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In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces.It can be thought of as the double integral analogue of the line integral.
Integral as area between two curves. Double integral as volume under a surface z = 10 − ( x 2 − y 2 / 8 ).The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.
Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration. [1]: 367ff
The definition of surface integral relies on splitting the surface into small surface elements. A surface integral generalizes double integrals to integration over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
Therefore, the integral may also be written as [] = ˙. This form suggests that if we can find a function ψ {\displaystyle \psi } whose gradient is given by P , {\displaystyle P,} then the integral A {\displaystyle A} is given by the difference of ψ {\displaystyle \psi } at the endpoints of the interval of integration.
In 1760 Lagrange extended Euler's results on the calculus of variations involving integrals in one variable to two variables. [50] He had in mind the following problem: Given a closed curve in E 3, find a surface having the curve as boundary with minimal area. Such a surface is called a minimal surface.
Although this formula provides a closed expression for the surface area, for all but very special surfaces this results in a complicated double integral, which is typically evaluated using a computer algebra system or approximated numerically. Fortunately, many common surfaces form exceptions, and their areas are explicitly known.