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. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.
An illustration of a single surface element. These elements are made infinitesimally small, by the limiting process, so as to approximate the surface.
calculus, a surfaceintegral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue...
curve connecting two points in space. In a surfaceintegral, the curve is replaced by a piece of a surface in three-dimensional space. The first documented...
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear...
the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form...
a scalar quantity, defined as the surfaceintegral of the perpendicular component of a vector field over a surface. The word flux comes from Latin: fluxus...
closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surfaceintegral of a vector...
Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surfaceintegral to obtain the value of the solution...
The following are important identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)}...
The integral symbol: ∫ (Unicode), ∫ {\displaystyle \displaystyle \int } (LaTeX) is used to denote integrals and antiderivatives in mathematics, especially...
the magnetic flux through a surface is the surfaceintegral of the normal component of the magnetic field B over that surface. It is usually denoted Φ or...
simplify the calculation of the surfaceintegral. If the Gaussian surface is chosen such that for every point on the surface the component of the electric...
infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes...
calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially...
b}\;dA} To make this demonstration, we need to express this surfaceintegral as a volume integral. In flat space-time, we would use Stokes theorem and integrate...
_{q}(\mathbf {r} )\,d\ell } similarly a surfaceintegral of the surface charge density σq(r) over a surface S, Q = ∫ S σ q ( r ) d S {\displaystyle Q=\int...
to the line integral of the vector field over the surface boundary. The second fundamental theorem of calculus states that the integral of a function...
point x0 is defined as the limit of the ratio of the surfaceintegral of F out of the closed surface of a volume V enclosing x0 to the volume of V, as V...
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...
\mathrm {d} \mathbf {A} =0} where the integral is a surfaceintegral over the closed surface S (a closed surface is one that completely surrounds a region...
the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium rule Integral of the...
the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function...
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