Page 252 - Linear Algebra Done Right

Volume

−r sin θ
cos θ
,
sin θ
r cos θ
as you should verify. The determinant of the matrix above equals r, 243
thus explaining why a factor of r is needed when computing an integral
in polar coordinates.
Finally, when n = 3, we can use the change of variables induced by
spherical coordinates. In this case σ is defined by
σ(ρ, ϕ, θ) = (ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ),
where we have used ρ, θ, ϕ as the coordinates instead of x 1 ,x 2 ,x 3
for reasons that will be obvious to everyone familiar with spherical
coordinates (and will be a mystery to everyone else). For this choice
of σ, the matrix of partial derivatives corresponding to 10.39 is
 
sin ϕ cos θ ρ cos ϕ cos θ −ρ sin ϕ sin θ
 
 sin ϕ sin θ ρ cos ϕ sin θ ρ sin ϕ cos θ  ,
cos ϕ −ρ sin ϕ 0

as you should verify. You should also verify that the determinant of the
2
2
matrix above equals ρ sin ϕ, thus explaining why a factor of ρ sin ϕ
is needed when computing an integral in spherical coordinates.