Vector Fields and Gradient Fields

1.

A vector field $\mathbf{V}$ is simply a way of assigning, to points (x,y) in the plane, vectors $\mathbf{V}(x,y)$. For example, the vector field $\mathbf{V}(x,y)=(4xy-1)\mathbf{i+}(2x^{2}+2y)\mathbf{j}$ assigns the vector $\mathbf{V}(2,3)=23\mathbf{i}+14\mathbf{j}$ to the point (2,3).

2.

Given a function f(x,y), its gradient $\nabla f=\left( \frac{\partial f}{\partial x}\right) \mathbf{i}+\left( \frac{\partial f}{\partial y}\right) \mathbf{j}$ is a vector field. For example, suppose f(x,y)=2x²y+y²-x+2. Then $\nabla f=(4xy-1)\mathbf{i+}(2x^{2}+2y)\mathbf{j}$ (see immediately above), so $\nabla f$ assigns the vector $23\mathbf{i}+14\mathbf{j}$ to the point (2,3).

3.

We can picture vector fields by selecting a grid of points in the plane and drawing the vector $\mathbf{V}(P)$ at the point P. Matlab does this with the command quiver (a ``quiver'', you may recall, is a carrying case for arrows). Here is an m-file which draws the gradient field for the function above, as well as a contour plot for the same function

$$\begin{tabular}[c] {c}\texttt{grad1.m}\\ \hline \multicolumn... ...x,y,px,py), contour(xx,yy,zz), hold off}}\\ \hline\end{tabular}$$

Here are some comments about this m-file.

• Note first that there are two grids, one for the gradient field and the other for the contours. This is because you want a fine grid (every 0.2) for drawing smooth contours, but such a fine grid would draw so many arrows for the vector field that the screen would be too cluttered. Therefore we use the gap 1.0 instead of 0.2 for the gradient.
• The arrays have to match up: [px,py] goes with x and y, zz goes with xx and yy; otherwise you'll get an error message.
• You can get a more colorful plot by replacing the contour command with contourf. However, if you do this, you'll want to draw the arrows in black, so use quiver(x,y,px,py,'k').
• Matlab scales the arrows so that they fit nicely in the screen and don't touch each other. They can also be scaled manually -- look it up by typing: help quiver at the Matlab prompt.

4.

Combined gradient and contour plots can be helpful in classifying critical point (places where the gradient is zero). These are of three types:

• Maxima: nearby gradient vectors all point toward the critical point (function increasing)
• Minima: nearby gradient vectors all point away from critical point
• Saddles: some nearby gradients point toward, others away.