Proofs involving the Laplace–Beltrami operator

Main article: Laplace operator

div is adjoint to d

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.

Laplacede Rham operator

One may prove that the Laplacede Rham operator is equivalent to the definition of the LaplaceBeltrami operator, when acting on a scalar function f. This proof reads as:

where vol_n; 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 Laplacede Rham operator is actually minus the LaplaceBeltrami 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

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

Proof

where f and h are scalar functions.

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