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## The Divergence Theorem

This is a second, very, very important statement of the Fundamental Theorem: In this expression should be read in your mind as over the open volume bounded by the closed surface '', and is an arbitrary vector quantity, typically a vector field like or or a vector current density such as . Note well that the right hand side you should be reading as the flux of the vector function out through the closed surface S''.

You might also see this written as: where is read as the surface bounding the volume ''. This is slightly more compact notation, but a student can easily be confused by what appears to be a partial differential in the surface limits.

A simple consequence of the divergence theorem is: Proof: Assume then so that Since is constant and arbitrary, we can factor it out from the integral: Since this has to be true for any nonzero , we can essentially divide out the constant and conclude that: Q.E.D.

You should prove on your own (using exactly the same sort of reasoning) that: There thus is one such theorem for (acting on any scalar ), (acting on any vector function ) or acting on any vector function . We can use all of these forms in integration by parts.    Next: Stokes' Theorem Up: The Fundamental Theorem(s) of Previous: A Scalar Function of   Contents
Robert G. Brown 2017-07-11