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 componentsof 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 multipleintegrals 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=∫∫∫Rf(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,