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 [latex]{\bf{v}}[/latex] is the velocity field of a fluid, then the divergence of [latex]{\bf{v}}[/latex] 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 [latex]P[/latex] measures the tendency of particles at [latex]P[/latex] to rotate about the axis that points in the direction of the curl at [latex]P[/latex].
A vector field with a simply connected domain is conservative if and only if its curl is zero.

Key Equations

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

curl: the curl of vector field [latex]{\bf{F}}=\langle{P,Q,R}\rangle[/latex], denoted [latex]\nabla\times{\bf{F}}[/latex] is the “determinant” of the matrix [latex]\begin{vmatrix}{\bf{i}} & {\bf{j}} & {\bf{k}}\\ \frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\P & Q & R\end{vmatrix}[/latex] and is given by the expression [latex](R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}[/latex]; 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 [latex]{\bf{F}}=\langle{P,Q,R}\rangle[/latex], denoted [latex]\nabla\times{\bf{F}}[/latex] is [latex]P_{x}+Q_{y}+R_{z}[/latex]; it measures the “outflowing-ness” of a vector field