curvilinear coordinates

Cartesian axes are straight, perpendicular grid lines everywhere. But a problem with spherical symmetry (a charged sphere) or cylindrical symmetry (a long wire) is far easier if the coordinate lines follow the symmetry. Curvilinear coordinates are any coordinate system whose lines are curves -- like the r, theta, phi of spherical coordinates -- chosen so the boundaries and symmetries of the problem become simple.

A coordinate system (u1, u2, u3) is curvilinear when the position vector r(u1, u2, u3) has coordinate lines that curve. At each point one builds local basis vectors and scale factors (the Lame coefficients) h_i = |partial r / partial u_i|; these measure how much physical length corresponds to a step du_i. For orthogonal systems the vector operators pick up these factors: the line element is ds^2 = h1^2 du1^2 + h2^2 du2^2 + h3^2 du3^2, the gradient's i-th component is (1/h_i) partial f / partial u_i, and the Laplacian becomes laplacian f = (1/(h1 h2 h3)) sum over i of partial_i( (h_j h_k / h_i) partial_i f ). For spherical coordinates h_r = 1, h_theta = r, h_phi = r sin theta.

This is why separation of variables works: Laplace's equation in spherical coordinates separates into a radial equation and an angular one, generating Legendre polynomials and spherical harmonics; in cylindrical coordinates it generates Bessel functions. Choosing the right curvilinear system is often the single most important step, and the special functions of physics are essentially the fingerprints of the common coordinate systems.