Top 10 similar words or synonyms for laplace_operator
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Top 30 analogous words or synonyms for laplace_operator
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Laplace operator | As a second-order differential operator, the Laplace operator maps functions to functions for . The expression (or equivalently ) defines an operator , or more generally an operator for any open set . |
Laplace operator | Since this holds for all smooth regions , it can be shown that this implies |
Laplace operator | If denotes the electrostatic potential associated to a charge distribution , then the charge distribution itself is given by the negative of the Laplacian of : |
Laplace operator | This is a consequence of Gauss's law. Indeed, if is any smooth region, then by Gauss's law the flux of the electrostatic field is proportional to the charge enclosed: |
Laplace operator | Another motivation for the Laplacian appearing in physics is that solutions to in a region are functions that make the Dirichlet energy functional stationary: |
Laplace operator | The spectrum of the Laplace operator consists of all eigenvalues for which there is a corresponding eigenfunction with |
Laplace operator | The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. The Laplace–Beltrami operator, when applied to a function, is the trace () of the function's Hessian: |
Laplace operator | The additional factor of in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the direction were measured in meters while the direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that in order to simplify the equation. |
Laplace operator | where the latter notations derive from formally writing |
Laplace operator | The left-hand side of this equation is the Laplace operator. The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation. |