General expressions for gradient, divergence, curl and laplacian in 3D

In this post I have enlisted general expressions for gradient, divergence, curl and laplacian in 3-dimensional (orthogonal) curvilinear co-ordinate system (x₁, x₂, x₃). These operators are frequently needed in classical mechanics, electrodynamics and quantum mechanics in the form of Laplace or Poison equation, Schrodinger equation, fourier transform, wave equation, diffusion equation, etc. The following formulae are very helpful while solving problems in 3-dimensions and handy during entrance exams.

A line element in an arbitrary 3-dimensional orthogonal curvilinear coordinate system (x₁, x₂, x₃) can be written as

 Here (h₁, h₂, h₃) are known as scale factors and are, in general, functions of (x₁, x₂, x₃). For example, a line element in Cartesian co-ordinate system, spherical polar co-ordinate system and cylindrical co-rdinate system can be expressed as

Thus, for Cartesian co-ordinate system (x₁, x₂, x₃) = (x, y, z) and (h₁, h₂, h₃) = (1, 1, 1), for spherical polar co-ordinate system (x₁, x₂, x₃) = (r, θ, ϕ) and (h₁, h₂, h₃) = (1, r, r sin θ).

The following are general expressions for gradient, divergence, curl and laplacian in 3-D curvilinear orthogonal co-ordinate system. These expressions can be readily obtained (for any number of dimensions) using tensor calculus in a more general form. A nice geometrical approach is given in appendix section in Introduction to Electrodynamics (3^rd edition) by D. J. Griffiths. (In the following expressions, ϕ is a scalar function and is not to be confused with  angle.)

 

Using these results, we can obtain the expressions for gradient, divergence, curl and laplacian in the three widely used co-ordinate systems viz. Cartesian, Spherical and Cylindrical co-rdinates simply by replacing the general co-rodinates (x₁, x₂, x₃) with the appropriate one. For instance, to obtain curl in spherical co-rdinates replace the general co-ordinates with (r, θ, ϕ) and corresponding parameters (h₁, h₂, h₃) = (1, r, r sin θ).