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# Divergence Theorem

The divergence theorem, more commonly known especially in older literature as Gauss's theorem (e.g., Arfken 1985) and also known as the Gauss-Ostrogradsky theorem, is a theorem in vector calculus that can be stated as follows. Let be a region in space with boundary . Then the volume integral of the divergence of over and the surface integral of over the boundary of are related by

 (1)

The divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary.

A special case of the divergence theorem follows by specializing to the plane. Letting be a region in the plane with boundary , equation (1) then collapses to

 (2)

If the vector field satisfies certain constraints, simplified forms can be used. For example, if where is a constant vector , then

 (3)

But

 (4)

so

 (5) (6)

and

 (7)

But , and must vary with so that cannot always equal zero. Therefore,

 (8)

Similarly, if , where is a constant vector , then

 (9)

Curl Theorem, Divergence, Gradient, Green's Theorem

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## References

Arfken, G. "Gauss's Theorem." §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.Morse, P. M. and Feshbach, H. "Gauss's Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 37-38, 1953.

## Cite this as:

Weisstein, Eric W. "Divergence Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DivergenceTheorem.html