These are notes on how to express the Laplacian in spherical coordinates.

The first step is to define the coordinates via the following identities:

Here is the radius from the origin, is the angle from the -axis and is the azimuthal angle from the -axis. The map is a one-to-one invertible map from onto minus the -axis. The mapping is singular on the -axis since the value of is undetermined there.

If we use (1) to express in terms of , then the chain rule leads to the identities:

In principle we could use these identities to express the Laplacian in terms of by first inverting the formulae to obtain , , , in terms of , , etc. and then squaring. However, this computation is boring and not very enlightening. There is a more geometric approach.

To obtain this approach it is useful to think first about the *gradient operator *We are used to thinking of the gradient of a function as the vector field

where , , are unit vectors in the directions of the positive , and axes respectively. However, it is also useful to think of somewhat differently. Specifically given and ** **in , gives the directional derivative of at in direction , i.e.,

We will now express the gradient operation using a moving frame that is more suited to the spherical coordinates. Namely

while

That is , , is an orthonormal frame. It is a *moving *frame because the choice of frame depends on our position in configuration space.

Combining eqs. (2) and (3) we see that

Since , , is an orthonormal frame, it follows that

In other words, we can express the gradient operator as follows

Now to find the Laplacian we need only express as

However in expanding this expression we need to be mindful of the fact that , , is a *moving* frame and thus certain derivatives of these vectors are non-zero:

Consider, for example, the term When this acts on a function we have, by the product rule,

since and . Thus

When expanded fully, the right hand side to (4) has nine terms:

Adding these up leads to the expression

Observing that and this can be re-expressed as