Definition:Vector Field

Definition

Let $F$ be a field which acts on a region of space $S$.

Let the point-function giving rise to $F$ be a vector quantity.

Then $F$ is a vector field.

Classification

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:

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

Examples

Velocity of Fluid

In a moving fluid, the velocity $\mathbf v$ of the fluid is an example of a vector field.

That is, the velocity $\mathbf v$ at a point $P$ in the fluid is the velocity of the particle which is situated at $P$ at a given instant.

Electric Field Strength

Let $R$ be a region of space in which there exists an electric field.

The electric field strength in $R$ gives rise to a vector field over $R$.

Magnetic Field Strength

Let $R$ be a region of space in which there exists an magnetic field.

The magnetic field strength in $R$ gives rise to a vector field over $R$.