Intro to Multivariable Functions: Graphs, Contour Plots, and Partial Derivatives

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So, so far we've seen things about vectors, equations of planes, motions in space, and so on. Basically, we've done

geometry in space. But, calculus really is about studying functions. So, now we are going to

actually move on to studying functions of several variables. So,

this new unit that what we'll do over the next 3 weeks or so will be about functions of

several variables and their derivatives. Okay? So,

first of all, we should try to figure out how we are going to think about functions.

So, remember if you have a function of one variable, that means, you know, you have a

quantity that depends on one parameter, maybe F depends on a variable X. And for example, you know, the function

that you all know is f of x equals sin x. And the way we would represent that is

using maybe by plotting the graph of a function. So, the graph of a function, we plot

y equals f of x. And in the case of the sine function, that looks

like this. Okay? So, now let's say that we have actually a

function of two variables. So, that means that the value of F depends actually on two different

parameters. Say, maybe variables are X and Y or they could have any names you want.

Okay? So, given values of the two parameters X and Y,

the function will give a will give us a number that we'll call f of xy that depends on

x and y according to some formula. Okay? Not very surprising so far. So,

for example, you know, I can give you the function f of xy equals x

squared plus y squared. And of course, as with functions of one variable, we don't need things to be

defined everywhere. Sometimes there's a domain of definition. So,

this one is defined all the time, but if I tell you, say, f of xy equals square root of y, well, this is only

defined if y is non-negative. Or if I tell you

f of xy equals 1 over x plus y, well, that's only defined

if x plus y is not zero, and so on. Okay? Now, so these are, you know,

mathematical examples given by explicit formulas. And of course, there's physical examples

or examples coming from real life. For example, you can look at, you know, the temperature

at a certain point on the surface of the Earth. So, you use maybe longitude, latitude, that's X and Y. And then you

have f of xy equals the temperature at that point. Or in fact, because temperature depends

also maybe on how high up you are. It depends on elevation, so it's actually a function of maybe xyz.

And it also depends on time, so in fact, maybe it's a function of t for time, xyz coordinates in space.

So, you see that real world functions can depend on a lot of variables. So, our goal will be to understand how

to deal with that. Okay. So,

now you'll see very soon that actually it's already tricky enough to picture a

function of two variables. So, we are going to focus on the case of functions of two variables.

And then we'll see that if we have more than two variables, then it's harder to plot the function. We cannot draw what

the graph looks like anymore. But, the tools are the same. The notion of the partial derivatives, gradient

vector, and so on. All the tools that we will develop work exactly the same way, no matter how many variables you have.

So, I'll say for simplicity, we'll focus mostly

on two or sometimes three variables.

But, it works the same in any number of variables. Okay, so the first question is how do we

visualize a function of two variables? So, the first thing we can do is try to draw the graph of F. So, maybe

I should say F, which is a function of two variables.

So, the first answer will be we can try to look at its graph. And the idea is the same as with

one variable, namely we look at all the possible values of the parameters X and Y, and for each of them, we plot

a point whose height is the value of a function at these parameters.

So, we'll plot let's say z equals f of xy. And now that will become actually

a surface in space. Okay? So, for each value of X and Y, here we have X Y in the XY plane,

then we'll plot the point in space at position X Y and z equals f of xy.

Okay? And if we take all of these points together, they will give us

some surface that sits in space. Yes? Oh, a function of two variables.

Shorthand. Well, let's say how to visualize a function of two variables if you want.

Okay? So, how do we do that concretely? Say that I give you a formula for F. How do

we try to represent it? So, let's do our first example. Let's say

that I give you the function f of xy equals negative y. Okay?

So, I mean, it looks a little bit silly because it doesn't depend on x, but that's not a problem. It's still a valid

function of x and y. Just happens to be constant with respect to x. So, to draw the graph,

we look at the surface in space defined by z equals minus y. What kind of surface is that?

It's a plane, okay? And if we want to draw it, z equals minus y,

we'll look, well, that's the y-axis, that's the x-axis, that's the z-axis. If I look at what happens in the yz

plane, in the plane of the blackboard, it will just look like a line that goes downward with slope one.

Okay? So, it will do this. And what happens if I change x? Well, if I change x, nothing happens because x

doesn't appear in this equation. So, in fact, if instead of setting x equal to zero, I set x equal to one, I'm

in front of the blackboard, or minus one at the back, well, it it still looks exactly the

same. So, I have this I have this plane

that actually contains the x-axis and slopes downwards with slope one. It's kind of hard to draw.

Um now you see immediately what the big problem with graphs will be, that these

pictures are hard to read. But, so that's our first graph. Okay, any questions so far?

Okay. So, let's say that we have a slightly more complicated function. How do we see it?

So, let's do another example. Let's say I give you f of xy equals 1 minus x squared minus y squared.

So, we should try to picture what the surface Z equals 1 - X squared - Y squared looks

like. So, how do we do that? Well, maybe you're

actually you know, very fast and you've already figured out what it looks like. But, if

not, then we need to work, you know, piece by piece. So,

maybe it will help if we understand first what it does in the plane of the

blackboard. Okay? So,

if we look at it in the YZ plane, that means we set X equal to zero. And then,

Z becomes 1 - Y squared. What is that? It's a parabola pointing downward and starting at one.

Okay? So, we should draw maybe this downward parabola.

Starts at one and it cuts the Y axis at one. Okay? When Y is one, that gives us zero.

What do we see? Let's, you know, so we might have an idea of what it might look like or maybe not.

Let's get more slices. Let's see what it does in the XZ plane.

This other vertical plane that's coming towards us. So, in the XZ plane,

as we set Y equal to zero, and we get Z equals 1 - X squared. It's again a parabola coming downward.

Okay? So, I'm going to try to draw a parabola that goes downward, but now to the front

and to the back. So, we are starting to have a slightly

better idea, but we still don't know whether the cross section of this thing might be, you know, round, square,

uh something else. So, if we want more confirmation, we might want

to also figure out maybe where does this surface intersect the XY plane?

So, we hit the XY plane when Z equals zero. That means

1 - X squared - Y squared should be zero. That becomes X squared + Y squared

equals one. That is a circle of radius one.

That's the unit circle. So, that means that here we actually have

the unit circle. And now you should imagine that you have this thing that when you slice it by a

vertical plane looks like a downward parabola, and it's actually a surface of revolution. You can rotate it around the

Z axis. Okay? Now, if you stare long enough at that equation, you'll actually see that

yes, actually a posteriori, we know that it had to be like that. But,

see, so these are useful ways of trying to guess what the graph looks like. Of course, we have other ways to just

ask your computer to do it. And then, you know, you'll get that kind of formula.

Okay. Well, I can leave it on if you want, but Uh no, because I plotted a different

function that I will show you later. So, it goes this way. Uh I mean, if you want 1 - I mean, it's really going

downward. Yes, I agree that the sheet is upside down, but that's because I plotted something else.

Okay. So, in fact, well, let me Yeah. So, what I plotted in my computer

was actually X squared + Y squared. That looks like that. See, it's the same with a parabola going

upwards. Uh if you want to see more examples, I have

various examples to show. Well, here's the graph of a function Y squared minus

X squared. See, so that one is kind of interesting. It looks like a saddle.

Uh if you look at it in the YZ plane, then it's a parabola going up, Z equals Y squared.

And that's what we see to the left and to the right. But, if you put it in the XZ

plane, then it's a parabola going downwards, Z equals minus X squared. So,

we have a parabola going downwards in one direction, upwards in the other one, and together they form this surface.

And of course, you can plot, you know, much more complicated functions. So, this one, I well, if you can read very

small things, then you can see the formula, but it doesn't matter. Just to show you that

you know, you can put the formula into a computer and it will show you a picture. Okay.

So, that's pretty good, but I mean, you can see that it can get a bit cluttered because maybe there's

features that are hidden behind or maybe we have trouble seeing if we don't have a computer. You know, that looks very

readable, but this is kind of hard to visualize sometimes.

So, there's another way to plot functions of two variables. And

that's called the contour plot. So, the contour plot is a very elegant elegant solution to the problem that

it's difficult to draw space pictures on a sheet of paper or on a blackboard. So, instead, let's try to represent a

function of two variables by just a map. You know, the same way that when you walk around, you have actually

geographical maps that fit on a piece of paper that tell you about what the real world looks like.

So, what the contour plot looks like is something like this. So, it's

an XY plot, and on it you have a bunch of curves. And what the curves represent

are the elevations on the graph. So, for example, this curve might correspond to

all the points where f of XY equals one. And that curve might be all the points where

f equals two and f equals three and so on. Okay?

So, when you see this kind of plot, you're supposed to think that the graph of the function

is, you know, somewhere sits in space above that. It's like a map telling you how high things are.

And you know, what you would want to do with a function really is be able to tell

quickly what's the value at a given point. Well, let's say that I want to look at that point. I know that f is

somewhere between one and two. You know, actually it's much faster to read than the graph. On the graph, I might have to

look carefully and then measure things and so on. Here, I can just read the value of f by comparing with the nearby

lines. Okay, so let me try to make that more precise. So, this

shows all the points So, so

where f of XY equals some fixed values.

Some fixed constants. And these constants typically are chosen at regular intervals.

For example, here I chose, you know, one, two, three, and I could continue with zero, minus one, and so on.

So, one way to think about it, how does this relate to the graph?

Well, that's the same thing as cutting I mean, if we say we slice

the graph by horizontal planes. So, horizontal planes have equations of

the form Z equals some constant, Z equals zero, Z equals one, Z equals two, and so on.

So, maybe the graph of my function, you know, will be some sort of blob out there. And if I slice it by

the plane Z equals one, then I will get

the level curve, which is the points where f of XY equals one.

And so, that's called a level curve of f. Okay? And so,

we repeat the process with maybe Z equals two, and we get another level curve.

And so on. And that's when we squish them all on the map, and that's how we get the contour plot.

Okay? So, each of these lines, you know, imagine that this is like some mountain or something that you're hiking on. Each

of these lines tells you how you could move to stay at a constant height, you know, if you want to get to the other

side of a mountain but without ever going up or down, you just walk along that path. And it will get you there

without effort. So, in fact, if you I've been talking about hiking on mountains, well,

that's exactly what a topographical map is about.

Sorry, so I need to zoom a bit. So, at the topographic map, this one is from the US

Geological Survey, um shows you basically all the level curves

of a altitude function on you know, a piece of land.

So, you know that if you walk along these curves, you will stay at the same height. And you know that if you walk

towards Oh, these don't have numbers. Yeah, let me find a place with numbers. Here, there's a 500 right

in in the middle. Um

So, you know that if you walk on the line that says 500, you stay always at 500 m in elevation. If you

walk towards a mountain that I think is below it, then you will go up and so on. So, you can see, for example,

here there's a peak and while here there's a valley with actually the river in it and the altitudes go down and then

back up again on the other side. Okay, so that's an example of a contour plot of a function. Of course, we don't

have a formula for that function. Uh but we have a contour plot. And that's what we need actually to

understand what's going on there. Okay, any questions? Nope.

Okay, so another example of contour plots, well, you've probably seen various

versions of these, you know, temperature maps. So, that's supposed to be how warm it is

right now. Um So, you know, this one is color-coded

instead of having curves, it has all of these colors, but the effect is the same. If you look at the separations

between consecutive colors, these are the level curves of a function that tells you the temperature at a given

point. Okay, so these are examples of contour plots in real life.

Okay, um no questions?

Nope. Okay, so basically one of the goals that one should try to achieve at this

point is, you know, becoming familiar with the plot, the contour plot, Sorry, the the contour plot, the graph,

and how to view and deal with functions. Oh. Oh, oh.

Okay. So, let's do an example.

Well, let's do a couple of examples. So, let's start with f of xy equals -y. I'm I'm going to take the same two

examples as that to start with so that we, you know, we see really the relation between

the graph and the contour plot. So, let's try to plot it.

So, where is the level curve f equals zero for this one? Well, f is zero when y is zero.

So, that's the x-axis. Okay, so that's the level zero. Where is the level one? Well, f is one

when -y is one. That means when y is -1. So, I'll go to -1

and that will be where my level one is. And so on. F is two when y is -2. F is -1 when y is one.

And so on. Okay? Is that convincing?

Do you see how we got that? Okay, let me do it again. I don't see anybody nodding, so that's

kind of bad news. Um so, if I want to know where is the level curve, say, one,

I try to set f equal to one. Let's do this one. F equals one

means that -y is one means that y is -1

and y equals -1 is this horizontal line on this chart. Okay.

And same with the others. So, you can see on this map that the value of a function doesn't depend on x.

If you move parallel to the x-axis, nothing happens. If you move in the y direction, it decreases at a

constant rate. That's why the contours are evenly spaced. See, how how spaced apart they are tells

you actually how steep things are. So, that corresponds exactly to that picture, except that here we draw x

coming to the front and y to the right, so you have to rotate the map by 90° to get to that. It's an unfortunate

consequence of the usual way of plotting things in space. Okay, so these horizontal lines here

correspond actually to horizontal lines. Yeah.

Okay. Second example, let's do 1 - x² - y².

Okay? Or maybe I'll rewrite it as 1 - x² + y². It's really the same thing.

So, x, y. Let's see, where is this function equal to zero?

Well, we said f is zero on the unit circle. Okay. So, the zero level.

Well, let's say that this is my unit. That's where it's zero. Uh what about f equals one?

Well, that's when x² + y² equals zero. Well, that's only going to be here, so you know, that's just a single point.

What about f equals -1? That's when x² + y² equals two. That's a circle of radius

square root of two, which is about 1.4, so it's somewhere here. Then minus So, let's say -1. -2

similarly will be x² + y² equals three. Square root of three is about 1.7. And then -3 will be

of radius two and so on. So, what I want to show here is that So, that's -3. That's

-2. What I want to show here is that they're getting closer and closer apart. Okay,

so first it's concentric circles that tells us that we have a shape that's a solid of The graph is going to be a

surface of revolution. Things don't change if I rotate. And second, the contour the level lines

the level curves are getting closer and closer to each other. That means it's getting steeper and steeper because I

have to travel a shorter distance if I want my altitude to change by one. and steeper. And that's what we observe on

that picture over there. Okay. So,

just to show you a few more. Where did I put my So, for x² + y², the contour plot

looks like this. Maybe actually I'll make it Okay. So, it looks exactly the same as

this one, but the difference is if you can see the numbers, which are not there, so you can

see them, uh then you would know that instead of decreasing as we move out, this one is increasing as we go out.

Okay, so that's where we use actually the labels on the level curves that tell us

whether things are going up or down. But the contour plots look exactly the same.

For the next one I had I think was y²

- x². So, the contour plot Well, let me actually zoom out.

So, the contour plot looks like that. So, the level curve corresponding to zero is actually two diagonal lines.

And if you look on the plot, s- say that you started with the saddle point in the middle and you try to stay

at the same level. So, it looks like a mountain pass, right? Well, there's actually four

directions from that point that you can go staying at the same height. And they're

actually on the map, they look exactly like these two crisscrossing lines. Okay, so these are things that go on the

side of the two mountains that are to the left and right and stay at the same

height as the mountain pass. On the other hand, if you go along the Y direction to the left or to

the right, then you go towards positive values.

And if you go along the X axis, then you get towards negative values. Okay.

Uh the equation for the the function was Y squared minus X squared. Okay.

So, you can try to plot them by hand and confirm that it does look like that, but I trust I trust my computer.

And finally, this one. Well,

so the contour plot looks a bit complicated, but you can see two things. In the middle, you can see these two

regions with these concentric circles. They're not really circles, but you know, these closed curves

that are concentric and they correspond to the two mountains. And then, at some point in the middle,

we have a mountain pass and there we see the two crossing lines again like on the plot of X squared minus Sorry, Y squared

minus X squared. And so, at this saddle point here, if we go

north or south, then we'll go down on either side to the valley. And if we go east or west, then we'll go towards

the mountains. We'll go up. Okay, does that make sense? A little bit?

Okay, so you know, reading plots is not easy, but hopefully,

we'll get used to it very soon. Okay, so actually, let's say one.

Okay, so I want to point out one thing. The contour plot tells us actually what

happens when we move, when we change X and Y. So,

if I change the value of X and Y, that means I'm moving east, west, or north, south

on the map. And then I can ask myself, does the value of the function increase or decrease in each of these situations?

Well, that's the kind of thing that the contour plot can tell me very quickly.

So, Okay, so say for example that I have a piece of contour plot

that looks, you know, like that. Maybe this is F equals one

and this is F equals two and here this is F equals zero. And let's say that I start

at a point say at this point. Okay, so here I have X0, Y0

and the question I might ask myself is if I change X or Y, how does F change? Well, the contour plot tells me

that if X increases

and I keep Y constant, then what happens to F of XY?

Well, it will increase because if I move to the right, then I go from one to a value bigger than one. I don't know

exactly how much, but I know that somewhere it's somewhere between one and two. It's more than one.

If X decreases, then F decreases. Okay, and similarly, I can tell that if Y increases,

then F Well, looks like if I increase Y, then F will also increase.

And if Y decreases, then F decreases.

And you know, that's the kind of qualitative analysis that we can do easily from the contour plot.

But maybe we'd like to actually be more precise than that and tell how fast F changes if I change X or Y.

Okay, so to find rate of change, that's exactly where we use derivatives. So,

we are going to have to deal with partial derivatives. So, I will explain to you soon why

partial. So, let me just remind you first, if you have a function of one variable,

then so let's say F of X, then you have a derivative F prime of X

is also called dF dX and it's defined as a limit when delta X goes to zero

of the change in F. Uh sorry, it's not going to fit. I need to go to the next line.

It's going to be the limit as delta X goes to zero of the rate of change.

So, the change in F between X and X plus delta X divided by

delta X. Sometimes you write delta F for the change in F divided by delta X.

Then you take the limit of this rate of increase as delta X goes to zero. Now, of course, if you have a

formula for F, then you know, at least you should know uh

I suspect most of you know how to actually take the derivative of a function, you know, from its formula.

So, now, how do we do that? Oh, sorry, and I should tell you also

what this means on the graph. So, if I plot the graph of a function, you know, and I have my point X and here

I have F of X, how do I see the derivative? Well, I look at the tangent

line to the graph and the slope

of the tangent line is F prime of X. Okay. And truth in advertising, not every

function has a derivative. You have functions that are not, you know, regular enough to actually have a

derivative. So, in this class, we are not going to actually worry too much about

differentiability, but it's good at least to be aware that you can't always take a derivative.

So, yes, and what else do I want to remind you of?

Well, we also have an approximation formula which says that, you know,

if we have the value of F at some point X0 and then we want to find the value at a nearby

point X close to X0, then our best guess is that it's F of X0 plus the derivative F

prime at X0 times delta X or if you want, X minus X0. Okay.

Uh is this kind of familiar to you? Yeah, I mean, maybe with different notations. Maybe you called that delta X

or something. Maybe you called that X0 plus H or something, but it's

it's the usual approximation formula using the derivative. If you put more terms, then you get the

the graded Taylor approximation, but I know that you guys don't like, so. Okay.

So, the question is, how do we do the same for a function of two variables, F of

XY? So, the difficulty there is we can change X

or we can change Y or we can change both. And presumably, the manner in which F

changes will be different depending on whether we change X or Y. So, that's why we have several different

notions of derivative. So, Okay, so we have a notation.

Okay, so this is a curly D and it is not a straight D and it is not a delta. It's a D that kind of curls backwards like

that. And the way you read this symbol is partial.

Okay, so it's this it's the special notation for partial derivatives. And essentially what it means is we are

going to do a derivative where we care about only one variable at a time. That's why it's partial, it's missing

the other variables. Okay, so a function of several variables doesn't

have a usual derivative, it has only partial derivatives for each variable. So, the partial derivative

DF partial F partial X at X0 Y0 is defined

to be the limit when I take a small change in X delta X of

the change in F divided by delta X.

Okay, so here I'm actually not changing Y at all. I'm just changing X and looking at the rate of change with

respect to X. And I have the same with respect to Y partial F partial Y is the limit

so I should say at a point X0 Y0 is the limit

as delta Y tends to zero of So, this time I keep X the same, but I

change Y. Okay, so that's the definition of a partial derivative.

And we say that a function is differentiable if these things exist. Okay, so most of the functions we'll see

are differentiable and we'll actually learn how to compute the partial derivatives without having to do this.

Of course, we'll just use the usual methods for computing derivatives. So, in fact, I claim you already know

how to take partial derivatives. So, let's see what it means

geometrically. So, geometrically my function can be represented by its

graph. And you know, I fix some point X0 Y0. And then I'm going to ask myself what

happens if I change the value of well

if I change the value of X keeping Y constant. So, if I keep Y constant and change X, it means I'm

moving forward or backward parallel to the X axis. So, that determines for me a vertical

plane. Okay? Parallel to the XY plane

when I fix Y equals constant. And now if I slice the graph by that

I will get some curve that goes

you know, it's a slice of the graph of F. And now what I want to find is how F

depends on X if I keep Y constant. That means it's the rate of change if I move along this curve.

So, in fact if I look at the slope of this thing So, if I draw the tangent line to this

slice then the slope will be

partial F over partial X. I think I have a better picture of that somewhere.

Yes, here it is. Okay, that's the same picture just with different colors.

So, I take the graph, I slice it by a vertical plane, I get a curve, and now I take the slope of that curve, and that

gives me the partial derivative. And to finish, let me just tell you how

Sorry, and I should say partial F partial Y is the same thing, but slicing now by a plane that goes in the YZ

directions. Okay, so I fix I fix um X equals constant, that means I slice by a plane that's parallel to the

blackboard. I get a curve, and I look at the slope of that curve.

Okay, so it's just a matter of forgetting one variable, setting it constant, and looking at the other one

only. So, how to compute these things? Well

we actually to find Oh, there's a piece of notation I haven't told you yet.

So, another notation you will see, I think this is what one uses a lot in

physics, and this is what one uses a lot in applied math, which is the same thing as physics, but with different

notations. Okay, so there's two different notations, partial

F partial X or F subscript X, and they're the same thing. Well

we just treat Y as a constant and X as variable.

And vice versa if we want to find partial with respect to Y. So, let me just finish with one quick

example. Let's say that I give you F of XY equals X cubed Y

plus Y squared. Then partial F partial X

well, let's take the derivative. So, here it's X cubed times a constant. Derivative of X cubed is 3 times X

squared times the constant plus What's the derivative of Y squared?

Zero, because it's a constant. If you do instead partial F partial Y then this is actually a constant times

Y. The derivative of Y is 1, so that's just X cubed. And the derivative of Y squared is 2Y.

Okay, so it's fairly easy. You just have to keep remembering which one is a variable, which one is not.

Okay, so more about this next time, and we'll also learn about maxima and minima in

several variables.