Computer Lab: Gradient Fields

Introduction

The aim of this lab is to visualize some of the properties of the gradient, and to use them to interpret the behavior of a function of two variables.

Review of the Gradient

1. Suppose is a scalar-valued function of two variables, so . The gradient of f is the vector of functions: .

2. The gradient of at the point is the vector:

3. points in the direction, from , in which the function increases most rapidly.

4. is perpendicular to the contour or level curve for through .

Vector Fields and Gradient Fields

1. A vector field is simply a way of assigning, to points in the plane, vectors . For example, the vector field assigns the vector to the point .

2. Given a function , its gradient is a vector field. For example, suppose . Then (see immediately above), so assigns the vector to the point

3. We can picture vector fields by selecting a grid of points in the plane and drawing the vector at the point . Here is a Maple session which draws a combination of a gradient plot and contour plot for the function .

   > with(plots):

   > F:= x^2-2*x*y+3*y^2;

   > P1:=contourplot(F,x=0..4,y=0..4,contours=20, grid=[30,30]):

   > P2:= gradplot(F,x=0..4,y=0..4,arrows=SLIM, grid=[30,30]):

   > display(P1,P2);

   
   
   

   Note that the parameter grid = [30,30], in the contour plot, determines how finely the contours are plotted (if they come out rough, increase the numbers, but don't make them much bigger than 50). In the gradient plot, the grid determines how many arrows are plotted. Both of the plot commands end in a colon (":") so they don't display anything, but save their data in variables P1 and P2, which are plotted simultaneously in the display command.

4. Combined gradient and contour plots can be helpful in classifying critical points (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 (function decreasing)

   • Saddles: some nearby gradients point toward, others away.

Exercises to Hand In

1. Draw a gradient/contour plot for the function in the rectangle (usually written as simply ). Use this to locate, approximately, the critical points. Classify them: max, min or saddle; explain how you know.

2. Do the same as exercise 1, except use the rectangle and the function:

   .

3. Do the same as exercise 1, except use the rectangle and the function: .

4. Do the same as exercise 1, except use the rectangle and the function: