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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.
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks ...
Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use.
A parametric surface is a surface in the ... a ≤ t ≤ b represents a parametrized curve on this surface then its arc length can be calculated as the integral: ...
Curve–surface integrals. In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ...
It is an arbitrary closed surface S = ∂V (the boundary of a 3-dimensional region V) used in conjunction with Gauss's law for the corresponding field (Gauss's law, Gauss's law for magnetism, or Gauss's law for gravity) by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of ...
A related law governs the rate of change of the surface integral. The law reads = where the /-derivative is the fundamental operator in the calculus of moving surfaces, originally proposed by Jacques Hadamard.