Divergence Theorem

The Divergence Theorem states that under suitable conditions, the outward flux of a vector field equals the triple integral of its divergence over the solid region $D$ enclosed by the surface.

$$\iint \_S\vec{F}\cdot \vec{n}d\sigma =\iiint \_D\nabla \cdot \vec{F}dV$$

• $\vec{n}$ outward unit-normal vector field

Divergence

The divergence of a vector field (medium) is interpreted as its rate of expansion or compression in the defined space. It is the flux per unit volume flux density.

$$\text{div}F=\nabla \cdot F=\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y}+\frac{\partial P}{\partial z}$$

• $\nabla$ Curl- and Divergence Component

Important Identities

Divergence of Curl is zero

The divergence of the curl of a vector field is always $\text{div}(\nabla \times \vec{F})=\nabla \cdot (\nabla \times \vec{F})=0$ Interesting interpretations arise from this statement that govern the field that is the curl of a function $\vec{F}$, $\vec{G}=\text{curl}\vec{F}$.

• Its divergence must be zero or else it isn’t the curl of $\vec{F}$
• Its outward flux across any closed surface $S$ must be zero as well, or else $G$ cannot be the curl of $\vec{F}$ So if there exists a closed surface for which the surface integral of $\vec{G}$ is nonzero, $\vec{G}$ is proven to not be the curl of $\vec{F}$