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 V be a region in space with boundary partialV. Then the volume integral of the divergence del ·F of F over V and the surface integral of F over the boundary partialV of V are related by

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 S be a region in the plane with boundary partialS, equation (1) then collapses to

If the vector field F satisfies certain constraints, simplified forms can be used. For example, if F(x,y,z)=v(x,y,z)c where c is a constant vector !=0, then

But

so

and

But c!=0, and c·f(v) must vary with v so that c·f(v) cannot always equal zero. Therefore,

Similarly, if F(x,y,z)=cxP(x,y,z), where c is a constant vector !=0, then

See also

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.

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Divergence Theorem

Cite this as:

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

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