Multivariable Calculus

Curl and Divergence

Info

Curl tells us the differential rotation and divergence tells us the differential expansion. Therefore, the divergence of a curl is always zero. $\nabla \times (\nabla f)=0$ $\nabla \cdot (\nabla \times \mathbf{F})=0$

Curl

Calculating the cross product of the vector differential and the vector field provides the curl of the vector field.

$$Curl\mathbf{F}=\nabla \times \mathbf{F}$$

If $Curl\mathbf{F}=0$, then $\mathbf{F}$ is both conservative and irrotational due to no differential rotation.

Note: See Curl Derivation.

Divergence

Calculating the dot product of the vector differential and the vector field provides the divergence of the vector field.

$$Div.\mathbf{F}=\nabla \cdot \mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$$

If $Div.\mathbf{F}=0$ and is $C^1$, then $\mathbf{F}$ is the curl of $\mathbf{G}$.

$$\mathbf{F}=\nabla \times \mathbf{G}$$

Explanation

Basically, if taking the divergence of $\mathbf{F}$ is non-zero, then there exists points where field lines within $\mathbf{F}$ originate and terminate i.e. sources and sinks.

When we get the curl of vector field, $\mathbf{G}$, we are getting a new vector field, $\mathbf{F}$, which contains only the rotation information and has no sources and sinks.