Laplacian in Spherical Coordinates

July 9 2016

Verify Laplace's Equation in Spherical Coordinates

We verify the following equation is correct by expanding `u_(r r)`, `u_r`, `u_(theta theta)`, `u_theta`, and `u_(phi phi)` in the right side of the equation.

`u_(x x) + u_(y y) + u_(z z) = u_(r r) + (2 / r) u_r + (1 / r^2) u_(theta theta) + (cos(theta) / (r^2 sin(theta))) u_theta + (1 / (r^2 sin^2(theta))) u_(phi phi)`

Derive Laplace's Equation in Spherical Coordinates

Laplacian for general coordinates is defined with covariant derivative in Riemann geometry.

`Delta = g^(i j) grad_i grad_j`

This formula is transformed as follow.

`Delta = (1 / sqrt(g)) del_i sqrt(g) g^(i j) del_j`

In the following program, we calculate Laplacian using this formula.


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