Chapter 21

Functions of Many Variables21.1 Graphs

In general, you really can’t graph functions of many variables because we see in threedimensions. If you have a function of three variables, you would need four dimensions tograph it. However, in the case that z = f (x,y) a scalar valued function of two variables, youcan do so fairly well, especially with a computer algebra system. You graph y → f (x,y)for many values of x and x → f (x,y) for many values of y. This will result in a nice pictureof a surface. For example, consider the graph of z = f (x,y) where f (x,y) = x2 − y2.

-10

4

0

2 4

z=f(x,y)

20

10

0-2 -2-4 -4

z

To use MATLAB, to draw such a graph, modify the following syntax which was usedfor the above problem. Remember to get to a new line, you type shift enter.

[x,y]=meshgrid(-3:.5:3,-3:.5:3);z=x.ˆ2-y.ˆ2; surf(x,y,z,’LineWidth’,2)

21.2 Review of LimitsRecall the concept of limit of a function of many variables. When f : D(f)→ Rq one canonly consider in a meaningful way limits at limit points of the set D(f).

Definition 21.2.1 Let A denote a nonempty subset of Rp. A point x is said to be alimit point of the set A if for every r > 0,B(x,r) contains infinitely many points of A.

Example 21.2.2 Let S denote the set{(x,y,z) ∈ R3 : x,y,z are all in N

}. Which points are

limit points?

This set does not have any because any two of these points are at least as far apart as 1.Therefore, if x is any point of R3,B(x,1/4) contains at most one point.

Example 21.2.3 Let U be an open set in R3. Which points of U are limit points of U?

They all are. From the definition of U being open, if x ∈U , There exists B(x,r)⊆Ufor some r > 0. Now consider the line segment x+ tre1 where t ∈ [0,1/2]. This describesinfinitely many points and they are all in B(x,r) because |x+ tre1 −x|= tr < r. Therefore,every point of U is a limit point of U .

The case where U is open will be the one of most interest, but many other sets havelimit points.

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