Divergence and curl of the vector field

$$\nabla \cdot \vec{F} = \sum_i \frac{\partial F_i}{\partial x_i}$$
It is so called divergence, because it measures the tendency of a vector field to diverge (positive divergence) or converge (negative divergence). In particular, a vector field is said to be incompressible (or solenoidal) if its divergence is zero.

Examples:
$\vec{F} = (x,y,0)$, $\vec{G} = (x,-y,0)$ and $\vec{H}=(-x,-y,0)$, we have
$$\nabla \cdot \vec{G} = 1-1 = 0$$
$$\nabla \cdot \vec{F} = 2$$
$$\nabla \cdot \vec{G} = -2$$
The arrows on the plot of $\vec{F}$ diverge and on the plot of $\vec{H}$ converge.

$$\nabla \times \vec{F} = (\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z})\vec{i} +  (\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x})\vec{j} + (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y})\vec{k}$$
The curl of a vector measures its tendency to rotate. A vector field is said to be irrotational if its curl is the zero vector.



Examples:
$\vec{F} = (-y,x,0)$, $\vec{G} = (y,x,0)$ and $\vec{H}=(y,-x,0)$, we have
$$\nabla \times \vec{F} = 2 \vec{k}$$
$$\nabla \times \vec{G} = 0$$
$$\nabla \times \vec{H} = -2 \vec{k}$$
The coefficient of k in curl F being positive indicates anticlockwise rotation.

Reference:
http://www.maths.gla.ac.uk/~cc/2A/2A_notes/2A_chap4.pdf

https://en.wikibooks.org/wiki/Calculus/Vector_calculus_identities