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- As we continue on in our understanding of multivariable functions, it probably is about time for us to talk about what do we mean when we talk about multivariable functions? We've kind of dipped our toe in some sense where we talked about parametric curves, but that was where we had multiple variables in the output. And x of t, y of t, z of t, there's a single input.
0:24
We wanna get to the point where we say, "Look, we are gonna have multiple inputs." Which is good 'cause that's really, it's in the name of the class, right? Multivariable calculus. So we should get comfortable. So we need to start by talking about what are multivariable functions and do some basic, how do we represent them? How do we talk about things like domain range? And so forth and so on. So let's begin. And the best place to begin is talking about, first off, what are functions?
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And the way we think about a function is very simple. See, a function is a rule. You put something in and you get something out. That's it. That's what a function is. You put something in, you get something out on the other side. There's one important thing
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and that is consistency. So that is say, look, if I plug in a two this time and I get an answer, and I come back a year later, I plug two into the same function, I'm gonna get the same answer as I did before. That's what we mean when we talk about consistency. Same inputs always produce the same outputs. Now, if you took a pre-calculus course, you may have heard something called the vertical line test. And the vertical line test said, "Well,
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"let's check to see what's happening for a particular input "and see if I ever get more than one answer." 'Cause if you did you lack consistency and therefore it wouldn't be a function. Now we oftentimes think of functions in math as, oh, I'm plugging in a number, or maybe several numbers and I get a number out. But functions are much, much more broad. All we need to do is have a consistency. So whatever we plug in, we'll get the same consistent answers you plug out.
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There's one function that calculus students really love, they're very passionate about. It's perhaps their favorite function to talk about. And it's called the curve.
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Well, what's the curve? Well, that's assigning the grades. So what happens? Well, a student comes in, they take the course, and they get certain grades on various assessments and they get a final grade. And so where you're gonna say, "Look, "as long as the student gets gets these scores "on this assessment, this is the grade that they'll get." Now this, I'll tell you, this function right here, that's like I said, the most important function for most calculus students.
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Well, our goal of course is to learn and then we don't have to worry about the curve. Now, there are lots of different functions and we're gonna talk about several of them as we go on. So we're used to the one variable functions, y is a function of x. Well, now we're getting into two variables. So we're gonna have two inputs. Traditionally we call them x and y. There's nothing special about the symbols x and y. It really is tradition. You'll see in some problems we might have different symbols.
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That's fine, no problem, No problem. So we can say, "Look, z is a function of x and y." Okay. And what if we have more than two? Absolutely, you can have as many inputs as you want. And it makes sense to have many inputs because a lot of processes depend on multiple things, which can vary. So anytime you have a contribution to a process that's allowed to vary, well that's a variable. And you might have a process that involves dozens and dozens of things which vary.
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So you can have very, very complex functions. So for functions of three variables, we like to think of our symbols, x, y, z as our variables. And since we ran out of the alphabet, we like to think of the next symbol as being w. So w as a function of x, y, and z. So that's what we have. Now we have various ideas, there's the the range of a function. What is that? Well, if you think about the range, it says the following, if I have a function, the range says what comes here?
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What are all the possible outputs? So that's the question about the range. Now, how do you find range? Well, the answer is very carefully. It's very tricky to find range. And oftentimes on even simple looking problems, you have to pull out a little bit of calculus because you have to say, "Well, "what are the largest or smallest values "that are achievable?" In other words, find some extreme points.
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So we're not gonna oftentimes worry too much about range because unless our function is very special, it's not so easy to find. Now the domain, okay, that's a little bit simpler. What's the domain? Well, the domain are all the inputs. So if you go back to our picture, when we talk about the domain, we're asking the question, what can go into our function? Now when we're thinking about functions, which we express as some formula, there tend to be three potential problems
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that we need to worry about.
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So if we're thinking about a formula, we say, "Look, as long as we avoid our problems, "life is good and we're ready to go." So what are they? Well, no surprise, divide by zero, division by zero is bad and we just can't handle it. So if we ever have a division by zero anywhere in our expression, we say, "Look, "we can't make any progress, we're stuck." Square roots of negatives. We can't take a square root of a negative number. We have to assume that our square root says
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either a zero or something greater than equal to zero.
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Also, in a similar fashion, log. Log has issues and we'll talk about log of non-positives. So in other words, you can't put a negative number into a log.
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Now these aren't the only issues, but these are the primary ones. You might have things such as tangent, for instance. You say, "Aha, what about tangent? "Well, that doesn't define for our values." And you're right, there are some restrictions, but if you think of tangent as a ratio, sine over a cosine, you say, "Oh yeah, "that's just the same as saying one is cosine zero." Which is when a zero division by zero problem. So there's sort of a hidden division by zero going on there.
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But these are the primary things to look at. So remember when you're looking for a problem that says find the domain, think about, all right, I have three potential problems, let me make sure I avoid them. And then life is good.
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Now before we go on, I do want to talk a little bit about how do we think of functions? Because oftentimes when you say, "Visualize a function," you don't think, oh, there's this machine and I put something in here (imitates machine whirring) and something comes out there. You know, that's not how we think of a function. If someone says, "Hey, think of a function," the way we do it is we say, "Oh, okay, "I'll think of it in sort of this way." I have my input down in one axis,
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I have my output on another axis, and then I have some sort of curve. So when we think of a function, this is how we oftentimes think of a function, something like this. And we call this as a function, y equals f of x. So when we talk about this sort of representation, what do we really mean?
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Well, what we mean is we've gone through and said, "Hey, for every potential input, "so in other words, down here, there's lots of x values. "And for everyone that makes sense, we come up "and we get a point." And we just repeat this over and over and over again so that we get many, many, many points. And so what you see is when you see a curve like this, it's really the collection of all points that have the form x comma f of x. And it forms this nice shape, at least if our curve is reasonable.
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But we like to work with reasonable functions, so that's not too surprising. So what are we to think of for functions of two variables? All right, well for that we should go up and say, "All right, same setting, right?" We should have a two dimensional input. So we're gonna think of x, y, our floor if you will, the xy plane. Those will correspondence to our inputs, and then z, that will correspond to our output. And now what do we have? Well, we do the same thing that we did before,
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which is to say the following, "Look, "you give me a point down here in the xy plane,
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"then it will come up to some height "and that will be a point on my function, one point." Well, but then you repeat, repeat, repeat, repeat. And what ends up happening is for every single point down in your domain, you're gonna get something upstairs. And so you're gonna build up a surface.
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And so this is our surface that we form. So every point down here in our domain corresponds to some point up here in our function. And we call this z equals f of xy.
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So now our new setting is you're not thinking of curves, we're thinking of surfaces and space. And okay, well already we can see it's getting more complicated.
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And they might say, "Well wait, what about here? "Functions of three variables, how do we draw those?" And you're probably thinking, oh, okay, well that's not so bad, right? Well, what do you do? Well, here you do a one dimensional thing, 'cause it's one dimensional input, one thing that could vary. Here there was two dimensions that could vary, so of course you just say, "Look, "here's my three dimensions, I'll call them x, y and z. "And here's my fourth dimension, w, and what do I get?"
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Well, you get some sort of crazy thing,
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which really means you can't really visualize it very well. It's not so clear what it looks like because I mean what's happening? For a function of a single variable, you get a curve, which is really like a one dimensional thing,
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sitting in two dimensional space. When I had a function of f of xy, I got a surface, sort of a bumpy, but again, it's a bumpy, two-dimensional object sitting inside a three dimensional space. So for a function of three variables, now we're gonna have sort of some weird three dimensional object sitting inside of a four dimensional space, which, ah, we really have a hard time visualizing. So great news, we don't try to visualize them,
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at least not directly. We have a hard time doing that. Now that's not to say that we can't interpret things and we'll actually see there are some fairly reasonable interpretation assigned to our functions of three variables as we move along. So primarily when we think about plotting, we'll think about a function of one variable or a function of two variables.
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But let's talk about one way we can understand a function of three variables a little bit better. And that is through what are called level curves. You also might hear these referred to as contour lines.
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So sort of let's visualize what it might look like. And what you might see is some sort of surface. So here's a surface, if you look at it, it's a paraboloid that's been turned upside down.
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And we say, "Okay, so here's our surface." So instead of trying to understand the picture in three dimensions, what we're doing is we're slicing it at various heights. And then for every single height we say, "Hey look, "you can go straight down." So for example, this layer here corresponds with this layer there
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and it's true for every single layer. So our next layer up, here, would correspond to the picture here and so forth and so on. So that when we look at these contour lines, we're really getting is sort of a sense of elevation. If you've ever been out hiking, you can buy something called a topographical map. And what the topographical map does is it says, "Well, where are the various elevations at "and how do they all connect together?" And so you can use that information to get a sense
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of what's a 3D picture of the place where you're out hiking. And so now of course some topographical maps are more informative than others. You can imagine for instance, that we were to go out and to get a topographical map of some place such as, say, Kansas and it would look something like that, which is very flat. Well that's not giving us a lot of information.
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So let's step back, I've been talking about level curves. So let's be really explicit about what a level curve is. Our goal in level curves is trying to find a way to visualize something in three dimensions in a two dimensional way. So what we do is we say, "Well look, let's fix a height." So we come up and we're talking about slicing, we're really talking about taking a plane, something of the form z equals k. So that's this plane here. And then we have our surface, this is our surface,
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z equals a function of x of y. The intersection is where they line up. So if we have z equals a constant and we have z equals the function of x and y, that becomes the function of x and y equals the constant. So that is what's being shown here. Now the nice thing is you say, "Wait, look, there's no z." So that defines a curve down in the plane. So that's why when you come down here, there's this curve down in the plane. So that's what these contours are doing. And so you can get information
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and so we can sketch for instance, let's just do a quick sketch of one possible contour map. And we might have something that's... (pen scratches)
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I'd say that's what our contour map does. It's possible, it's possible. Now you'll be a little bit skeptical here 'cause I haven't marked something.
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Well if you just see the lines you don't quite have all the information that you need. The other thing you need to know is the heights. And so the height would correspond to sort of at what level you took. So it's like the elevation in terms of a topographical map. You might also see this, by the way, in weather maps, if you've taken a look at a weather map for certain days, you'll see that they've color coded everything according to different heats. And then there's these nice various bands.
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So you should also label, so suppose we label things, so we'll label this one as two. Why not? Now you notice if you ever have two lines crossing, they have to have the same value, so these are all twos over here. And we'll mark this one as a one. This is a zero. Here is a three, a four, a five, and a six. And, oh this can be a three and a four. This can be a three and this will be a three as well. Okay, so there we have information and now you can actually start to read information,
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for instance. You can imagine, for instance, if these lines represented different elevations where you see things really bunched together, that's where the curve, or I rather should say the surface, is very steep. It would correspond to something like a cliff. Whereas where if their spaced further apart, that's more of a gentle slope, something you might want to hike up. You can also say, "Ah, look, "I see that I'm going higher and higher up." So this is a high point, or we might call a max.
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And here we're going down, down, down. This is a low point, that's a zero, that's a min and so forth and so on. So by looking at this two dimensional image, we can get a sense of what's happening with the three dimensional surface that's happening above.
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Now that's level curves also called contour lines. We'll see them again. There's a nice theory about contour lines.
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What about functions of three variables? And the answer is you can do something basically the exact same. So what's the difference? Well, instead of having level curves,
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we're gonna have level surfaces. So in other words, we're gonna take our cross sections where we fix our output. So we have h of x, y, z equals something, a constant. And so this becomes some sort of surface, maybe not necessarily an a surface where I write z as equals to something. It might be something more that wraps around,
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but this gives us a way to think about what's happening with our four dimensional surfaces. So that's level surfaces. So again, if you think of level curves or level surfaces, just think of it, oh, I'm starting my function equal to a constant. So I'm fixing my output to be a certain value and I'm trying to understand what's going on. All right, one quick topic then we're done.
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We're gonna have need over the course of the next few sessions to start talking about certain phrases. And we just wanna make sure we all agree on what we mean when we say things. So some terminology.
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So I'm gonna think of there being a region in the plane. So let's just use this region as an example. You'll notice that some parts are dotted, that's indicating we don't include, some parts are solid, that indicates we do include, and it doesn't have to be, you know, necessarily a square or a circle, it can be more interesting. This looks kind of like a pair of of goggles, in some sense.
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So what are our phrases? Well, we're gonna talk about things like interior our inside. So to be on the inside just means everything around me is also on the inside. So in other words, I can put a little tiny circle around me
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and all the points around me are also on the inside. So everything nearby is also in. Okay, well that's the interior. All right, well great. How about exterior? Well, exterior should be kind of a similar idea and that should be what? Well, it should say, "Hey look, if I pick a point here, "everything nearby is also out." So I'm out. But everything else near me is also out. And of course when we say near, we say sufficiently near. 'Cause you might say, "Well,
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"what if you had drawn a huge circle there, Steve? "Then you'd have a part." Well, yeah, but that's if I draw a huge circle, the point is I can find some small circle that completely misses it. And as long as I can find a small enough circle that's not zero, has to have something to it, then that's something that's on the outside.
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Now there's the interesting thing, okay, you're in, you're out or you can be on the edge. We call that the boundary. So what's the boundary point? Well, boundary just says, "Look, I'm not in "and I'm not out." But no matter how small I draw, I always hit a little bit of both. So for example, here, if I draw a circle, no matter how small I draw that circle, there are some points on the bottom that are in, there's points on the top. Now you'll know it, I can do something similar over here.
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No matter how small a circle I draw, there's points that are in, there's points that are out. So boundary points don't have to be in your set. They don't have to not be in your set. They may or may not be in your set. So that's something to keep in mind.
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Now, some fun terms. And I will say a lot of the terms that we introduce now aren't super critical for what we're talking about. We just have to have a really basic passing understanding. But if you love calculus and you're like, "Boy, I wanna know more," these are the types of terms that you'll start to see as you go further and further into the exploration. Because a lot of the tools that underpin calculus, you gotta get into the machinery. So this is sort of getting
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to the really base fundamental things. So an open set just says, "Look, "everything is on the inside."
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Now that means no matter what I have,
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everything nearby is also in. Now you can, for example, an open set could look like the following, take a circle but don't include the boundary, don't include everything on the inside. That's an open set because no matter what point you pick, I can say, "Look, I can always put a small enough circle "that stays completely inside the larger circle." You might say, "Well wait, "what if you're close to the edge?" Yeah, but I'm not on the edge and therefore there's still a little bit of space,
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which means I can still put a little bit of a circle there. It doesn't matter. So that's an open set.
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A closed set says what? Says, "Hey look, I always include my boundary." So this set, which is an example of an open set, but it's not closed. The boundary will be not in our set. So a nice example of a closed set
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would be something that looks like this, same sort of basic setup, but now I just include the boundary along with it. Why? Because, well look. The boundary is the outer edge and lo and behold, everything that's in my set, that's also in there as well. Now the last term that we wanna talk about is bounded.
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And bounded, basically, you can think and say, "Look, "I don't get too big, everything stays reasonably close." And reasonably can be within a million.
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Another way to think about it is just think about what does unbounded mean? Says, "Look, I can be arbitrarily far away." That's what we mean when we say unbounded. And so to say, bounded says, "I can't be arbitrarily far away. "Everything is within some finite distance "of a central point." Now there's a basic fundamental idea, and we're gonna see this as we go on, but the basic fundamental idea says, "If something is closed and bounded, "this is a good situation."
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We like things which are closed and bounded. And we'll get into why that is in a later session. Or at least we'll hint at why that is, to fully appreciate it, well, that's just a few more years of study down the road. And then you'll be like, "Ah, I get it, that's beautiful." But for now, we're done. Time for us to do a little bit of practice, exploring functions. We're gonna focus on mostly domain and range for right now. But don't worry, we're gonna start doing some really interesting stuff.
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Calculus is just around the corner. See you again.
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