# Famous Theorems of Mathematics/Applied Mathematics

## -div is adjoint to d[edit]

The claim is made that −div is adjoint to *d*:

Proof of the above statement:

If *f* has compact support, then the last integral vanishes, and we have the desired result.

## Laplace-de Rham operator[edit]

One may prove that the Laplace-de Rahm operator is equivalent to the definition of the Laplace-Beltrami operator, when acting on a scalar function *f*. This proof reads as:

where ω is the volume form and ε is the completely antisymmetric Levi-Civita symbol. Note that in the above, the italic lower-case index *i* is a single index, whereas the upper-case Roman *J* stands for all of the remaining (*n*-1) indices. Notice that the Laplace-de Rham operator is actually minus the Laplace-Beltrami operator; this minus sign follows from the conventional definition of the properties of the codifferential. Unfortunately, Δ is used to denote both; reader beware.

## Properties[edit]

Given scalar functions *f* and *h*, and a real number *a*, the Laplacian has the property:

### Proof[edit]

where *f* and *h* are scalar functions.