Math Genius - The Divergence Theorem lyrics

So you've been taking surface integrals of vector fields to figure out the vector flux through the surface. You may have noticed by now that the math behind this process isn't very easy. Enter the Divergence Theorem. When we're asked to calculate the flux of a vector field through a closed surface, we can think about the problem in a slightly different way. Taking the integral of a vector field over the boundary it's going through (∫∫SF dS) is the same as taking a vector field's divergence over the interior of the region through which it flows (∫∫∫V (div F) dV). Here's the identity to remember: ∫∫ S F· n dS = ∫∫∫ V∇· F dV If you're already well-versed in Green's Theorem, you'll recognize that this is simply a three-dimensional application of the same concept. Remember: just because the theorem is true doesn't mean you have to use it to solve every problem. Apply it whenever it's convenient to make the integration simpler.