Mathematical Methods for Physics and Engineering - Matematica.NET

MULTIPLE INTEGRALSwhich agrees with the result given in chapter 10.If we place the sphere with its centre at the origin of an x, y, z coordinate system thenits moment of inertia about the z-axis (which is, of course, a diameter of the sphere) isI =∫ (x 2 + y 2) dM = ρ∫ (x 2 + y 2) dV ,where the integral is taken over the sphere, and ρ is the density. Using spherical polarcoordinates, we can write this as∫∫∫(I = ρ r 2 sin 2 θ ) r 2 sin θdrdθdφ= ρV∫ 2π0dφ∫ π0dθ sin 3 θ= ρ × 2π × 4 × 1 3 5 a5 = 8 15 πa5 ρ.Since the mass of the sphere is M = 4 3 πa3 ρ, the moment of inertia can also be written asI = 2 5 Ma2 . ◭∫ a0dr r 46.4.4 General properties of JacobiansAlthough we will not prove it, the general result for a change of coordinates inan n-dimensional integral from a set x i to a set y j (where i and j both run from1ton) isdx 1 dx 2 ···dx n =∂(x 1 ,x 2 ,...,x n )∣ ∂(y 1 ,y 2 ,...,y n ) ∣ dy 1 dy 2 ···dy n ,where the n-dimensional Jacobian can be written as an n × n determinant (seechapter 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 propertiesof Jacobians can be given as follows. Other readers should turn straight to theresults (6.16) and (6.17) and return to the proof at some later time.Consider three sets of variables x i , y i and z i , with i running from 1 to n foreach set. From the chain rule in partial differentiation (see (5.17)), we know that∂x in∑ ∂x i ∂y k=. (6.13)∂z j ∂y k ∂z jk=1Now let A, B and C be the matrices whose ijth elements are ∂x i /∂y j , ∂y i /∂z j and∂x i /∂z j respectively. We can then write (6.13) as the matrix productn∑c ij = a ik b kj or C = AB. (6.14)k=1We may now use the general result for the determinant of the product of twomatrices, namely |AB| = |A||B|, and recall that the JacobianJ xy = ∂(x 1,...,x n )= |A|, (6.15)∂(y 1 ,...,y n )206