# Definition:Calculus of Variations

## Definition

**Calculus of variations** is the subfield of analysis concerned maximizing or minimizing real functionals, which are mappings from a set of functions to the real numbers.

## Also see

- Results about
**calculus of variations**can be found here.

## Historical Note

Some sources suggest that, in a sense, the earliest problem in the calculus of variations arose in one of the legends of the founding of Carthage; the city was granted as much land as could be enclosed by a given length.

The calculus of variations emerged as a branch of mathematics as a result of investigations into the cycloid in the $18$th century.

The first systematic investigation of the topic was given by Joseph Louis Lagrange in his earliest and most important works, together with Leonhard Paul Euler, who coined the term in $1766$.

Karl Weierstrass ushered in a new era of precise reasoning with his lectures in $1879$ on the subject.

One of his students, Oskar Bolza, took on the subject and developed the Chicago school of the calculus of variations.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**calculus of variations** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**calculus of variations**