Multivariate Functions

In Precalc, Calc I, Calc II, and just about all math classes prior to Calc III, you study functions mapping \(\mathbb R \to \mathbb R.\) Calculus III, though, is all about increasing the dimension in which we work. In the study of parametric/vector functions, we study functions mapping \(\mathbb R \to \mathbb R^n\), i.e. we increase the dimension of the range. We often think of the input as time so that we’re studying motion.

Now, we’re going to study functions mapping \(\mathbb R^n \to \mathbb R\), i.e. we increase the dimension of the domain.

Bivariate functions

A bivariate function is a real valued function of two variables, i.e. \(f:\mathbb R^2 \to \mathbb R\). Let’s start there.

Graphs of bivariate functions

The graph of a bivariate function is the set \[ \{(x,y,z)\subset\mathbb R^3: z = f(x,y)\}. \] In general, this looks like a surface:

basic_bivariate_pic

You can use the menu below to examine several of these things.

viewof select1

comments

graph3d

Contour plots

A contour plot offers another way to visualize bivariate functions. Given \(f(x,y)\), the idea is to consider the family of curves \[f(x,y) = k,\] for a select set of values of \(k\). If \(f(x,y)=x^2+y^2\), then each contour is a circle centered at the origin, for example.

The select menu below allows you to examine several of these things. It might be instructive to examine the common values in both menus.

viewof select2

comment

contour_pic

The peaks function

The folks at Mathworks created a fun function called the peaks function that’s interesting to view as both a graph and as a contour plot. The formula for the peaks function is \[ f(x,y) = 3 \, (1-x)^2 e^{-x^2-(y+1)^2}-10 \, e^{-x^2-y^2} \left(-x^3+\frac{x}{5}-y^5\right)-\frac{1}{3} \, e^{-(x+1)^2-y^2}. \] Yeah, it’s complicated. Effectively, this is a combination of shifted normal distributions in two variables \(e^{-(x^2+y)^2}\).

The graph of the peaks function looks like so:

peaks_graph

Here’s a contour plot:

peaks_contour_pic

And here’s another graph that emphasizes the relationship between those two:

peaks3D

The peaks function was built to illustrate some of the challenges involved in multivariate optimization.

Higher dimensions

You can talk about functions of as many variables as you might want.

Examples

Three variables: The ideal gas law relates pressure \(P\), volume \(V\), amount (i.e. number of moles) \(n\), and temperature \(T\) via \(PV=nRT\). Note that \(R\) is a constant called the ideal gas constant. We could solve this for \(P\) to get pressure as a function of the three variables \(n\), \(T\), and \(V\): \[P = \frac{nRT}{V}. \]

Four variables: The mass of a box of uniform density is the product of its length \(\ell\), width \(w\), height \(h\), and density \(\rho\): \[ m = \ell \times w \times h \times \rho. \]

Five variables: The force between two objects with masses \(m_1\) and \(m_2\) with a displacement vector \(\langle x,y,z \rangle\) between them can be expressed as a function of five variables: \[ F(m_1,m_2,x,y,z) = G \frac{m_1 m_2}{x^2 + y^2 + z^2} \]

I could go on.

Even more variables

The graph of a function of three variables lives in four dimensional space so I guess we don’t draw those.

We can draw the contours of functions of three variables, though, and they are often called level surfaces in that context. Here are a few examples:

viewof exampleSurface

exampleSurface.comment

pic

Better, we can plot collection of level surfaces. When doing so, it might make sense to slice them in some way to reveal things that would otherwise be obscured. We could also include a fourth variable using color to yield an animation. The animation below, for example, indicates how a solid, hot sphere might cool when dropped in an ice bath.

cooling_sphere

play

This example uses partial differential equations and you can learn more about the implementation in this dynamic notebook.

Limits and continuity

The basics of limits and continuity don’t really change when we move to functions of more variables.

What is a limit?

When we write \[\lim_{(x,y)\to(a,b)} f(x,y) = L\] we mean that we can make the value of \(f(x,y)\) as close to \(L\) as we want by taking the point \((x,y)\) as close as necessary to the point \((a,b)\).

Often, you can compute a limit simply by plugging in the values \(x=a\) and \(y=b\). For example, \[ \lim_{(x,y) \to (2,\pi)} (\sin(xy) + x^2 - y) = \sin(2\pi) + 2^2 - \pi = 0 + 4 - \pi. \] I mean, if \(x\) is close to \(2\) and \(y\) is close to \(\pi\), then \(xy\) is close to \(2\pi\) so that \(\sin(xy)=0\), etc.

The limit laws

Pushing this further, all the basic limit laws we know and love are applicable. For example, \[ \lim_{(x,y)\to(a,b)} f(x,y) = L \: \text{ and } \lim_{(x,y)\to(a,b)} g(x,y) = M, \] then \[ \lim_{(x,y)\to(a,b)} (f(x,y)+g(x,y)) = L+M. \] These limit laws are detailed as Theorem 4.1 in your text.

A curious limit

Of course, limits really become interesting when we have a division by zero. The corresponding limit law, \[ \lim_{(x,y)\to(a,b)} \frac{f(x,y)}{g(x,y)} \;=\; \frac{L}{M}, \quad \text{when } M \neq 0, \] effectively says that you’ve got to somehow avoid division by zero. In Calc I, you learn some algebra tricks to deal with this.

In two dimensions, a further complication arises because \((x,y)\) can approach \((a,b)\) from one of many directions. A function that’s often used to illustrate this is \[ f(x,y) = \frac{xy}{x^2+y^2}. \] Here’s the graph of that function:

pic3d

You can see that the function varies widely near zero. The typical way to analyze a situation like this is to consider \[ \lim_{(x,y)\to(a,b)} f(x,y) \] where the point \((x,y)\) is restricted to some curve. For our function, it might make sense to restrict \(f\) to lines that go through the origin. For example, along the line \(y=x\), we have \[f(x,x) = \frac{x\times x}{x^2+x^2} = \frac{1}{2}.\] Along the line \(y=-x\), though, we have \[f(x,-x) = \frac{x\times (-x)}{x^2+(-x)^2} = -\frac{1}{2}.\] Thus, from one direction the limit is \(1/2\) while, from another direction, the limit is \(-1/2\). We can see this clearly reflected in the contour diagram, as well:

strange_contour_pic

As far as the unrestricted limit goes, we’ve got to say that it does not exist, i.e. \[ \lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y^2} \: DNE. \]

Using Desmos

It’s pretty easy to plot functions of two variables with Desmos3D:

• A wavy checkerboard{target}
• Varying paraboloids
• The Peaks function

You can also plot level surfaces with Desmos3D:

• A two-holed torus
• A family of ellipsoids indexed by size

And you can plot good old level curves with plain old Desmos:

• Varying paraboloid contours
• Contour of the Peaks function

Next up

The next topic you’ll probably explore would be partial derivatives, which is explored in Chapter 3 of your text.

Problems

1. The contour diagram of \(f(x,y)=(x + y)\cos(x)\cos(y)/2\) is shown in Figure 1. Mark the locations of any maxima, minima, or saddle points that you see.

2. Match the groovy function below with the groovy graph shown in Figure 2.

   1. \(\vec{p}(t) = \langle 2\cos(t), \sin(t) \rangle\)
   2. \(\vec{p}(t) = \langle 2t\cos(t), t\sin(t) \rangle\)
   3. \(\vec{p}(t) = \langle 2\cos(t), \sin(t), t/4 \rangle\)
   4. \(f(x,y) = 1 - (4 x^2 + y^2)\)
   5. \(f(x,y) = e^{-(4 x^2 + y^2)}\)
   6. \(x^2 + 4 y^2 + 4 z^2 = 4\)

3. Sketch contour diagrams of the following functions:

   1. \(f(x,y) = 9x^2 + y^2\)
   2. \(f(x,y) = 9x^2 - y^2\)
   3. \(f(x,y) = 9x - y\)
   4. \(f(x,y) = x\,y\)

Figure 1: A contour diagram for \(f(x,y)=(x + y)\cos(x)\cos(y)/2\)

Figure 2: Some groovy graphs

import { 
    basic_bivariate_pic,
    peaks_graph,
    peaks_contour_pic,
    style
  } from '3a85784b45b30ba1';
import {
  viewof select1,
  comments,
  graph3d,
  modify
} from '@mcmcclur/functions-of-two-variables';
import {
  viewof select2,
  comment,
  contour_pic
} from '@mcmcclur/contour-plots';
import {
  peaks3D
} from '@mcmcclur/gradient-ascent';
import {
  pic3d,
  strange_contour_pic
} from '@mcmcclur/a-strange-discontinuity';
import {
  pic, 
  viewof exampleSurface
} from '@mcmcclur/examples-of-level-surfaces';
import {
  play,
  cooling_sphere
} from '@mcmcclur/cooling-of-a-sphere'

modify

style