∫2π0 dφ ∫π0 F (θ,φ) r2 sin θ dθ.
In every case the domain of integration must be chosen so as to include the whole surface.
(ix.) In three-dimensional polar coordinates the Jacobian
| ∂(x, y, z) | = r2 sin θ. |
| ∂(r, θ, φ) |
The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is
∫a0 dr ∫2π0 dφ ∫π0 F (r, θ, φ) r2 sin θ dθ.
(x.) Integrations of rational functions through the volume of an ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 are often effected by means of a general theorem due to Lejeune Dirichlet (1839), which is as follows: when the domain of integration is that given by the inequality
| ( | x1 | )a1 + ( | x2 | )a2 + ... + ( | xn | )an ≤ 1, |
| a1 | a2 | an |
where the a’s and α’s are positive, the value of the integral
∫∫ ... x1 n1−1 · x2 n2−1 ... dx1 dx2 ...