Divergence Theorem

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The divergence theorem states that the overall (net) flux through the surface of an object equals the summation of all the changes in flux inside the object. Stated more formally, the net flux through a closed surface equals the sum of all the changes in flux throughout the volume enclosed by the surface.

In mathematical terms, the divergence theorem establishes that the volume integral (triple integral) of the derivative (divergence) of a vector function over a three-dimensional shape (V) equals the double integral of the perpendicular component of same function with respect to a two-dimensional surface that encloses the space (S):


For example, the volume integral of the electric field caused by a charge within an electric sphere equals the surface integral of that same field.

This is also known as the Gauss Theorem (1813), and it is the simplest of the three main integral theorems in advanced or multivariable calculus, the other two being Green's Theorem (1825) and Stokes' Theorem (1850).