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The Beltrami identity, ... An example of an application of the Beltrami identity is the brachistochrone problem, which involves finding the curve = () ...
v. t. e. 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 ...
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.
Principal symbol. The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature. The principal symbol of the map assigns to each a map from the space of symmetric (0,2)-tensors on to the space of (0,4)-tensors on given by.
In the calculus of variations and classical mechanics, the Euler–Lagrange equations[1] are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Beltrami vector field. 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 is solenoidal - that is, if such as for an incompressible fluid or a ...
The Laplace–Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian: = (()) where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields , by ...
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.