Dirichlet energy

English

Alternative forms

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Etymology

Named after German mathematician Template:W (1805-1859), who made significant contributions to the theory of Fourier series.

Noun

1. Template:Lb A quadratic functional which, given a real function defined on an open subset of ℝⁿ, yields a real number that is a measure of how variable said function is.
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Usage notes

• In mathematical terms, given an open set $\mathrm{\Omega }\subseteq {ℝ}^n$ and a function $u:\mathrm{\Omega }\to ℝ$, the Dirichlet energy of $u$ is $E[u]=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }\Vert \mathrm{\nabla }u(x){\Vert }^{2}dx$, where $\mathrm{\nabla }u:\mathrm{\Omega }\to {ℝ}^n$ denotes the gradient vector field of $u$.

Translations

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Further reading

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