Summary of Divergence and Curl

The divergence of a vector field is a scalar function. Divergence measures the “outflowing-ness” of a vector field. If ${\bf{v}}$ is the velocity field of a fluid, then the divergence of ${\bf{v}}$ at a point is the outflow of the fluid less the inflow at the point.
The curl of a vector field is a vector field. The curl of a vector field at point $P$ measures the tendency of particles at $P$ to rotate about the axis that points in the direction of the curl at $P$ .
A vector field with a simply connected domain is conservative if and only if its curl is zero.

Key Equations

Curl
$\nabla\times{\bf{F}}=(R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}$
Divergence
$\nabla\cdot{\bf{F}}=P_{x}+Q_{y}+R_{z}$
Divergence fo curl is zero
$\nabla\cdot(\nabla\times{\bf{F}})=0$
Curl of a gradient is the zero vector
$\nabla\times(\nabla{f})=0$

curl: the curl of vector field ${\bf{F}}=\langle{P,Q,R}\rangle$ , denoted $\nabla\times{\bf{F}}$ is the “determinant” of the matrix $\begin{vmatrix}{\bf{i}} & {\bf{j}} & {\bf{k}}\\ \frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\P & Q & R\end{vmatrix}$ and is given by the expression $(R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}$ ; it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point

divergence: the divergence of a vector field ${\bf{F}}=\langle{P,Q,R}\rangle$ , denoted $\nabla\times{\bf{F}}$ is $P_{x}+Q_{y}+R_{z}$ ; it measures the “outflowing-ness” of a vector field