Laplacian in Polar Coordinates
Verify Laplace's Equation in Polar Coordinates
We verify the following equation is correct by expanding `u_(r r)`, `u_r`, and `u_(theta theta)` in the right side of the equation.`u_(x x) + u_(y y) = u_(r r) + (1 / r) u_r + (1 / r^2) u_(theta theta)`
Derive Laplace's Equation in Polar 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.