The Divergence Theorem

Divergence

Divergence at a point is what that point is contributing to the flux

$$\text{div}\vec{F}=\vec{\nabla }\cdot \vec{F}$$

Using

Over Line

$${\Phi }_{\text{out}}={\oint }_{\partial D}(\vec{F}\cdot \vec{n})ds={\iint }_D\left(\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}\right)dA$$

Over Surface

1. $E$ is a bounded solid
2. $\partial E$ is the boundary of $E$ (surface) oriented outwards by $\vec{n}$
3. $\vec{F}$ has continuous partial derivatives in a solid that contains E (includes boundary)

$${\Phi }_{\text{out}}=\oint {\int }_{\partial D}(\vec{F}\cdot \vec{n})dS={\iiint }_E\text{div}\vec{F}dA$$

Changing Surfaces

$${\iint }_{S1}\vec{F}\cdot \hat{n}dS+{\iint }_{S2}\vec{F}\cdot \hat{n}dS={\iiint }_E\text{div}\vec{F}dV$$

• Where $\partial E=S_1\cup S_2$
• This is for outward orientation, switch signs of flux integrals for inwards