Vector Field

In simple terms, a vector field assigns every point (x,y) in a region a vector pointing in a certain direction, and when you input a point into a vector field, you get a vector function.

Definition: Let $R$ be the region in the $xy$ plane. A Vector Field $\vec{F}$ assigns every point $(x,y)$ in $R$ a vector $\vec{F}(x,y)$ with two components.

$$ \vec{F} = M(x,y)\hat{i} + N(x,y)\hat{j} $$

This plane vector field involves two functions of two variables. They are the components $M$ and $N$, which vary from point to point. A vector has fixed components; a vector field has varying components.

Intution

A real-life example of a vector field is wind blowing down the street. Every point in the street has its own wind velocity. If you walk with the wind, it does positive work; if you walk against it, it does negative work; and if you walk sideways, you’re not affected (zero work relative to the wind).