Mathematics 41, section 1 – Analysis 3 Lab Day 6 – Local Linearity for f(x, y) March 3, 1998

Background

Yesterday in class we saw that one possible way to say that a function f(x, y) has a “total derivative” at (x_{0}, y_{0}) might be to borrow the idea of local linearity from one variable calculus. There, for g : R → R, if g′(x_{0}) exists, then as we “zoom in” closer and closer to the point (x_{0}, g(x_{0})), the portion of the graph y = g(x_{0}) we see becomes indistinguishable from a straight line of slope g′(x_{0}). The analogous property for a function f(x, y) would be that as we zoom in closer and closer to a point (x_{0}, y_{0}, f(x_{0}, y_{0})), then the graph should be become indistinguishable from a plane in R^{3}. Moreover, we saw that if this local linearity property is true, the plane should be defined by z = f(x_{0}, y_{0}) + \frac{∂f}{∂x}(x_{0}, y_{0})(x − x_{0}) + \frac{∂f}{∂y} (x_{0}, y_{0})(y − y_{0}). Today, we will use Maple to

• investigate several examples,

• determine whether they are locally linear near given points, and

• see how local linearity of the graph z = f(x, y) relates to local linearity of the slices of the graph in vertical planes.

Maple Function Definitions

Start your Maple session by loading the plots package as usual.

To be able to “zoom in” on a graph or its slices conveniently, we will need to have functions that generate plots of the graph or of a slice, in which we can easily fill in new functions, and control the range of values that are plotted. Here are two such function definitions.

1. Zooming in on the graph z = f(x, y) near a given (x_{0}, y_{0})

For this, we can use:

           funcplot:=(f,h,x0,y0) -> plot3d(f(x,y),x=x0-h..x0+h,
               y=y0-h..y0+h,axes=BOXED,scaling=CONSTRAINED);

Take a look at the plot3d command on the right of the arrow first. You should note the meaning of h in defining the ranges of x and y values being plotted. To “zoom”, we can take a sequence of decreasing h values such as h = .1, then h = .01, then h = .001 generate plots for each, and compare! The options in the plot3d command will produce axes with equal scales in all plots so you can see the effect of zooming. Finally, the function f can be any function you have defined previously in a session. The name you supply when you use funcplot does not have to be f either. For instance if you had defined a function using

                         g := (x,y) -> x^2 - y^2;

then you could enter

                            funcplot(g,.1,1,1);

to plot the graph z = g(x, y) for .9 ≤ x ≤ 1.1 and .9 ≤ y ≤ 1.1. Note the order of the information supplied when you use funcplot – the name of the function to be plotted comes first, then h, then the coordinates of (x_{0}, y_{0}).

2. Zooming in on a slice of the graph z = f(x, y) near (x_{0}, y_{0})

For this, we want to use the spacecurve command. To specify both a range of values and a slice, we need an h value as above, and a direction vector (u_{1}, u_{2}) in the xy-plane. (This determines the slice in the vertical plane containing the vectors (u_{1}, u_{2}, 0) and (0, 0, 1).) Here is an appropriate command:

           sliceplot:=(f,h,u1,u2,x0,y0) -> spacecurve([x0+t*u1,
          y0+t*u2,.002*h+f(x0+t*u1,y0+t*u2),t=-h..h],color=black,
                     thickness=2,scaling=CONSTRAINED);

This should be relatively self-explanatory from last week’s lab. The only new feature is a “trick” in the definition of the space curve to be plotted. Note the .002*h added on to the z-coordinate. This has the effect of “raising the slice” off the graph slightly for better visibility! Note the order of the information supplied when you use sliceplot – the name of the function to be plotted comes first, then h, then the components of the direction vector specifying the slice, then the coordinates of (x_{0}, y_{0}). For example,

                         sliceplot(g,.1,1,0,0,0);

would plot the slice of z = g(x, y) in the plane with u = (1, 0), h = .1, through the point (x_{0}, y_{0}) = (0, 0).

The output from funcplot and sliceplot can be combined using display3d as in last week’s lab.

Lab Questions

A) Investigate z = x^{2} − y^{2} near (x_{0}, y_{0}) = (0, 0). 1) Are the slices for u = (1, 0), u = (0, 1) locally linear?

2) Is the graph z = f(x, y) locally linear? Explain.

B) Same questions for z = −\sqrt{x^{2} + y^{2}} at (0, 0). C) Same questions for z = −\sqrt{x^{2} + y^{2}} at (1, 2). D) Same questions for

z = \sqrt{|xy|} sin^{2}(arctan(y/x))

at (0, 0). (In Maple format, this function is sqrt(abs(x*y))*sin(arctan(y/x))^2). E) Is it possible for the slices of a graph to be locally linear but for the graph itself not

to be locally linear? Explain. Are any of the graphs above examples of this?

Assignment

Lab write-ups due Wednesday, March 18 (the week after Spring Break).