When to use the Jacobian in spherical coordinates?

Amaelle · Dec 14, 2021

Greetings!

here is the solution which I undertand very well:

Typed math solution: parametrization of the unit sphere octant in spherical coordinates, x = cos u sin v, y = sin u sin v, z = cos v, with the surface normal cross product and its norm sin v, followed by the integral setup.

Typeset calculus derivation: I = integral over K of f(sigma(u,v)) times norm of N(u,v) du dv equals integral over K of f(cos u sin v, sin u sin v, cos v) times norm of N du dv, then integral of cos u sin^2 v du dv evaluated as product of integrals, giving pi/4.

my question is:
if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv?

thank you!

ergospherical · Dec 14, 2021

It's sort of already baked in. The surface element comes from a little parallelogram with sides ##d\boldsymbol{l}_u = \frac{\partial \boldsymbol{\sigma}}{\partial u} du## and ##d\boldsymbol{l}_v = \frac{\partial \boldsymbol{\sigma}}{\partial v} dv## and therefore area
\begin{align*}
dS = |d\boldsymbol{l}_u \times d\boldsymbol{l}_v| = \left| \frac{\partial \boldsymbol{\sigma}}{\partial u} \times \frac{\partial \boldsymbol{\sigma}}{\partial v} \right| du dv
\end{align*}The part ##\frac{\partial \boldsymbol{\sigma}}{\partial u} \times \frac{\partial \boldsymbol{\sigma}}{\partial v}## is what your solutions denoted by the vector ##\mathbf{N}##.

Amaelle · Dec 14, 2021

thank you so much!

When to use the Jacobian in spherical coordinates?

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