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The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals of the form. I a b {\displaystyle I [u]=\int _ {a}^ {b}L [x,u (x),u' (x)]\,dx\,,} where and are constants and . [1]
Laplace–Beltrami operator. 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. It is named after Pierre-Simon Laplace and Eugenio Beltrami.
Giovanni Frattini. Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant ...
Calculus. The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. [a] Functionals are often expressed as definite integrals involving ...
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ¯ =. for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L 2, and where μ is a given complex function in L ∞ (U) of norm less than 1, called the Beltrami coefficient, and where / and / ¯ are Wirtinger derivatives.
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian - or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that. Thus and are parallel vectors in other words, . If. {\displaystyle \mathbf {F} } is solenoidal - that is, if.