Vector Fields

Let’s start off with the formal definition of a vector field.

Definition

A vector field on two dimensional space is a function

\vec{F}

that assigns to each point

(x, y)

a two-dimensional vector given by

\vec{F}(x,y)

That may not make a lot of sense, but most people do know what a vector field is, or at least they’ve seen a sketch of a vector field. If you’ve seen a current sketch giving the direction and magnitude of a flow of a fluid or the direction and magnitude of the winds then you’ve seen a sketch of a vector field.

The standard notation for the function

\vec{F}

is,

\vec{F}(x, y) = P(x, y)\vec{i} + Q(x, y)\vec{j}

The functions

P

and

Q

are somtimes called

\textbf{scalar functions}

Check your knowledge

A two dimensional vector field is a function from...

R¹ to R²

R² to R²

R² to R³

Select an answer...

Visualization

We can visualize vector fields by drawing the vector "outputs" at a sampling of points.

Adapted from Paul's Online Math Notes.