Definition:Vector Field/Classification

Classification of Vector Fields

Let $R$ be a region of space.

Let $\mathbf V$ be a vector field acting over $R$.

Conservative Vector Field

$\mathbf V$ is a conservative vector field if and only if its curl is everywhere zero:

$\curl \mathbf V = \bszero$

Solenoidal Vector Field

$\mathbf V$ is defined as being solenoidal if and only if its divergence is everywhere zero:

$\operatorname {div} \mathbf V = 0$

Illustration

The following diagram gives a visual impression of the properties of vector fields with various combinations of properties relating to divergence and curl: