 All right guys, it's time. How are you guys doing? Good? Did you guys have a good summer? Yeah? No? Come on. Are you excited about the new semester? Yeah. Yeah? That's what I want to hear. So by the way, did you know that this course is going to be a podcast? Video and audio? Do I look okay? Thank you. Tell me if my hair doesn't look so good. We only want to show the best to the worldwide audience. So in case you're wondering, this is Math 53, multi-variable calculus. And this is my name, Edward Frankel. I'm a professor here at the master's department. I've been here quite a while, actually, and I've taught this course right here in this room. And I love this place. So it's good to be back. And you all look great. So, you know, I feel good energy. So now, let's see. There are a few things that I want to tell you about this course. And then we'll start. We'll start with the material. So what do I want to tell you? Video podcast. That's a great thing, so great resource for you. We also have other resources. This is the URL for the course webpage, where you can find the syllabus, various information about the exams and such, as well as the assignments, the reading and homework assignments. But roughly the way it will work is that the homework assignments will be due on Mondays in your sections, although there is one section which means on Tuesdays and Thursdays. So I'm not going to repeat each time, but if I say Monday, it means for that section Tuesday. All right? Not too complicated. Hold your question for a second. Ask me later, okay? So homework assignments will be due on Mondays, except this coming Monday. Because it's only one class, okay, so it's too short. So then normally it would have to be due the following Monday. But the following Monday is Labor Day, okay? So it's going to get much more complicated than this, so you have to keep up, okay? Now this is actually easy to figure out. So the very first homework assignment will be due on Wednesday, which is a week from the coming Wednesday in your sections, okay? And Wednesday in this case means Tuesday for those sections which meet on Tuesdays and Thursdays. But this is the only holiday. No, well, there is another one later on, but let's not worry about it. So there is the only one on Monday, maybe. But whenever it's not a holiday, it will be due on Mondays, all right? So that's one thing you should know. But anyway, all of this stuff is on the web here and also on B-Space. So there is this thing now which is called B-Space, which you know you have to forgive me because I haven't taught for a couple of years, so I'm new to this stuff, but it looks really cool. And it's up and running, so you can easily find it. And some of you have already, I know. And it has lots of stuff, so first of all it has all the same stuff which is here, but also it has like chat and forums and all this stuff. So I encourage you all to go there and discuss with your fellow students, and I'll be logging in every once in a while and answering questions and things like that. So you're welcome also to ask me questions through that space, through that B-Space. All right. Next, this is a large class, as you can see, okay? And I personally would like to accommodate everyone. And actually with the video podcast, it becomes more realistic, I suppose. Now, however, I don't handle any enrollment issues. I would love to, but I just have no control over it. So please don't ask me about switching sections, about moving from the wait list, et cetera. I just don't know anything about this, all right? There is a person in the math department who is in charge of this. Her name is Barbara Peavey, and please address all these queries to her. You can find more detailed information about her here. Next, we will have weekly quizzes, which will be administered in sections. We will have two midterm exams, and we'll have a final. Please check the dates. I can tell you now, October 1st and November 5th are the midterms. And the final is on December 19th, which is like the last day of the exam session. I know, but what can I do? Please read the rest on my homepage. I don't want to waste too much time, because all of this is there. The grade, how the grade is derived, and things like that. So now, I think it will be beneficial to everyone if we try to keep the noise down. And besides, there is competition to get into this class. So the worst offenders will be replaced by people on the wait list. So be careful. Now, what else? Homeworks. I want to say about homeworks. Homeworks get pretty heavy, and this is intentional. So don't, you know, curse me and say, okay, you gave us so many problems and so on. This is for your own benefit. The more homework problems you do, the better you do in class. In fact, I might take some of the problems on the midterms directly from the homework. So the more time you spend on homework, the better chance you have of getting a good grade. Now, are there any questions? Is that one? For the webcast, for the webcast, there is a, it's going to be, there is a page at Berkeley, which I will put there. I haven't put the link yet, but I will tonight, okay? I'm sure that it's up and running, but it's also on YouTube and iTunes, okay? So there are all kinds of ways to look at me if you don't get enough. Well, I wish the camera was pointing at you, but, you know, unfortunately there is only one. You know, these days I just find myself in front of the camera all the time and, you know, what can I do? What can I do? Now, any other questions? There are two people next to each other. Yeah. Office hours. Office hours. Oh, yes. It's all there, but it's Tuesdays and Thursdays after the class. Okay? Any other questions? Is that what? This site? Yeah. Yeah, it does work. No. This math is capitalized. No, no. This math is capitalized. Well, why don't you do this? Just go there and you will see a nice picture of me. And then there is a link to the... Because if you try to do it, you might misspell it and so on, but this you should be able to spell. Or go there and it will send you. Anything else? Yes? Quizzes, yeah. It will be based on homework and the working sections and the lectures, right? Not directly. I mean, it's all going to be much more subtle. But it's based on the material. Yeah, of course. There will be no tricks, you know? Whatever you learn here and in the book and in the sections will be tested on quizzes, homework, and exams. There is no... There will be no tricky questions. Everything will be just from... Taking from the kind of stuff that we do here. Yes? Pop quizzes. What's a pop quiz? It's like unexpected. No, it's okay. I don't want to trick you into anything, you know? It's all straightforward. It's all laid out for you, you know, from the beginning and we'll just follow that. Okay? Yes? Where? Where? Evans Hall. But all this information is on the web. Since our time is limited, I would love to answer all the questions, but I'll take one more if there is one. Okay, you go. I'm sorry? Ask your TAs. They will explain this in more detail. I'm sorry? Okay, I will instruct them to give you a better answer next time. Okay. All right. No, don't worry. I'm sorry? I hope so, is it? Are we filming? Yes. We are. Well, hello, worldwide audience. And welcome to Math 53. I think they'll cut the first part. So, that's where we start. So, shall we do it again? Let me tell you about this course. So, this course is called multivariable calculus. And it is a continuation of math courses you've taken before, or must have taken before, because there are prerequisites for this course, Math 1A and 1B, or equivalents, in which you studied single variable calculus. Okay? So, the difference is that, in this course, we are going to talk about objects which live not in one dimension, or functions in one variable, but in two and three dimensions. Okay? But we will build on the kind of material that you learned in the previous courses when you studied the single variable calculus. So, that's the first thing that I want to tell you. The second thing I want to tell you is that what I will try to do in this lectures is that I'll try to explain to you the main points, and I will give you examples. Okay? Don't expect that I will cover 100% of the material. Think of these lectures as a component of the course, and there are other components, other resources. First of all, the book, which is Stuart's book, Multivariable Calculus. You can either use the custom edition for UC Berkeley, which I prefer because it's lighter, or if you just happen to have the sixth edition of this book. I'm told it's okay. It's exactly the same material. So, you may use the sixth edition. So, what I'm going to do, actually, what I will try to do is to tell you the kind of stuff that you will not find in the book. Let's emphasize some of the points that you don't find in the book. Talk about the big picture, so to speak. But, of course, my time is limited, so certain things which I find perhaps the simplest or the easiest to understand, you will have to learn on your own by reading the book or in sections or by asking your TAs or asking me. So, there are many different resources, and you should use all of them. Now there is B-space and all this stuff. All right? So, I also encourage you to read ahead, actually. I think that's a great idea. If you just read maybe even the beginning of the section, even if you don't understand it, when you come to the lecture and I start to talk about this stuff, you will find it much easier to absorb and to process. So, if you have a chance to read it beforehand, you will do yourself a favor. Now, mathematics. This course is a beautiful course, actually, which I really enjoy teaching, because it has a lot of beautiful applications to all areas of science, engineering, physics, biology, chemistry, you name it. In other words, there are many problems in those areas which you cannot even approach without using the kind of material that we are going to study here. So, it's really important. But apart from this, I want to say that for me personally, I think it's also very important to appreciate and to see and appreciate the beauty of this material. And I know that for some of you, this might sound strange because mathematics and beauty, two things which are sort of far away, perhaps you think. But, and I know because that's the popular misconception and also people think math is boring and, you know, not interesting and it's sort of all carved in stone and it doesn't change and it's all been around for centuries, which is all false, okay? It is actually a live organism which is moving, which is growing all the time. You know, every day there are dozens of papers written by mathematicians around the world, every day there's some new theorems proved, every day some new stuff is discovered and some old stuff perhaps disproved, you know? So, it's really a really beautiful subject and I think one of my purposes here and one of my goals is to unlock this beauty for you. I would like to show it to you, I would like to prove it to you, to prove to you that it's not just that we study it for convenience because we need to solve some practical problems. But in fact, it has inherent beauty of itself, mathematics, okay? So, now, I am actually a practicing mathematician, as you might say. In other words, I do my research and I think that puts me in a good position to actually tell you about some of the things which perhaps would not be so easily seen to kind of someone who doesn't do research, mathematical research on a regular basis. If you are interested in what kind of stuff I work on, you can find information here. And I'll be happy to talk to you about this as well. Okay, it's not an easy course by any stretch of imagination and I'm not promising to make it easy for you, it's hard. You have to practice, you have to, a lot of things, you just have to practice on and on and on, okay? But what I do promise you is that I will try to make it interesting for you. I will try to make it worse your time here, okay? Now, what are the main topics of this course? And I want to kind of put it in perspective and to talk about the connection between this course and the math courses you have studied previously, okay? So you kind of see the big picture. The main topics are we studied geometric objects which are now defined in two and three dimensions. So now what are dimensions, okay? So this is something which, you know, we learn from everyday life. One-dimensional objects are things which are like a line or a curve, things which, in which you kind of can move only in one direction. So I'm just simplifying, I'm giving you a very rough definition, intuitive definition, okay? So if we didn't care about the thickness of this wire, we could say it's one-dimensional because you can only move this way but you can't move in any other direction, okay? So that's one-dimensional. What's two-dimensional? Well, this blackboard is two-dimensional because I can go this way or I can go this way. Someone might say you can also go this way. Yeah, but this is the kind of superposition of moving in this direction and this direction. So there are only two independent directions, in fact, like this one and this, okay? And finally, this classroom is three-dimensional, okay? Because in addition to, you know, we have the length and width and the height, right? So there are three dimensions here. And also there is time. So actually, after Einstein, we know that you can't separate space and time and, in fact, you should think in terms of four-dimensional space-time. Time is like a force dimension, which we can't really visualize so easily but it's there, right? So that's the notion of dimension that I was talking about. And we'll talk about it more. I just want to give you an intuitive idea what dimension is. And in this course, we will mostly work with things which are embedded in two- or three-dimensional space. In other words, the objects themselves might have dimension one, two, or three, but they are embedded into a plane like a blackboard or three-dimensional space. So that's the idea. So what kind of objects? First, the simplest ones are the one-dimensional objects, which is like this wire. If you ignore the fact that it has some thickness, okay? Someone might say, actually, there are also zero-dimensional objects, namely points, just points. That's also a geometric object, right? You can't move within it in any direction. I mean idealized point. No cell phones, by the way. I turned off mine, and you should do the same. Lest our worldwide audience gets offended. Now, but points, they're not so interesting, right? They don't have inner structure. So that's why we start with one-dimensional objects, which are curves. So we study curves like this. And where can a curve live? Okay, where can a curve live? It can live on a plane. It can live in two-dimensions, in other words, in the two-dimensional space, or it can live in a three-dimensional space like this. This wire, you know, it doesn't live in any particular plane. It is actually, I don't want to unroll it, but you see what I mean. It is sort of, it is embedded in a three-dimensional space, but it doesn't live in any particular plane in that space. So the first thing we'll study are curves in two-dimensional, two-dimensional, and three-dimensional space. That's the first thing. We'll talk about how to represent them mathematically, how to learn things about them, like tangent lines and various surface areas that we can get out of those curves. Okay? And what kind of formulas will help us, will facilitate obtaining these kind of results. That's number one. Number two, next in line are surfaces. Surfaces are two-dimensional objects. And because they are two-dimensional, they cannot live in the, most of them will not live inside a plane, because the only plane which will live inside a plane will be plane itself, or maybe some piece of the plane. So most surfaces, the most beautiful surfaces, like a sphere, they will live in a three-dimensional space. So surfaces will be in three-dimensional space. And again, we will talk about how to represent them, how to analyze them, and things like that. That's the second subject. And not necessarily in that order. I'm just kind of giving you an overview in a kind of logical, in a logical way, rather than chronological order. Now, and next we studied tools of calculus. And what are the tools? There are two, in some sense, opposite procedures. One is differentiation. And the other one is integration. And of course you know that, because you have studied that, but in the case of one-dimensional, it's a two-dimensional case for functions in one variable. And what we're going to do is we're going to expand this to functions in two and three variables, so that we can understand things which live in the two-dimensional and three-dimensional spaces. So the first comes differential calculus. Differential calculus. And this is about taking derivatives, roughly. It's about taking derivatives geometrically. It's about finding tangent lines and tangent planes and things like that. It has very important applications, for example, to finding maximum and minimum functions. And the second tool is integration. Just say integration. And that has to do with finding things, like volumes and surface areas, et cetera. Okay? And then we'll talk about the interplay between the two. Interplay between the two. So what do I mean by the interplay? I just want to tell you about the kind of stuff that you talked about in Math 1A1B, in a single variable calculus. In single variable calculus, in single variable calculus, you have functions in one variable. So you have something like f of x, we call it. f of x, function in one variable. For example, f of x is equal to y squared minus y plus 1. I'm sorry, x squared minus x plus 1. Okay? So it's a function in one variable. And what you learned in the single variable calculus is that you can differentiate it and you can integrate it. And so derivative f prime of x has to do with the slope of the tangent line. So I'll just say roughly tangent lines. Okay? And they also have the integral f of x dx. And that has to do with the area. So if your function is something like this, and this is a and b, this is a graph of the function, yx, and this is y equals f of x, what this integral counts, it computes, is the area under the graph, roughly. Okay? So you see that these are two completely different procedures. Derivative tells you about tangent lines. Next time I'll bring colored chalks, but for now let's just... like this. In other words, it tells you about what happens on a very small scale around the particular point here. Okay? Tangent lines. But integral tells you about something global. It tells you what happened on the entire interval from a to b. You can find the area under the graph, for example. So a priori, there is no connection between the two procedures. One is differentiation, one is integration. And so here comes the beauty, which I promised you. The beauty is that these two procedures are actually inverse to each other. Something which if you didn't know about this, you wouldn't be able to see it right away. And people did not see it for many centuries. It took some really, you know, big brains to realize that. And that's what we call now the fundamental theorem of calculus. The fundamental theorem of calculus, of single variable calculus, is a statement that if you integrate the derivative of a function, you will get the difference between the values of the function. In other words, see, you first apply the derivative, and then you take the integral, and you get back your original function, evaluated at the endpoints. So what's beautiful here? What's beautiful is the understanding that you can use a procedure, differentiation, which a priori has nothing to do with taking the area, computing the area, and use it to be able to express this area in a very simple way. So more generally the way, this is usually formulated, you say that if you have some function, if you put here some arbitrary function g, let's say, of x, dx, then this will be the anti-derivative of this function. But that's the same thing, because that's just saying that that g of x is a derivative of, so that this is what we call the anti-derivative. So I have summarized for you essentially the entire single variable calculus, the main result. So what are we going to do? We are going to do roughly, we are going to do an analog of this, but in two and three dimensions. And because we are now in two and three dimensions, it becomes much more interesting, because we can now integrate not only over an interval, if you are on a line, that's the only thing you can integrate over, over an interval. Now we are going to integrate over curves, over surfaces, and we are going to relate different types of integrals to different types of derivatives and things like that. So that's what I mean about the interplay between the two. And it's going to be really beautiful stuff. I'll give you, I'll tell you once a story. So I was teaching this class and then one guy told me that he was so fascinated by this. When he went home for Thanksgiving, he explained it to his grandmother. So that was the best compliment that I could receive. But it is in fact, when you get to the bottom of it, when you really start understanding what it's all about, the structure, the inner structure of it, you will appreciate the beauty of it. And that's what I will try to show you and to unlock for you. Any questions? So that's the outline. That's the rough picture of what we are going to do throughout this semester. So what we're going to start now is we're going to study curves first in two dimensions and then later on we'll talk about three dimensions. So we'll gradually build the theory, starting with the simpler things and then going gradually to something more complicated and so on and so forth. And we're going to use the intuition that we gain by looking at the simpler things. And so we have to start somewhere and we start with curves on the plane. So that's the simplest object out of all of this which we have discussed briefly here. Curves on the plane. So by a plane I mean a geometric object which you can think of as this blackboard extended to infinity in all directions. So usually to help us navigate this plane, we introduce a coordinate system on it. So that once we do that every point gets a name. Because every point now is going to have a coordinate, an x coordinate. Usually we call this x and y. So it's going to have an x coordinate, maybe x0. It's going to have a y coordinate, y0. So we can talk about a point as being defined by these two coordinates. It's like the address of this point. It's a uniquely defined address. It's a point which has this address. And conversely for any pair of numbers x0, y0 you have a well defined point. So that's the way that we will talk about points on the plane. We'll think of them in terms of their coordinates. Now very soon we will learn other coordinate systems. We will learn other ways of representing points. For example polar coordinates are coming up next week. We'll talk about that. But for now we will think of points as being represented by their coordinates, x0 and y0. So now we've learned how to represent points. So for instance let's say this is 1 and this is 2 and this is 3 and so on. And this will be 1, 2 and so on. So for example if you have a point with coordinates 3 and 1, that's this point. So that's coordinate 3, 1. So first goes the x coordinate, then goes the y coordinate. So now the question is how do we represent a curve on this plane? And again what is a curve on this plane? Well it's something that I can draw with a chalk without sort of just moving once, without removing the chalk from the blackboard. So that's a curve as opposed to something which I would say a strip or something which I would not be able to do in one stroke. How would we represent a curve? And this is a very important question because already here you see the main tools for representing geometric objects mathematically. Essentially there are two ways. One is to write equations and the other way is to parameterize your object. So two ways to represent one is by an equation. And second is in parametric form. So what do I mean by representing by an equation? This is actually something which we have already, which I have kind of used implicitly in this discussion because I talked about the graph of a function. And what is a graph of a function but a curve? If you look at this picture you see that curve. It is just like the kind of object that I'm talking about here. And it sort of goes without saying after the single variable calculus that a graph of a function in one variable is such a curve on the plane. So in other words, let me erase this. So let's say one, let's call this one and let's call this two. So for one, an example would be an equation like this, y equals f of x. In other words, it's a graph of the function f of x. So where's the equation? This is the equation. When we write this, we are imposing a constraint. In other words, we are not looking at all points x, y, but we are only looking at the points which have the form x, 0, say, for some x, 0, and f of that x, 0. So for instance, in this case, if f of x is x squared minus x minus 1, let's say if I put x, 0 to be equal to 2, say, then f of x, 0 is going to be 2 squared minus 2 minus 1, which is 4 minus 2 minus 1, which is 1. So that means that if x is 2, y has to be 1. So this point is going to belong to our curve. But this point, for example, will not. In other words, no other point with x coordinate 2 will belong to our curve, only the one for which the y coordinate is equal to this function f in which I substitute this value. So that's what I mean by imposing the equation. By imposing the equation, I mean considering just the points x, 0, y, 0, which have this special relation, that the second coordinate is a given function f, like this function, evaluated at the first coordinate, x, 0. So when I impose this condition, I get a curve. I get that curve we call graph of the function. Any questions about this? So graph of a function is the simplest example of a curve on the plane, and it is represented in the first form by an equation, which actually is a good time to talk about how dimensions change depending on how many equations and how many independent variables we have. Because this is a point which I think could be confusing. So there is sort of a very important rule here. You see, I start with two independent variables, namely x and y, and I have one equation, right? This. I only have one equation. So the dimension of the object I get is the difference between the two. This is the dimension of the object that I get. So it is indeed a curve. And this is the general rule. The dimension of your object is the difference between the number of independent variables, which in our course is going to be two or three, minus the number of equations you impose. So here I start with a plane, so I only have two variables, and I have one equation, so the dimension is one. I get a curve. But more generally, I could start with a three-dimensional space, right? So if I start with three-dimensional space, this is something we'll do later, but I just want to use it by way of showing that the same formula works in much higher generality. If I start with a three-dimensional space, then usually we put coordinates x, y, z. So that's like the space of this classroom. If I impose one equation, I will get two. So that's going to be a surface. If, on the other hand, I put two equations, I'm going to get one-dimensional object, which is a curve. So there is always this calculus, this simple formula. The difference between the two tells you the dimension. So that's the first way of representing. That's not the only way. Perhaps one more thing. Not all curves, not all equations look like this, right? It's a very special form of an equation in which on the left-hand side you have the variable y, and on the right-hand side you have something which is independent of y. It just depends on x, on the first variable. You can have equations which are not like this. And the simplest equation, which is not like this, maybe not the simplest, but one of the simplest equations, is this equation. Do you know what this represents? Circle. That's right. So this is again one equation, which we write on two variables, x and y, and the geometric object which it represents is a circle of radius one. If I were to put here r squared for some number r, I would actually get a circle of radius r. So what does it mean when I say that this equation represents a circle? It simply means that if on the plane I look only at the points x, y, which satisfy this equation, then I will just get this circle and nothing else. That's all I'm saying. That's what I mean when I say that this equation represents this object. So this is a very efficient way to represent geometric objects. You can represent graphs like parabolas, like in this case, but you can also represent a circle, which is actually not a graph of a function, because here you can't really express very easily y. You can try to express y in terms of x, but what you get is y squared is 1 minus x squared. So you have to extract the square root, right? And so it's not so nice, because first of all there are two square roots, so it kind of starts looking ugly. There are other ways to see that this is not a graph of a function, I don't want to get too deep into this. My point is that you can write equations more general than y equals f of x, and this way you get more general curves. And that's all about the first type of representation of a curve in on the plane. We can represent it by an equation, but there is also a second way, which is equally important, if not more important. And that's called parametric form. And parametric form has a very nice intuitive explanation. The idea is that instead of trying to find a constraint which x and y satisfy, like x squared plus y squared equals 1, we introduce an additional variable, which usually we call t, like time, and we think of the curve as being traversed by, say, a point, or some, you know, little object, a point-like object, and you can think of this extra variable, t, as time, and you just look at the position of this object on the plane as time goes by. So at the moment t equals 0, say it's here, at the moment, you know, 1 second is here, 2 seconds is here, and so on. But really for any, for every value of t is going to be somewhere. So parameterizing this curve means introducing an additional variable in such a way that you can label all points which sit on this curve by some value of this additional parameter. Two ways to think about it. One is time. So think of this as a trajectory of some point-like particle, and this, at each moment in time, it is somewhere. A second way, think of this as a kind of a rope. So just this, you know, this microphone wire. I can take, you know, I can cut a piece, and I can label, you know, I can say it's going to be one yard. And I, so for each point, I know exactly the distance from the one of the ends to this point, right? So I parameterize this wire, and then I throw it on the plane. Well, I can't throw it because it's vertical, but I just put it here and I just throw it like this. So now each point on the wire which has a certain address, which has a certain coordinate within the wire, is going to be, to map somewhere on the plane, right? So in other words, let me, let me draw it. In other words, so here's my wire. So it's going to be, say, from zero to one. I could actually take as long as I want, but let's say it's only one yard. So I can't really take more. So each point here has a certain coordinate, let's say t0. So let's say this point has coordinate t0. I will think of this interval as sitting on a line, which has coordinate t. Now, I assume that it is completely flexible, like this wire, and I just throw it on this plane, okay? What do I mean by this? Well, this point will go here, this point will go here, and each point, for example, this one, which I call t0, will go somewhere here. So t0 will go here. So this will be, more generally, this will be some a and b, if you want. So this will, a will go here, b will go here, and t0 will go here, and so on, right? But now, since I threw it on the plane, each of these points is going to have an x coordinate, which will be some expression of t0. So this t0 got, it just happened to be here. But when we look at it and say, this is a point on the plane, so it has two coordinates. Let's call it x0 and y0. But as t0 moves, as t0 moves here, these two coordinates are going to change. So these are going to be two functions of t0. So then what I do, I just write x is f of t in y's g of t, and where t goes between a and b. And that's called parametric representation of this curve. Is this clear? Any questions? So see, yes? Well, you see this? This has become this, right? It's like I take the piece of wire, and I just make it like this. Now, on this interval, I have a coordinate, which I call t, which goes from a to b. When I throw it on the plane, for each value of t, I'm going to get a pair of numbers, which are the coordinates x0 and y0. But when t changes, or t0 within it changes, these two coordinates are going to change, depending on t. Which I symbolically write as a function of t. For first, I call it f, and for the second, I call it g. So I get this expression. So that's the idea. That's right. I have two coordinates, but these are not equations in this sense. This was an equation which was a constraint. I was constraining my variables x and y, and saying that I only look at points which satisfy this. And I was writing just one formula for this, right? Which, see, it involves only x and y, but no auxiliary variables. But here, I have an auxiliary variable, and by using this auxiliary variable, I write the first and the second coordinate as a function of this auxiliary variable. So it's a different way of representation. But maybe it's better to look at a concrete example to see how it works. So the example will be... Here's an example. Here's an example. x is equal to t minus 1, and y is equal to 3t minus 2. So this is f, and this is g. In other words, I have written two formulas just like this, but I have chosen some concrete functions f and g for it. The first one is t minus 1, the second one is 3t minus 2. So the question is, what does it represent? So you can just approach it in a very basic way, in a very naive way, and you say, all right, professor, you taught us that for each value of t, I'm going to get a point, and what these formulas tell me are the coordinates of this point. So we can just plot a few points, for example. So the curve is going to be somewhere. We don't know yet what it is, but what we can try to do is just plot some of the points. So what we can do is we can just make this diagram. We just make t, x, and y. So let's say t equals 0. The point is going to be negative 1 and negative 2. When t is 1, this is going to be 0 and 1. And t is 2 is going to be 1 and what? 4. I'm getting slower as time goes by. OK, we're actually good. Is it one hour or one hour and a half? I'm just kidding. So I've got three points. I have found three points. Let's plot them. So for t equals 0, it's negative 1, negative 2. So let's call this A. This is B and this is C. So this is A. So we don't know yet what the curve is, but we already know one point on this curve. Second point is 0, 1. This is 0, this is 1. So that's B. And the third point is when this is 2 and this is 4. So maybe a little longer. It's 1, 2, 3, 4, something like this. I'm sorry, yes, you're right. I'm just checking if you are awake. And you are. Some of you are. That's good. Perhaps more than me. So this is 4. And this is a point C. So now it's tempting to say, so these are three points, it's tempting to say that the curve itself is just this. So far it's just a guess. It's intuition, because in principle, it could be like this. Both of these curves share these three points. So for now, I have not proved that it is this line. So in other words, this is a useful tool, but it's very limited. It gives you an idea. It can help you to guess what the object is. But just by using this, you cannot say for sure what the curve is. To say for sure what it is, you have to use more sophisticated methods. Well, which in this case actually is not very sophisticated. But in general, it can become more complicated. So what's the method? The method is you can try, there are several methods. But one of them is to try to eliminate this variable. In other words, to try to find the equation on X and Y. So I gave you a curve now in the parametric form, which is number two. This is number two representation. What you can try to do is to rewrite it as number one, a type one representation by an equation. How to do this? You can try to express this auxiliary variable T in terms of X or Y. And then plug it in the other formula. So express T in terms of X or Y and plug in the other formula. So what does it mean in this particular case? Well, if I access T minus one, that means that T is X plus one. And now I substitute this into the second formula. And what I obtain is that Y is equal to three times X plus one minus two. Which means three X plus one. So the end result of this is that Y is equal to three X plus one. So you see, we started out with a parametric representation, but we ended up with an equation like this. Or like that. Because we have been able to express Y as a function of X. And now, of course, we recognize this because this is a graph. This is already a graph of a function in one variable. So to understand what this is, you don't need to study multivariable calculus. It's enough to study single variable calculus. Because it's a graph of a function in one variable. And what is it? It's a line, right? It's a line which has a slope three. And we actually see it. So the first guess was actually correct. It is a line with slope three. Right? But this is a proof. Because we reduced this problem, the problem of describing this curve to something we already know. Because we know from the previous class that when you have a linear function, in other words, a function which only involves X and the scalar, but not X squared and so on, and neither does it involve any more complicated functions, then the graph is a line. And we can easily draw that line because you know a line as soon as you know two points on this line. In fact, here we know three points on this line. So it's even better because we can also test ourselves whether we got those points correctly. Clear? Any questions? All right. So, by the way, you can see the kind of interplay between a single variable and a multivariable. And it sort of comes a little bit of a surprise because here it's really about a function one variable, and yet we obtain a graph which lives in two dimensions. There's a very simple reason for this. Even if we study functions in one variable, when we want to talk about a graph of this function, we have to talk not only about the argument of the function but also the value of the function. So implicitly, there are two variables already. Once we start talking about functions in one variable, there are already two variables involved. One is the actual variable X on which the function depends, but then there is also a second variable we can think of which represents the value. That's why the graph of a function in one variable actually is going to be a curve which lives on the plane. So in some sense, you already had some elements of a multivariable calculus when you started single variable calculus. And likewise, when we will talk about a function in two variables, a function in two variables to really understand them properly, we will have to introduce, throw in one more variable, a third variable. For example, a graph of a function in two variables is going to be a surface in three dimensions. So it's good to keep track of all of this. In other words, there are different things. There's like independent variables, there are a number of equations, there are auxiliary parameters, there are also graphs which somehow bump everything, the dimensions all by one for the ambient space. So it's important to keep this clear in mind. Which are we talking about? We're talking about a function, of how many variables are we talking about, a graph of this function, and things like that. All right. Another example, a little bit more complicated, but basically the same in the same vein. Let me try something. All right. Second example. This was one. This is example two. X is equal to t minus one, but y now is equal to t squared minus t minus one. So same idea. Again, express t in terms of x, x plus one, and substitute in this formula. Right? So what you get is x plus one squared minus two times x plus one minus one x squared plus two x plus one minus two x minus two minus one. And that's what? That's x squared minus two plus. So we end up with y equals x squared minus two which again is a graph of a well-known function which is called parabola. So parabola, if we had it like this, it would be parabola. Parabola would be like this, but because we have minus two, we have to shift everything by two. So instead of this picture, we'll have pictures starting at minus two, negative two. So it's going to be, for example, negative one. So I'll be one. And then here we'll be at two. So it's something like this. Does it look familiar? I hope my drawing is okay. So this is called parabola, right? So again the same idea. Now let's play with this a little bit. Suppose that I wrote, I switched the variables. Suppose I write x is t squared minus two t minus one. And y is equal to t minus one. Suppose I write this. Same thing, but I switch x and y. I'm allowed to write any functions whatsoever. In fact, you're lucky I'm writing this. I could write some logarithms and exponential functions and what are you going to do? But like I said, I start with simple examples. But here's a little twist. I just switch the order, switch the two variables. So now x is a complicated function, now we've learned the trick how to deal with this kind of parametric curves and the trick was to express t in terms of x and then substitute. But in this particular case, that would be unwise because the formula for t would be, if I try to express t in terms of x would be very complicated. It would involve square roots and so we don't want to do that. Instead it's much wiser to use the second formula to eliminate t. So we write t is equal to y plus 1 and then we substitute in here. So then what I end up with I don't have to calculate the second time, I just take the formula I had before and I switch x and y. So what I end up with is x equals y squared minus 1. So see here, minus 2. So see here I had y equals x squared minus 2 and I said this is great, this is the graph of the function x squared minus 2. So what I got now is not really a graph of a function in the usual sense because usually we write the formula for our graph of a function by writing y equals f of x. It's not in this form because I could try to express y in terms of x but I would get like square roots and things like that which I prefer not to have. But we should not we should not insist on a curve being a graph of a function because before we looked at graphs of functions because we were interested only in functions in one variable. So the only curves on the plane that we could possibly get this way or curves like this, y equals f of x. But now we've grown, we look at general curves in the two dimensional space where actually the variables x and y become equal. We cannot say that x is more important than y or conversely. So this should present no problem for us because even if it's not a graph of a function it's very clear what the curve is because we got this curve just by exchanging x and y. So what does it mean? Geometrically exchanging x and y simply means a flip with respect to this diagonal. So if I flip what do I get? I get exactly the same kind of thing but it's now aligned like this. So it starts at negative 2 but now in x coordinate and it will also have points like 1, negative 1 and negative 1, 1. Right? And we just go like this. So this is precisely the mirror image of that curve with respect to that diagonal. And it's no more complicated than this. But this is a good illustration of how more general our curves are now than they were before. We don't insist anymore that they would look like this but we also allow things like this. And you should be open to kind of playing with x and y as if they were completely independent. Not trying to necessarily fit the same profile as y equals f of x but, for instance, also allow x equals f of y. Okay? Is that clear? So let's do a slightly more complicated example. How about this? Let's write x is a cosine of t and y is a sine of t. So, as I said, in principle we can use any functions whatsoever. I could write some really ugly functions and it will be a bona fide curve. Of course, in general it's going to be very difficult to draw it, to plot it. You can plot some points and next week we will talk about other methods which will give you a better idea kind of qualitative ideas at least about what the picture looks like because we will also talk about tangent lines, what tangent lines look like. So, we have a bunch of tools, tricks and tools which would allow us to get some understanding of what these curves look like in general. But in these simple examples, actually, we can somehow see the whole thing. And, for instance, here, how do we see the whole thing? Well, the point is that these two functions, cosine and sine, satisfy a very simple equation, right? Which we know from trigonometry. We know this cosine squared plus sine squared is equal to 1. Right? So, we can use it to our advantage. So, we can say that this is exactly this is equivalent to this equation. And we already discussed the fact that this equation represents a circle. So, in other words, we start with this parametric representation and we end up with this equation. So, this is equivalent forms over-presenting the circle. However, there is a small caveat. There is a small subtlety here, which is something which I haven't really kind of emphasized up to now, but now I am forced to do it. Because, remember, when I talked about the most general setup, remember when I wrote the formula for a general curve, I said the general curve is parametrizing a curve by a segment. And the segment had two ends, A and B, right? So, when I wrote general formula, when I wrote I'm switching X and Y today. I also wrote this where A and B are some numbers. So, you have to remember this. Because if you are given this in the discussion up to now, I was not paying attention because I never wrote any limits. And if you don't write any limits by default, it means that it can take an arbitrary value. In other words, in the example of this parabola, I didn't write any limits, right? And so, what I'm talking about is the entire parabola. The entire parabola is parametrized, if you will, by T going from minus infinity to plus infinity. If, however, you are doing homework and in the homework it says T goes from negative one to one. You don't draw the parabola. You draw the part of the parabola which corresponds to the values of T between these two limits. And what is it? Well, for T equals negative one that's what? So, that's it's minus T equals negative one. Let's do it. In fact, like we did before, right? So, I see I don't have the answer right away. I have to count. So, you have minus one. So, you have one plus two minus one. Right? So, that's two. And for y, it's negative two. Right? So, x is two and y is negative two. Right? So, it's this point. Now, everything, this which goes on this side is going to have T less than negative one. So, I raise it. And likewise, next I look at T equals T equals one. So, that would be what? One minus two minus one. So, that's negative two. Right? And if I put T equals one here, it's zero. So, it's this point. So, if I'm told to look only at the parametric curve for these limits this is what I have to draw. This is what I should look at. Okay? Now, let's see how this plays out in the case of a circle. I was a little bit cavalier when I said that this formula just represents a circle. The point is that let's say if you start from T equals zero for T equals zero you get this point. And as you increase T I hope you remember from the denominator that x is going to decrease and y is going to increase and for example here you reach this point when T is pi over two you reach this point when T is pi you reach this point when T is three pi over two and you come back here when T is equal to two pi. If you continue to increase your variable T you will actually start making the second round. Second circle. And so on. So, you're actually going to wrap the circle infinitely many times. Right? So, it's not correct to say that this is equivalent to this. What is correct is to say that if I impose the condition that T is between zero and two pi then this is really equivalent to the circle. Likewise, if I would if you want I can also say alpha plus two pi for any alpha. That would also correspond to this. But if for example I say that T is for example from zero to four pi it's not just a circle but our curve is not just a circle. Our curve is obtained by wrapping around the circle twice. If it's four pi. If it's four pi. If it's six pi three times and so on. Counterclockwise. Right? So, you have to be aware of this because parametric curve actually gives you more flexibility. Because parametric curve for example allows you to go through the same point twice and even through the same curve twice or three times or infinitely many times. So, you have to be careful when you when you start converting this parametric equations into this sort of more conventional equations relating x and y. You have to be careful to keep track of what the limits are. Any questions about this? Yes? In other words, what is the dimension of parabola and what is the dimension of the circle? Also one. That's why we call them curves. Right? Circle is one-dimensional and parabola is one-dimensional. Right? And I know that it sounds it could sound a little bit anti-intuitive. Right? Because you may think one could think that one dimension is like a line but as soon as it becomes curved right? It's somehow more than a line. Right? But see, the point is we have to... Let me backtrack. The point is to separate two things. To separate our geometric object from the ambient space where it lives. A circle lives on the plane. Right? It's embedded into a plane. But the plane is not a circle. The plane is the ambient space. It's like it's home. It lives there. But it doesn't mean that the circle is two-dimensional. Dimensionality is a measure not of the dimension of the ambient space. It's the measure of how many independent variables you have within your object. In this particular case, we were able to parameterize the entire circle by one variable, namely variable t. For each value of t from 0 to 2 pi, we got one and only one point on the circle. More precisely, for 0 and 2 pi, we get the same point but that's not worry about this. Right? So we have been able to identify the circle with an interval from 0 to 2 pi. And that's what parameterization is all about. It's about mapping or establishing a correspondence or a more proper mathematical term would be one-to-one correspondence between your curve and an interval. Now, the interval clearly is one-dimensional. And when we make this procedure, we identify this interval with the circle. Each point here goes to one and only one point here. So that's the reason why we say that this is also one-dimensional. And more concretely, this is what is expressed by this formula because we say that both x and y coordinates are expressed in terms of t. That's why it's one-dimensional. Put a different way. These are not all possible values of x and y but only values of x and y which satisfy this equation. But there is one constraint. So the rule is the number of independent variables minus the number of equations. That works almost always works but it's a good rule. So in all the situations we'll consider it will work. You see what I mean? Any other? Direction of the graph. So the question is, I have to repeat it for the worldwide audience. The question is about direction of the graph. That's a very good point. So in fact here as t increases from 0 to 2pi this is traversing the circle counterclockwise. If, for example, I made a change if I wrote like negative t, for example, it would go clockwise. So in principle if we have to if we want to be really pedantic and I guess we should be we should be we have to specify also which in which direction does this parameter traverse the curve. For example in this case it would be this way with negative sign it would be this way. Likewise here it was this way. Because negative one corresponded to this point and one corresponded to this point so clearly we were moving this way. So the answer is yes. Anything else? Okay. So now let's do one more and I think time is up. Because let's look at this parameterization. Okay? If t is positive negative t is negative and the negative angle will correspond to this moving like this. This is minus pi over 2. Negative pi over 2. I know that I know that the lecture is almost over. Okay? And everybody wants to get out of here. Now I will not keep you more than you have to be here. I promise. Five o'clock it's over. Even in mid-sentence. But if we start thinking about leaving now five minutes before it's not going to help because people start putting stuff or moving and so then it means that the lecture is not 50 minutes but it's 45 minutes, okay? So that might as well finish at five minutes before but then of course you'll start even five minutes before and so on. So that's by the way this is a mathematical argument. Now so to avoid this I would like to ask you not to move well not to make any unusual movements. She wouldn't normally do in other words act in the same way between the beginning of the lecture and the end of the lecture only when it's finished then you just a simple request. There are many people and this will help everybody. So we have one more example and then I'll let you go. Ah, maybe let me just do a small variation on this. I already did one so I put negative sign. Here's another variation. Now but I put one half here. If I put one half here this will not be true anymore because if I take x squared I will get one quarter of cosine squared, right? But there is a very simple way to compensate for this. Let's take 4x squared. If I take 4x squared it's going to be 4 times 1 over 4 so that's again cosine squared and everybody's happy. So that means the equation becomes 4x squared plus 1. And what does it represent? Well, what we have to do is we have to just stretch everything along the x direction by a factor of 2 because 2 is because of this this one half. That's right, sorry. I meant shrinking. Shrinking everything by a factor of 2 because of this one half. So what it's going to look like is that it will now cross the x line at the points one half and negative one half but the y line y axis it will still cross at negative one and one. And so it will look like this. And that's something which is called an ellipse. So you see this is not a circle but it's very similar to a circle and we use our intuition, our gained knowledge about the circle, namely that we knew what this was to represent this curve because we see that geometrically it's not going to be like a circle but it's just squished by or shrunk by a factor of 2. Okay. Good, so now just in the remaining two minutes I will begin and I will let you finish it on your own. Sometimes you're going to be asked to write the parametric representation for a curve which is described in a different way, in words for example. In other words people will tell you or in the book when you do the homework assignment you will see a description of a curve and you will have to represent that curve in a parametric form. And here is a very typical example which is called which is called cycloid. So this is a curve which you obtain in a following way. You take a circle of radius one so this is one. Okay, or think of a disc if you like. And you put it here, it's like a wheel think of it as a wheel and then you have a marked point there is one point here which is now at the bottom which is now exactly at the origin of this coordinate system and then rotate it to the right you know roll it roll it to the right think of it as a wheel and just roll it to the right and let's see what happens with this point look at the trajectory of this point this is a very nice example because I was telling you that one of the ways to think about parametric curves is to think that there is a point particle which is moving and as it moves it sweeps the trajectory and that's your parametric curve, right? And so what will happen is that it will go something like this sorry, it's not, it's too low it has to reach at some point it will reach the highest level then it will come back and then it will do the same thing qualitatively it will look like this but now you have to write a formula or more precisely two formulas you have to write x as a function of t and y as a function of t, right? So that's what you need to do and so this is actually an exam it's a good example, it is in the book and I will just give you a hint the first hint is to decide what is a good auxiliary parameter so when the circle moves, let's say in this way so then the point could be the point could be somewhere here it becomes somewhere here, right? and so so the question is which parameter are you going to use and so in this particular case the good parameter is the angle of rotation by what by how much has the wheel turned and then after this it's a simple trigonometry that coordinates x and y of this point as a function of that coordinate so there are two parts, right? you first find a good coordinate which in this case is angle and second you use some methods like trigonometry to express the x and y as a function as functions of this coordinate that's how you do it okay, that's all for today and we'll continue next time