Let * be the Hodge star, and \ delta the codifferential.
2.
Where d is the exterior derivative or differential and ? is the codifferential.
3.
Here } } is the codifferential, which can also be expressed using the Hodge dual.
4.
Where d is the exterior derivative or differential and ? is the codifferential, acting as on " k "-forms, where " is the Hodge star.
5.
Notice that the Laplace & ndash; de Rham operator is actually minus the Laplace & ndash; Beltrami operator; this minus sign follows from the conventional definition of the properties of the codifferential.
6.
On functions, the Laplace de Rham operator is actually the negative of the Laplace Beltrami operator, as the conventional normalization of the codifferential assures that the Laplace de Rham operator is ( formally ) positive definite, whereas the Laplace Beltrami operator is typically negative.
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