calculus, a surface integral 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 surface integral, the curve is replaced by a piece of a surface in three-dimensional space. The first documented...

a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. The word flux comes from Latin: fluxus...

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...

of C. A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of...

the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted Φ or...

closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector...

relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical...

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...

Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surface integral to obtain the value of the solution...

calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve...

the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form...

\iint _{\Sigma }} is a surface integral over the surface Σ, The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density...

b}\;dA} To make this demonstration, we need to express this surface integral as a volume integral. In flat space-time, we would use Stokes theorem and integrate...

The integral symbol (see below) is used to denote integrals and antiderivatives in mathematics, especially in calculus. ∫ (Unicode), ∫ {\displaystyle...

simplify the calculation of the surface integral. If the Gaussian surface is chosen such that for every point on the surface the component of the electric...

To solve multivariable contour integrals (i.e. surface integrals, complex volume integrals, and higher order integrals), we must use the divergence theorem...

The following are important identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)}...

point x0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x0 to the volume of V, as V...

Friedrich Gauss. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's...

is an imaginary surface S, then the surface integral of flux over S is equal to the amount of q that is passing through the surface S per unit time:...

{\displaystyle 0\leq \theta \leq 2\pi } . Then the surface area is given by the surface integral A x = ∬ S d S = ∬ [ a , b ] × [ 0 , 2 π ] ‖ ∂ r ∂ t...

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...

applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions...

_{q}(\mathbf {r} )\,d\ell } similarly a surface integral of the surface charge density σq(r) over a surface S, Q = ∫ S σ q ( r ) d S {\displaystyle Q=\int...

and has a zero path integral. See gradient theorem. The integral equation is true for any path ∂Σ through space, and any surface Σ for which that path...

\mathrm {d} \mathbf {A} =0} where the integral is a surface integral over the closed surface S (a closed surface is one that completely surrounds a region...

infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes...

line integrals, surface integrals and volume integrals. Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot...

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...