![]() ![]() The resulting operator is called the Laplace–de Rham operator (named after Georges de Rham). Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. It is named after Pierre-Simon Laplace and Eugenio Beltrami.įor any twice- differentiable real-valued function f defined on Euclidean space R n, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for R n. Section 1.18 Solid Mechanics Part III Kelly 165 and so i k j jk i g g Partial Derivatives of Contravariant Base Vectors. ![]() In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. 2 note that, in non-Euclidean space, this symmetry in the indices is not necessarily valid. Like the Laplacian, the LaplaceBeltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. In this chapter we will generalize the Laplacian on Euclidean space to an oper-. The Laplacian Operator is an equation that informs the how cameras and computers can understand what theyre looking at. Not to be confused with Beltrami operator. structed from the curvature of a Riemannian metric. ![]()
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