If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Multivariable calculus

### Course: Multivariable calculus>Unit 5

Lesson 11: Divergence theorem proof

# Divergence theorem proof (part 5)

Home stretch. Proving the Type I part. Created by Sal Khan.

## Want to join the conversation?

• Sal's "proof" of the divergence theorem is straightforward and fairly simple owing to the shape of the surface over which he was integrating, but does anyone know of a way to apply this method to more general surfaces? My intuition suggests that any closed region in space can be broken into many Type !, II, and III regions(which type we choose depends on which component of F we choose to focus on), and from there one can apply sal's method to each individual region. We then add all the regions to get the final result. This approach may not seem very rigorous, but at least it comes much closer to generalizing the divergence theorem.
• Yes, that's how you do it. so we take the sum of all the little surfaces/regions (depending on whether you're looking at the left hand side or the right of the divergence theorem) and the flux inside the region cancels out. Similar thing can be said about Sal's stokes theorem proof, he said it was a special case where z=z(x,y) and the reason why the inside stuff cancel out is because of the positive and negative orientations.