Laplacian in Polar Coordinates

---

• Verify Laplace's Equation in Polar Coordinates
• Derive Laplace's Equation 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)` -- -- This file has been auto-generated by egison-translator. -- def x := r * cos θ def y := r * sin θ def uR := ∂/∂ (u x y) r uR def uRR := ∂/∂ (∂/∂ (u x y) r) r uRR def uΘ := ∂/∂ (u x y) θ uΘ def uΘΘ := ∂/∂ (∂/∂ (u x y) θ) θ uΘΘ uRR + 1 / r ^ 2 * uΘΘ uRR + 1 / r * uR + 1 / r ^ 2 * uΘΘ

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.

"\"egison\" (line 23, column 12):\nunexpected \"_\"\nexpecting top-level expression"

Links

Back to the Table of Contents