Divergence Theorem

Divergence Theorem

$$\oint \oint \vec{F}\cdot \hat{N}ds$$

• Remember, this is on a closed surface!
• F dot N is meaning the flux in the N direction
• N is the normal vector to the surface (points outwards)

$$\nabla \cdot \vec{F}=\text{ Divergence of F}=\text{Rate spreading away from the point}=\text{Source (sink)}$$ $${\iiint }_{Vol}\nabla \cdot \vec{F}dvol=\text{Total source inside volume}$$

So, what we can say is that…

$${\iiint }_{Vol}\nabla \cdot \vec{F}dvol=\oint \oint \vec{F}\cdot \hat{N}ds$$

And this is the divergence theorem! Divergence theorem calculates the total flux of a field through a surface

In 2D, this would be Green’s Theorem!

• Reminder! GRADIENT IS THE N, Normal value