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06. Laplace

Laplace in Cartesian Coordinates

$L f(x):=⁄{∂}{∂x_i} ⁄{∂}{∂x_i} f$

Laplace in Spherical Coordinates

Let be the vector in old (cartesian) coordinates.

Let be the vector field in the new coordinates, as a function of the old coordinates.

Let be the scalar field to be Laplaced.

Then by the chain rule it follows that:

$⁄{∂f}{∂x_i}=⁄{∂f}{∂x̄_k}⋅⁄{∂x̄_k}{∂x_i}$

By substitution then follows:

$L f=⁄{∂}{∂x_i}(⁄{∂f}{∂x̄_k}⋅⁄{∂x̄_k}{∂x_i})$

$L f=⁄{∂x̄_l}{∂x_i}⋅⁄{∂}{∂x̄_l}(⁄{∂f}{∂x̄_k}⋅⁄{∂x̄_k}{∂x_i})$

By product differentiation it follows that:

$L f=⁄{∂x̄_l}{∂x_i}⋅((⁄{∂}{∂x̄_l}⁄{∂f}{∂x̄_k})⋅⁄{∂x̄_k}{∂x_i}+⁄{∂f}{∂x̄_k}⋅⁄{∂}{∂x̄_l}⁄{∂x̄_k}{∂x_i})$

$L f=⁄{∂x̄_l}{∂x_i}⋅((⁄{∂}{∂x̄_l}⁄{∂f}{∂x̄_k})⋅⁄{∂x̄_k}{∂x_i}+⁄{∂f}{∂x̄_k}⋅0)$

$L f=⁄{∂x̄_l}{∂x_i}⋅(⁄{∂}{∂x̄_l}⁄{∂f}{∂x̄_k})⋅⁄{∂x̄_k}{∂x_i}$

$L f=⁄{∂x̄_l}{∂x_i}⋅⁄{∂x̄_k}{∂x_i}⋅⁄{∂}{∂x̄_l}⁄{∂}{∂x̄_k}f$

Given the spherical transformations:

$x_1:=r⋅(\cos φ)⋅\sin ϑ$

$x_2:=r⋅(\sin φ)⋅\sin ϑ$

$x_3:=r⋅\cos ϑ$

We arrive at the reverse transformations:

$√{x_i⋅x_i}=r$

$\arctan ⁄{x_2}{x_1}=φ$

$\arccos ⁄{x_3}{r}=ϑ$

And their derivations:

$⁄{∂r}{∂x_i}=⁄{x_i}{r}$

$⁄{∂φ}{∂x_i}=⁄{x_1²}{x_1²+x_2²}⋅⁄{∂}{∂x_i}⁄{x_2}{x_1}$

$⁄{∂ϑ}{∂x_i}=-(1-⁄{x_3²}{r²})^{-⁄{1}{2}}⋅⁄{∂}{∂x_i}⁄{x_3}{r}$

Because:

$⁄{∂}{∂a} \arctan a=⁄{1}{1+a²}$

$⁄{∂}{∂a} \arccos a=⁄{-1}{√{1-a²}}$

With the following helpers:

$⁄{∂}{∂x_i}⁄{1}{r}=-r^{-2}⋅⁄{∂r}{∂x_i}$

$⁄{∂}{∂x_i}⁄{1}{r²}=-2⋅r^{-3}⋅⁄{∂r}{∂x_i}$

And labelling these as the new coordinates:

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .