Definite integral in the context of "Coordinate axis"

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).

Integrals of a function of two variables over a region in ${\displaystyle \mathbb {R} ^{2}}$ (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in ${\displaystyle \mathbb {R} ^{3}}$ (real-number 3D space) are called triple integrals. For repeated antidifferentiation of a single-variable function, see the Cauchy formula for repeated integration.

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functionsand functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).

Integrals of a function of two variables over a region in ${\displaystyle \mathbb {R} ^{2}}$ (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in ${\displaystyle \mathbb {R} ^{3}}$ (real-number 3D space) are called triple integrals.