A vector field assigns a vector to every point in a region. In three dimensions, it can be written

$$\mathbf F(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z)).$$

Unlike a scalar field, which assigns one number to each point, a vector field assigns both a magnitude and a direction.

Visualizing a field

A vector field can be pictured by drawing an arrow at many points in space. The arrow direction shows the local direction of the field and its length represents the local magnitude.

Examples include a velocity field in a moving fluid, a gravitational force field or an electric field.

A simple example

The planar field

$$\mathbf F(x,y)=(-y,x)$$

is tangent to circles centered at the origin. At $(1,0)$ the field is $(0,1)$; at $(0,1)$ it is $(-1,0)$. The arrows therefore circulate counterclockwise around the origin.

Fields can vary from point to point

The components $P,Q,R$ are ordinary scalar functions. Their partial derivatives describe how the vector field changes through space.

This local variation leads to two important differential quantities: divergence, which measures net outward behavior, and curl, which measures local rotational behavior.