When one is learning about Laplace and Poisson equations it can be frustrating to remember its form in different coordinate systems. But when one is introduced to four vectors, special relativity and so on, here is a simple way to remember the Laplacian in any coordinate system.

where we is the inverse of the metric , is the determinant of the metric . And one specifies the coordinate system by mentioning the form of the metric. Let’s look at how this works out:

**Cartesian Coordinates**

**Cylindrical Coordinates**

Noting that because they are independent variables, we get

**Spherical Coordinates**

Following the same approach as the Cylindrical and Cartesian coordinates, we get the following form for the Laplacian in Spherical coordinates,