# Article

# Article

- Mathematics
- Analysis (calculus)
- Calculus of variations

# Calculus of variations

Article By:

**Tompkins, Charles B. **Department of Mathematics, University of California, Los Angeles, California.

**Taylor, Jean E. **Department of Mathematics, Rutgers University, Piscataway, New Jersey.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.103500

- Theoretical basis
- Multidimensional derivatives
- Single-integral problems

- Problem of Bolza
- Critical points

- Multivariable problems
- Additional Reading

**An extension of the part of differential calculus which deals with maxima and minima of functions of a single variable.** The functions of the calculus of variations depend in an essential way upon infinitely many independent variables. Classically these functions are usually integrals whose integrand depends on a function whose specification by any finite number of parameters is impossible. For example, let *C* be a smooth bounded region of a space of *m* variables, *x*_{1}, *x*_{2},…, *x _{m}*, let

*y*be any function of some smooth class on

*C*and its boundary into real numbers or into

*n*−tuples of real numbers and taking specified values on the boundary, and let

*f*(

*x*,

*y*,

*p*) be a smooth function of 2

*m*+ 1 variables

*x*

_{1},

*x*

_{2},…,

*x*,

_{m}*y*,

*p*

_{1},

*p*

_{2}, …,

*p*. Then the integral, Eq. (1),

_{m}*y*to the real numbers, and this space of functions is infinite dimensional unless excessive restrictions are placed on it. Here

*y*denotes the derivatives ∂

_{x}*y*/∂

*x*, and throughout this article subscripts will be used to denote derivatives and occasionally where the context is clear to denote particular values.

The content above is only an excerpt.

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