Calculus of variations
Tompkins, Charles B. Department of Mathematics, University of California, Los Angeles, California.
Taylor, Jean E. Department of Mathematics, Rutgers University, Piscataway, New Jersey.
- 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, x1, x2,…, xm, 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 2m + 1 variables x1, x2,…, xm, y, p1, p2, …, pm. Then the integral, Eq. (1),
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