Vector fields

Here's a short review on vector fields.

A vector-valued function F : R² $\rightarrow$ 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),

etc. By plotting these arrows, we see that this vector field appears to rotate in a clockwise direction.

(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 $\rightarrow$ 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.

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