Vector fields

What is a vector field?

In general, a vector field is a function that assigns vectors to points in space.

A vector field in the $xy$ plane is a vector function of 2 variables:
\[
\mathbf{F}(x,y)=\left(F_1(x,y),F_2(x,y)\right)=F_1(x,y)\mathbf{i}+F_2(x,y)\mathbf{j}
\]

The best way to picture a vector field is to draw the arrow representing the vector $\mathbf{F}(x,y)$ starting at the point $(x,y)$. Of course, it's impossible to do this for all points $(x,y)$, but we can gain a reasonable impression of $\mathbf{F}$ by doing it for a few representative points in $\mathbb R^2$.


Similarly a vector field in 3-D is a vector function of 3 variables:
\begin{eqnarray*}
\mathbf{F}(x,y,z)&=&\left(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z)\right)\\&=&F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}
\end{eqnarray*}




Simulation

Some computer algebra systems are capable of plotting vector fields in two or three dimensions. They give a better impression of the vector field than is possible by hand because the computer can plot a large number of representative vectors. The simulations below are tools for plotting vector fields in two and three dimensions.

Click on the image (or link) to run the simulation

2D Vector Field
http://ggbm.at/cXgNb58T

 3D Vector Field
http://ggbm.at/KKB2Ndez