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In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
In physics (specifically electromagnetism), Gauss's law, also known as Gauss's flux theorem (or sometimes Gauss's theorem), is one of Maxwell's equations. It is an application of the divergence theorem , and it relates the distribution of electric charge to the resulting electric field .
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.
Divergence theorem (vector calculus) Dold–Thom theorem (algebraic topology) Dominated convergence theorem (Lebesgue integration) Donaldson's theorem (differential topology) Donsker's theorem (probability theory) Doob decomposition theorem (stochastic processes) Doob's martingale convergence theorems (stochastic processes)
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 ...
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
In mathematics, a Green's function ... For example, if = and time is the ... To derive Green's theorem, begin with the divergence theorem (otherwise known as Gauss's ...
Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives: = = ((+)) = (+) = (+) i.e., + = By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary: