| 释义 |
Calculus of VariationsA branch of mathematics which is a sort of generalization of Calculus. Calculus of variations seeks to find thepath, 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
 | (1) |
 has an extremum only if the Euler-Lagrange Differential Equation is satisfied, i.e., if
 | (2) |
 | (3) |
 with Continuous second Partial Derivatives, then
 | (4) |
 .See also Beltrami Identity, Bolza Problem, Brachistochrone Problem, Catenary, EnvelopeTheorem, Euler-Lagrange Differential Equation, Isoperimetric Problem, Isovolume Problem,Lindelof's Theorem, Plateau's Problem, Point-Point Distance--2-D,Point-Point Distance--3-D, Roulette, Skew Quadrilateral, Sphere withTunnel, Unduloid, Weierstraß-Erdman Corner Condition
References
 Calculus of VariationsArfken, G. ``Calculus of Variations.'' Ch. 17 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 925-962, 1985. Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, 1925. Forsyth, A. R. Calculus of Variations. New York: Dover, 1960. Fox, C. An Introduction to the Calculus of Variations. New York: Dover, 1988. Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992. Menger, K. ``What is the Calculus of Variations and What are Its Applications?'' In The World of Mathematics (Ed. K. Newman). Redmond, WA: Microsoft Press, pp. 886-890, 1988. Sagan, H. Introduction to the Calculus of Variations. New York: Dover, 1992. Todhunter, I. History of the Calculus of Variations During the Nineteenth Century. New York: Chelsea, 1962. Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974.
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