The Calculus of Variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.

A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown functions and their derivatives.

The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is precisely zero.

One example is finding a geodesic (or line of shortest length) joining two points. Another example is the Brachistochrone problem

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