Laplace Equation in 3D

PDEs · Axiom Academy

Exploring harmonic functions in three dimensions: from Cartesian to spherical coordinates, separation of variables, and spherical harmonics 1. Cartesian Form of Laplace Equation In three-dimensional Cartesian coordinates (x, y, z), the Laplace equation is the natural extension of the 2D version. We sum the second partial derivatives in all three spatial directions: This equation describes harmonic functions u(x,y,z) - functions where the average value at any point equals the value at that point. Solutions include gravitational and electrostatic potentials in free space. For problems with spherical symmetry (like planetary gravitational fields), spherical coordinates (r, θ, φ) are natural: In spherical coordinates, the Laplacian transforms to: This form reveals the natural separation into radial and angular parts, making it ideal for solving boundary value problems on spheres. 3. Separation in Spherical Coordinates We seek solutions of the form u(r,θ,φ) = R(r)Y(θ,φ), separating the radial and angular dependencies. Substituting into the Laplace equation and separating variables yields two equations: Here λ is the separation constant. The radial equation has power-law solutions R(r) = r ℓ or r -(ℓ+1) , where ℓ is a non-negative integer. For azimuthally symmetric problems (no φ dependence), the angular part reduces to Legendre's equation: Setting x = cos θ, this becomes: The solutions bounded on [-1,1] are the Legendre polynomials P ℓ (x) for ℓ = 0, 1, 2, ...

This is the written version of the interactive lesson above. See the full PDEs course.