MULTIPLE INTEGRALS
which agrees with the result given in chapter 10.
If we place the sphere with its centre at the origin of anx, y, zcoordinate system then
its moment of inertia about thez-axis (which is, of course, a diameter of the sphere) is
I=∫(
x^2 +y^2)
dM=ρ∫ (
x^2 +y^2)
dV ,where the integral is taken over the sphere, andρis the density. Using spherical polar
coordinates, we can write this as
I=ρ∫∫∫
V(
r^2 sin^2 θ)
r^2 sinθdrdθdφ=ρ∫ 2 π0dφ∫π0dθsin^3 θ∫a0dr r^4=ρ× 2 π×^43 ×^15 a^5 = 158 πa^5 ρ.Since the mass of the sphere isM=^43 πa^3 ρ, the moment of inertia can also be written as
I=^25 Ma^2 .
6.4.4 General properties of JacobiansAlthough we will not prove it, the general result for a change of coordinates in
ann-dimensional integral from a setxito a setyj(whereiandjboth run from
1ton)is
dx 1 dx 2 ···dxn=∣
∣
∣
∣∂(x 1 ,x 2 ,...,xn)
∂(y 1 ,y 2 ,...,yn)∣
∣
∣
∣dy^1 dy^2 ···dyn,where then-dimensional Jacobian can be written as ann×ndeterminant (see
chapter 8) in an analogous way to the two- and three-dimensional cases.
For readers who already have sufficient familiarity with matrices (see chapter 8)and their properties, a fairly compact proof of some useful general properties
of Jacobians can be given as follows. Other readers should turn straight to the
results (6.16) and (6.17) and return to the proof at some later time.
Consider three sets of variablesxi,yiandzi, withirunning from 1 tonforeach set. From the chain rule in partial differentiation (see (5.17)), we know that
∂xi
∂zj=∑nk=1∂xi
∂yk∂yk
∂zj. (6.13)
Now letA,BandCbe the matrices whoseijth elements are∂xi/∂yj,∂yi/∂zjand
∂xi/∂zjrespectively. We can then write (6.13) as the matrix product
cij=∑nk=1aikbkj or C=AB. (6.14)We may now use the general result for the determinant of the product of two
matrices, namely|AB|=|A||B|, and recall that the Jacobian
Jxy=∂(x 1 ,...,xn)
∂(y 1 ,...,yn)=|A|, (6.15)