A branch of mathematics that 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

(1)

has an extremum only if the Euler-Lagrange
differential equation is satisfied, i.e., if

(2)

The fundamental lemma of calculus
of variations states that, if

(3)

for all
with continuous second partial
derivatives , then

(4)

on .

A generalization of calculus of variations known as Morse theory (and sometimes called "calculus of variations in the large")
uses nonlinear techniques to address variational problems.

See also Beltrami Identity ,

Bolza Problem ,

Brachistochrone Problem ,

Catenary ,

Envelope Theorem ,

Euler-Lagrange Differential Equation ,

Isoperimetric Problem ,

Isovolume
Problem ,

Lindelof's Theorem ,

Morse
Theory ,

Plateau's Problem ,

Line
Line Picking ,

Roulette ,

Skew
Quadrilateral ,

Sphere with Tunnel ,

Surface
of Revolution ,

Unduloid ,

Weierstrass-Erdman
Corner Condition Explore
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References Arfken, 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. Jeffreys,
H. and Jeffreys, B. S. "Calculus of Variations." Ch. 10 in Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 314-332, 1988. Menger, K. "What is the Calculus
of Variations and What are Its Applications?" Part V, Ch. 8 in The
World of Mathematics, Vol. 2 (Ed. K. Newman). New York: Dover,
pp. 886-890, 2000. Sagan, H. Introduction
to the Calculus of Variations. New York: Dover, 1992. Smith,
D. R. Variational
Methods in Optimization. New York: Dover, 1998. 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. Weisstein, E. W. "Books about Calculus of Variations."
http://www.ericweisstein.com/encyclopedias/books/CalculusofVariations.html . Referenced
on Wolfram|Alpha Calculus of Variations
Cite this as:
Weisstein, Eric W. "Calculus of Variations."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/CalculusofVariations.html

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