Past exam of the mathematical methods II course of the University of Oxford / 2019 resit / 10 / a / ii

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- Solution ii

Solution

Take spherical polar coordinates with polar angle

θ

measured from positive

z

and azimuthal angle

ϕ

:

x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ .

The derivative matrix is

\frac{\partial ( x , y , z )}{\partial ( r , θ , ϕ )} = sin θ cos ϕ sin θ sin ϕ cos θ r cos θ cos ϕ r cos θ sin ϕ - r sin θ - r sin θ sin ϕ r sin θ cos ϕ 0 .

Expanding its determinant gives

r^{2} sin θ

. Thus

d V = r^{2} sin θ d r d θ d ϕ, det \frac{\partial ( r , θ , ϕ )}{\partial ( x , y , z )} = \frac{1}{r ^{2} sin θ} .

The inverse formula holds where

r > 0

and

0 < θ < π

; it excludes the origin and polar axis. These formulas use the general chain rule and determinant construction in solution.

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