-div is adjoint to dEdit
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 operatorEdit
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.
Given scalar functions f and h, and a real number a, the Laplacian has the property:
where f and h are scalar functions.