3× 3


determinant:


∂(x, y, z)
∂(u, v, w)














∂x
∂u

∂y
∂u

∂z
∂u
∂x
∂v

∂y
∂v

∂z
∂v
∂x
∂w

∂y
∂w

∂z
∂w













.

So, in summary, the relationship between the elemental volumes in multiple

integrals formulated in the two coordinate systems is given in Jacobian form by


dx dy dz=





∂(x, y, z)
∂(u, v, w)




∣du dv dw,

and we can write a triple integral in either set of coordinates as


I=

∫∫∫

R

f(x, y, z)dx dy dz=

∫∫∫

R′

g(u, v, w)





∂(x, y, z)
∂(u, v, w)




∣du dv dw.

Find an expression for a volume element inspherical polar coordinates, and hence calcu-
late the moment of inertia about a diameter of a uniform sphere of radiusaand massM.

Spherical polar coordinatesr, θ, φare defined by


x=rsinθcosφ, y=rsinθsinφ, z=rcosθ

(and are discussed fully in chapter 10). The required Jacobian is therefore


J=


∂(x, y, z)
∂(r, θ, φ)

=


∣∣



∣∣



sinθcosφ sinθsinφ cosθ
rcosθcosφrcosθsinφ −rsinθ
−rsinθsinφrsinθcosφ 0

∣∣



∣∣



.


The determinant is most easily evaluated by expanding it with respect to the last column
(see chapter 8), which gives


J=cosθ(r^2 sinθcosθ)+rsinθ(rsin^2 θ)
=r^2 sinθ(cos^2 θ+sin^2 θ)=r^2 sinθ.

Therefore the volume element in spherical polar coordinates is given by


dV=

∂(x, y, z)
∂(r, θ, φ)

dr dθ dφ=r^2 sinθdrdθdφ,
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