Curl- and Divergence Component

The curl of a vector field defines the amount of rotation of a vector field along an axis or point. It is used to calculate Circulation for example using Stokes Theorem.

The component itself is given by

$$\nabla =i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y}+k\frac{\partial }{\partial z}$$

Curl

The tendency of a vector field to rotate clockwise around a point.

$$curl\_F=\nabla \times \vec{F}=\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ M& N& P\end{array}\right|$$

or in pre-computed form

\= (\frac{\partial P}{\partial y}-\frac{\partial N}{\partial z})i+(\frac{\partial M}{\partial z}-\frac{\partial P}{\partial x})j +(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})k

Divergence

The expansion or compression of a vector field, hence its flux per unit volume.

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