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Calculus of Variations


A branch of mathematics which is a sort of generalization of Calculus. Calculus of variations seeks to find the path, curve, surface, etc., for which a given Function has a Stationary Value (which, in physical problems, is usually a Minimum or Maximum). Mathematically, this involves finding Stationary Values of integrals of the form


I has an extremum only if the Euler-Lagrange Differential Equation is satisfied, i.e., if


The Calculus of Variations Fundamental Lemma states that, if


tex2html_wrap_inline10337 with Continuous second Partial Derivatives, then


on (a,b).

See also Beltrami Identity, Bolza Problem, Brachistochrone Problem , Catenary, Envelope Theorem, Euler-Lagrange Differential Equation, Isoperimetric Problem, Isovolume Problem, Lindelof's Theorem, Partition Calculus, Plateau's Problem, Roulette, Skew Quadrilateral, Sphere with Tunnel, Unduloid, Weierstraß-Erdman Corner Condition


Calculus of Variations

© 1996-7 Eric W. Weisstein
Mon Feb 17 13:44:27 EST 1997

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