The Divergence Theorem // Geometric Intuition & Statement // Vector Calculus
In this video we get to the last major theorem in our playlist on vector calculus: The Divergence Theorem. We've actually already seen the two-dimensional analog when we studied the Divergence or Flux form of Green's Theorem. Now we upgrade to the three-dimensional situation where we have a closed, smooth surface and a vector field. The question is, to what degree is there outwards flux of the vector field across this surface? The divergence theorem allows this global property to be compared to a triple integral over the enclosed volume of the divergence of the vector field; that is, adding up a local property of the divergence. Much like Stokes' Theorem before it, the divergence theorem is another example where integrating a differential operator along an entire region gives us information (in this case outward flux) on the boundary of that region.
0:00 Divergence of a Field
2:49 Recalling Green's Theorem
4:40 Divergence Theorem
6:20 Conditions
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