Vector fields
Here’s a short review on vector fields.
A vector-valued function F : R2 → R2 can be visualized as a vector field. At a point (x,y), we plot the value of F(x,y) as a vector, such as in the following figure.
We repeat this over a set of points (x,y) so that we can visualize the entire vector field.
For example, consider the function F(x,y) = (y,-x). We calculate values of the function at a set of points, such as
| F(1, 0) | = (0,-1), | ||
| F(0, 1) | = (1, 0), | ||
| F(1, 1) | = (1,-1), | ||
| F(2, 0) | = (0,-2), |
(So that the vectors wouldn’t overlap, I drew the arrows at only 40% of the length they should have been. I scaled the length down so that the arrows would overlap; such scaling is typical in plots of vector fields.)
If we visualize the vector field F(x,y) = (x,y), it looks like an explosion emanating from the origin.
We can also plot vector fields in three dimensions, i.e., for functions F : R3 → R3. For example, the three-dimensional analogue of the above picture would be F(x,y,z) = (x,y,z). Again, it corresponds to an explosion from the origin, although only a portion of this vector field with x > 0, y > 0, and z > 0 is shown below.
We could get a rotation in three dimensions using F(x,y,z) = (y∕z,-x∕z, 0). This is similar to the first vector field we plotted. In this case, since we divided by z, the magnitude of the vector field decreases as z increases.
