 Okay, good to see you all. Aren't you glad it's Thursday again? It's just one day to go. Okay, I'll try to make it quick, okay? Or I will try to make it feel like it's quick. Not to bore you too much. All right, we're actually getting to more and more interesting stuff. You can probably tell. Last time at the end of the lecture, I talked about general curves in space. And I took as a prototype the simplest curves, that is lines, right? And lines we found convenient to represent in parametric form. Which means that we write each coordinate as a function of an auxiliary variable, which we usually call t, but not necessarily t, it could be any other letter of your choice. By default, we could t. And then each of the three variables is written as a function of this t. I'm hearing more people today than I'm used to. You've spoiled me in the previous lectures. You've been so quiet, so let's keep it this way. So these are arbitrary functions in t a priori. And in the book, you can look at various examples. There are some pretty cool curves that you can parameterize this way. For example, the spiral, which goes like this, and many other ones. I'm not going to go over these examples because they are fairly easy to just read about in the book. What I wanna emphasize is the fact that this is a natural generalization of the formulas we had for lines for which these functions had very special form, they were linear functions. Functions of degree one in the variable t. I can wait. Just knock them, yeah, thanks. So that's what we have. Now I want to look at it in a slightly different way. I don't wanna think of this as not just the way to parameterize a curve, but as a function with values and vectors. Because the point is that when you have three coordinates, when you have x, y, z in general, you can view them as coordinates of a point in space, right? x, y, z. But you can also draw a vector from the origin to this point. That's called the position vector of this point. So there is always this two ways of thinking about the triple of numbers x, y, z. This is a point and this, which we also write as xi plus yj plus zk is the position vector. And even though we realize that these are two different objects, we use essentially the same information to represent these objects. And so oftentimes it's convenient to interpret a triple of numbers in two different ways, as a point or as a vector. So if we interpret it as a point, we just think of this point as traversing this curve in three space. At each moment t, which actually you can think of as time, if you wish, it will be in a certain position. And as time goes by, it will trace this curve. But you can also view this three functions as components of a vector. So we get a vector value function, which we can denote as r of t. So that's a vector in which the first component will be f of t, the second component will be g of t, and the third component will be h of t. In other words, if this point lies on this curve, you can think of this point p moving along the curve, and so that its three coordinates are changing as functions of t, f, g, and h. But you can also think of the vector itself moving so that the end point goes along this curve. And that's what this function r of t represents. We are more used to thinking of vectors as static objects. We would usually talk about a vector. So there's a particular vector. But nothing prevents us from allowing the vector to change with time, say, or with some variable t. And then what we get is not a single vector, but we get a vector for each value of t. Each value of t gives rise to a particular vector, which we denote, which we simply denote as r of t. So the convenience of this is that if we think about, in this way, we can use various operations on vectors and apply them to this vector-valued functions. For example, we can add two vector-valued functions. If we have r1 of t and r2 of t, we can take the sum. So that would just mean taking the sum of the components. So for example, you'll have f1 of t plus f2 of t and the same for g and h. I just wanna save time. I'm not going to write it in detail. You get the idea. If you were thinking in terms of points, you wouldn't be able to do that. It wouldn't make sense because we cannot add up points. Points cannot be, we cannot add one point to another. But we can add vectors. We can use parallelogram rule or triangle rule, whatever you like. There is a rule for adding vectors. Vectors can be added up. So that's why when we convert this into a vector-valued function, we can use this operation and sometimes it's very convenient. What is perhaps even more important is that we can approach this as functions and apply other operations which we normally apply to functions which are valued in numbers, like differentiation and integration. These kind of functions can be differentiated and integrated just like normal functions. And what it means is simply applying this operation to each of the three components. So it's very straightforward. You don't really have to think about it too much. You just differentiate the first one, differentiate the second one, and you differentiate the third one. And this is something which actually has important applications because the derivative of this function will give you information about tangent lines and tangent vectors to your curve. So the derivative, that would be r prime of t. This is, we use the same notation as for normal functions, but I shouldn't forget this side which emphasizes that it is a vector. So that would be just taking the first derivative of f, the first derivative of g and the first derivative of h. And so this derivative has a meaning, namely, if we take, I want to emphasize that I want to evaluate the particular point which I'll call t zero. There is always this issue with notation. When I write something like r of t, there is a question, what do I mean by this? Do I mean a particular value for a particular t? Or do I mean all of these values for all values of t as a function of t? And usually it's clear from the context. But now I want to really emphasize that I take a specific value. That's why I call it t zero. So this will not be, I don't want to view this now as a function. I want to view this as a value of that function, the derivative at this particular value t zero. For example, t zero could be equal to zero, like in the example which I calculated at the very end of last lecture. So that would mean that we are evaluating the derivative at the point t zero. So if we do that, this is a specific vector. This is a specific vector corresponding to point, corresponding to the value of t, which I call t zero, which is specific value, particular number. So numeric value. And the upshot of all this is that this is, this is a tangent vector. I'm tangent vector to the curve, to the curve described by these equations, at the point, at the specific point, which I'll call say x zero, y zero, and z zero, which is f of t zero, g of t zero. And last time, last time at the very end of last lecture, we looked at the example of this, how this works at the very end of last lecture. So once you pick a value t zero, you have a particular point on your curve. Let me go back to this picture. So let's say this is a point p, so that would be the point x zero, y zero, and z zero. That's the point which corresponds to the value of t equal t zero. And then for other values, it's somewhere along this red curve. So if you wanna calculate the tangent vector to this curve at this point, that's going to look something like this. So this is precisely this r prime of t zero. Why does derivative give you a tangent vector? Well, that's clear because that should be clear from what we discussed before, because what is the derivative? The derivative, by definition, is obtained as follows. We take the value, so derivative of t zero is the value of t zero plus some small number. Let's call it h minus r at t zero divided by h. So let's say this is a point when t is equal to t zero. And let's say this is a point on the curve. That's the same red curve, so maybe I should use the same color. There's a point t equal t zero, our original point, which I call p on that picture. And this is a point p prime, which corresponds to some value t zero plus h. But remember, we also have the origin somewhere. So we have a coordinate system. I'm magnifying, this is, I blow up a small part of this picture. But so there's a coordinate system somewhere in the background. And there is the origin, the point o, right? So r, r of t zero plus h is going to be this vector. And r of t zero is going to be this vector. I have already drawn on this picture, this one. So, and I'm using the standard formula for derivative. The derivative is the increment divided by the change in the parameter. The increment in the values, when you change the parameter by h, divided by the parameter. So it's like velocity. You calculate velocity by taking the difference between positions and dividing by the time, right? So it's the same idea, it's the same formula as for normal functions. We apply exactly the same formula for vector functions. So what do we get as a result? We have to subtract this vector from this vector. And we know how to do this by triangle rule. It's just the vector connecting the two, you see? So the point is that the difference between these two vectors is just the vector connecting them. You see, because why is it the difference? Because if you take this plus this, you get this. So that means that if you take this minus this, you get this. So the green vector is the difference, is the numerator, this. And I divided by h. When I divide by h, I just shrink it. I just, well, it depends if h is very small, then actually I multiply by some large number. So this is done to rescale it because if I don't do this, this difference is actually going to disappear because as the point goes closer and closer and closer, this is going to become smaller and smaller and smaller. And actually what I want to do is I want to take the limit of this. When I wrote what I wrote was an approximation and to make it more precise, I have to say it's a limit. One h goes to zero of this ratio. So if I didn't divide by h, I wouldn't get anything meaningful. I would definitely get zero vector because this green thing will shrink to zero if h goes to zero because this point is going to be the same. So these two vectors are going to be the same. r minus r is zero. So what I do is I rescale by h. So I adjust for the fact that actually I'm getting closer and closer. And when I do that, it doesn't necessarily go to zero anymore, right? So in fact, it is going to stay finite and as the point gets closer and closer and closer, I'm actually going to get something which will be tangent, which is what I drew here in this picture. So that's the reason. That's the reason why it's a tangent vector. It's a tangent vector. In the following sense, there are many tangent vectors because if I have one, I can always multiply it by something, by any number. I can multiply it by five, by 10, by a million. It's still going to be a tangent vector because when I multiply a vector by something, I don't change its direction. I only change the magnitude. So if the direction is the same, if it was a tangent vector to begin with, it will remain tangent vector when I multiply by something. So this is slightly inconvenient because you might get one vector using particular parameterization of the curve and someone else may use a different parameterization and then get a different vector. They're not going to be that different because both will be tangent. So they will both have the same direction but one of them could be like this and then one longer or shorter. So it oftentimes is a good idea to agree on normalization. And so no matter which result we get, we might say, okay, let's find a vector which has a unique length. So then what we need to do, and that's a vector which is called t, t zero, is obtained by dividing this tangent vector, the raw tangent vector that we get by taking derivative and dividing by its length, by its magnitude. Because when you divide a vector by its magnitude, you're going to get something of magnitude zero, magnitude, sorry, magnitude one. I was jumping ahead of myself because what I was going to say next is that this only makes sense if this is not zero. But of course, but assuming that, only assuming that is not zero. It's possible that this vector is zero to begin with. So then we can't normalize it, it is zero. But if it is not zero, we can divide by its magnitude and get a unit vector. So this is called a unit tangent vector. It's still not exactly unique because there are two of them now. There could be this one or it could be this one, but the ambiguity is much less now. There are two possibilities. So it could be very useful sometimes. And at the end of last lecture, I used this derivative, this tangent vector to write down a parametric equation for the tangent line to this point. To the curve at this point. And these are very easy to write because we know what the initial point is. It's just x zero, y zero, and z zero. And all we need is the components of the tangent, of a tangent vector. We have found a tangent vector by taking the tangent vector. So we have found a tangent vector by taking the tangent vector, by taking the tangent vector, by taking the tangent vector, by taking the derivatives. That's a top line on this board. So that's going to be f prime of t zero. And now I have to choose an auxiliary parameter for the line. And it's better to choose a different parameter than for the curve because we don't want to indicate that they kind of go hand in hand. They don't. The tangent line we are talking about is this tangent line on which this tangent vector lies. It only has something to do with this curve at this particular point. A priori has nothing to do with this curve at this point, or this point, or that point. So there's no reason to call the parameter for this line also t, because t was used as the parameter for the original curve. It's better to separate the two things. We have so many different letters that we can use, so why not use a different letter? The first thing that comes to mind is letter s, just the next one, an alphabet. So that's why we use it, but you can use whatever you want. So as long as it's in the light and alphabet, I guess. As long as we can recognize what it is and you can recognize that it's the same one. So then you get these equations. And I explained how this works in a particular example last time. Are there any questions about this? Yes. Large t is the unit tangent vector. It's a vector which is obtained by taking the derivative vector and divided by magnitude. Provided that the vector is not zero. She needs to be absolutely precise, I should say like this. Does it make sense? Okay. Anything else? Yes. Not really, because yeah, I subtract the position. So the position has no connection to the derivative. You see? It's really this vector. But I think what you are saying is, so the question was about whether this vector has anything to do with the position vector. And this might be, the reason why you might think so is because I draw it starting from this point. See, but that's just a matter of convenience because this vector, I would like to say that this vector is tangent to this curve at this point. So it's better to draw it here to indicate that. But it doesn't mean that somehow I have to add the position vector to this. This is a vector of itself, right? I'm choosing to draw, to take as the initial point for this vector, the point P, whereas I chose as the initial point for the position vector, the origin. And so as I explained before, there is no reason to choose this or that point. It depends on the context. We can choose any point. In other words, a given vector can be drawn starting from any point. I could draw this vector r prime starting from this point, which has nothing to do with this picture at all. But it doesn't negate the fact that this vector is the same as this vector as long as they're parallel and have the same magnitude, you see what I mean? So just the fact that I draw it from here doesn't mean that I have to add to the position vector. That's the derivative vector. And well, that's the way it's defined. Does this answer your question? Yes, tangent vector is a velocity vector. That's right. The tangent vector can also be interpreted as a velocity vector. That's a good point. Okay, so let's move on. Let's move on. One more thing before we move on. We have discussed now in detail differentiation of vector-valued functions. And as I explained, differentiation of a vector-valued function like this simply means differentiating the first component, the second component, and the third component separately. You can also integrate vector-valued functions. And this is something which will be useful also for us a little bit later in this class. And again, to integrate a vector-valued function, you simply integrate the first component, the second component, and the third component. So it's very straightforward. By the way, in the homework assignment, which was posted before, for this particular section, 13.2, there was a bug in the HTML file. And I prefer to say it's a bug, but actually I made a mistake, so. So I made a mistake, and I cut a couple of problems from this line. So I put them back now, but please use the updated version of the homework assignment. I haven't made any other changes, just in 13.2. This particular section about derivatives of vector-valued functions. Don't worry, don't worry, this is just a couple of exercises, which are very simple. I just didn't want you to get the idea that, what's this term all about? Look, there were only exercises about derivatives, about tangent vectors for plane curves. And I didn't want you to get the idea that I don't want, you don't need to know about derivatives for three-dimensional things in three-dimensional space. So that's all. All right, so let's come down. And let's move on, derivatives. Okay, so next topic we will need to discuss is functions of two and three variables. By the way, one other thing I remember now. For plane curves, there is a connection to what we discussed. For plane curves, there is a connection to what we discussed at the beginning of this course. Plane, by a plane curve, I mean a curve for which everything happens on the plane instead of in space. So let's say in R2, you would have also vector value functions like this, which would correspond to curves on the plane in the same way in which the three-component vector value functions correspond to curves in space. And then again, the same thing happens, which is that the tangent vector could be found by taking derivatives at a particular point, which could be taken, found by taking derivatives of these components. Now, in what sense is this related to what we discussed before? You see, what we discussed before was not so much the tangent vectors because we didn't, at that time, we didn't talk about vectors. We discussed the slope. We discussed the slope. So we had a curve now on the plane, okay? And we talked about the slope, which is the tangent of the angle theta. And we found the slope, and this is again a point t equal t zero. And we found the slope as x prime, sorry, y prime divided by x prime, which or whatever, dy, if you want dy dt divided by dx dt, which was g prime of t zero divided by f prime of t zero. And now everything, I just wanna show you that everything is consistent because now we actually give a more detailed information, not just the slope, but actually the tangent vector itself. And what I'm saying is that in this vector, this vector has two components. One of them is the horizontal one is f prime of t zero, right? And the vertical one is g prime of t zero. So in fact, the vector, the tangent vector has these two components, f prime of t zero and g prime of t zero. But then of course, what is the slope? What is the tangent of this angle? Well, this is a triangle with the right angle. And so the tangent by definition is this side divided by this side, and low and behold, you get back that formula. So what we get now, there's more precise information about what the tangent vector is, is really consistent with the old information which was just about the tangent of this vector also known as slope. In the three dimensional space, we can't really talk about the slope because now our vector has three components. So we can talk about ratios of different pairs of components. There is not a particular pair of components that is special. But in two dimensions, there are only two coordinates and therefore we can talk about the ratio of the second one to the first one, which is what we call the slope. So thinking in terms of vectors kind of allows us to generalize the analysis we did for plane curves to the case of curves in three space. Okay, so that was my last comment on this subject. And now I wanna talk about functions in two and three variables. So the next topic is, the next topic is functions three variables. So in fact, I would like to put it in a more general context. Up to now, we have talked about different kinds of functions. We have talked about functions in one variable, which we usually write like this, f of x, like x squared or cosine of x or e to the x. This is a function one variable. And today, we talked in a little bit last time, we talked about vector value functions, which we write as r of t. And this vector value functions may take values in vectors in two dimensional space or in vectors in three dimensional space. So in both of these cases, the word function is used, which begs the question as to whether these objects really are similar in what way. What are functions? And this is really a very elementary and very intuitive, but we have to say it once and for all, so that there is no ambiguity left. So the question is, what is a function? What do we mean by a function in mathematics in general? Of course, in our everyday life, we use the word function. We say, our appetite is a function of how much we've eaten before. So what does it suggest? It suggests a logical connection. If I ate a lot before, I'm not so hungry. If I haven't eaten all day, I was busy, then I am hungry, right? So each cause has a certain result. So there's a certain rule, there's a certain logical connection. And that's exactly what we mean by function in mathematics. In mathematics, a function is a rule which takes each element of what we call the domain of the function, domain of the function, and transforms it to an element of what we call the range of the function. And the point is that we can choose as domain in range, we can choose all kinds of objects. And each time we make a choice, we get particular functions. For instance, to get a function in one variable, we choose as a domain R, the set of real numbers. And as a range, we also choose R, the same set of real numbers, right? So it's kind of boring in a way because we choose the simplest set in both cases, not the simplest one, but perhaps one of the most obvious ones. So then the rule, the rule takes the following shape. Given an element of R, which we'll call X, so it could be any number, zero, one, whatever, pi. This rule tells us what is the value of the function. So that's going to be another number because it's, well, in general, it's going to be an element of the range, of the set, which we call the range. But range now is R again. So the result of the rule should be an element of, so in other words, when we write the formula F of X, what do we mean by this? We mean that we have created a rule which assigns to any value X, we assign a certain value F of X. For example, say F of X is X squared. So that means that we can, yeah, I can ask you, I can give you X and you will tell me immediately what F of X is, say, X is zero, it's zero, X is one, it's one, if it's two, it's four, but it doesn't have to be an integer. Anything pi has a value, for any value, I can substitute it into this function and get this. In other words, it is just writing this formula. It's just a very economical way to describe this rule. Instead of making a table where for each value of X, I will write the value of F of X. And actually, I wouldn't be able to do it because there are infinitely many values. Instead of making this table, I just do it in a single formula because this formula, I can use this formula to convert any given value in the domain into a value in the range by simply squaring it. So one formula takes care of all elements in the domain which is the set of real numbers. That's what we mean when we say function of F of X. It's not one number. It's not even a rule which assigns to one number and not a number. It's a rule which assigns to any number here, which is R, some number here. Okay, is that clear? Now, sometimes when we write certain functions, sometimes the domain is not necessarily the entire R. For example, sometimes we write F of X is one over X. So in this case, we cannot say the domain is R because the value X equals zero is forbidden. This rule does not tell us what the value is at X equals zero because it's X equals zero. To get the rule from this formula, we would have to calculate one divided by zero and we know that one divided by zero is meaningless. There is no such number, one divided by zero. So we have to make some adjustments. And the simplest adjustment is to say that domain for this function for this function is not R, but it's R without this point zero. So all non-zero numbers. So, but the function doesn't have to be, doesn't have to have as a domain R and the range to have R. A function could have any domain and any range. For instance, let's say the domain is a set of all students in this class. And the range is letters A, B, C, D and F. So then the function is a grade, a final grade, okay? Because grade, well, it's actually a little more because there's like A minus, A plus, B plus. Okay, but let's simplify. So A, B, C, D, and I don't know why there's E because it's never used, so let's say F. Anyway, we'll hope that we don't have so many values like this. So in principle, any function can happen, right? In principle. And I personally, I wish that everybody gets an A. It's possible if everybody gets great results on the tests and everything. Everybody will get an A. Now, historically, that's not what has happened. So I don't, being realistically, I don't expect that. So this is a function is going to be, sometimes people call this a curve if you, because you can look at the scores and then you can try to draw the curve, looking at the number of people getting certain, and so on, right? So it's already suggest some connection to functions, but just think about it in a simple way. Each student gets a grade, so that's a function. Each element in the domain, which is a student, gets a grade, which is a letter from A, B, C, D, F, C. So that's a function too. We don't know yet what this function is. We will find out by the end of December. But this is also a good illustration, just by way of example, that anything where you have a certain input and then have a rule which transforms this input into some output is a function. Now, in this course, we look at domains and ranges, which are R to the N. So in other words, what we are going to do now, and what's different between this course in the previous calculus course, which had to do with single variable calculus, is that we are going to allow, not just R, you're going to R, but we will look at R to the N, going to R to the M, where N and M take values one, two, and three. You have a question? Second? Oh, this one. Okay, the question is about this notation. Is this notation familiar? No, okay. So I'm glad that you asked me about this. So this means belongs to, this means belongs to, or is an element of the set. And this means the same, but now in this direction. So I just switched it. Usually we write like this. I did this to make it look nicer because then I'd have to, otherwise if I switched them, we'd have to draw a curve like this. But it's the same. The three Ns is to where it belongs. So this means X belongs to R to the N and not the other way around. And here F of X belongs to R to the N. Is it clear now? Okay, good. So the domain and range that we are going to look at in this class are usually going to be R to the N and R to the M, which means R, R to R three. Sometimes we will have to look at some subset here. We'll have to remove something because function for example may not be well defined at a certain point, like this function. This function one over X is not well defined when X is equal to zero, or more precisely this rule does not tell us what to do at X equals zero. So that means this rule is really only defined for the domain R without the point zero. Something like this can happen here as well. Okay, and so now if we stand on this point of view, that there are these functions which have as domain R to the N and as range R to the M for some N and M, one, two, three, which could be arbitrary, not necessarily equal. Then we can describe all the previous examples as sort of object under the same umbrella because for instance vector valued functions we can now interpret as functions in which the domain is R. Domain is R and that's where our variable T is going to live. And the range is R2 or R3. So in other words, in the case of R2 you will have T going to F of T and G of T, an element of R, see you can also do this sign in this way. And then you get an element in R2. And likewise in R3, each T will go to F of T, G of T and H of T. So again a point in R will go to a point in R3. Note that there is a small difference which I already mentioned earlier which is that you can treat F of T and G of T as components of a point or as components of a vector. So now I'm really standing on the point of view that I'm getting a point in R2 instead of a vector but because these are so closely related we can think about both scenarios as functions. It's just, it just becomes really the matter of what do I mean by R2 or R3? Do I interpret R2 as a set of points or as a set of vectors? And in fact it's better to think of it as a set of vectors but we oftentimes think of it as a set of points. But the points and vectors are given by the same information by two components anyway. So that's why we really, here I'm really thinking of R2 as being collections of two numbers, X, Y. First coordinate, second coordinate. You see, and so I'm not worrying about the fact whether these are components of a point or components of a vector. I know that in each particular situation in any given context I will be able to interpret it in the right way. I'm just using the fact that it has two components. Okay, so vector value function now becomes an example of a function because it's just a rule assigning now to a single variable, to a single number which we usually call T, a pair of numbers like here or a triple of numbers like here. Does it make sense? Yeah? Okay, so that's what we mean by function with values and vectors. So in this case our domain is R, is always R. But the range is R3 or R2, you see. So we go sort of from the smallest one to the two big ones, R2 and R3. And now we can understand very easily what are functions in two and three variables. It's sort of the opposite situation where the domain is R2 or R3, but the ranges are. So in that sense it's no more difficult to understand as a concept than vector value functions or even just ordinary functions from R to R. It's all part of the same story, of the same concept of functions. So going back to the title of this topic, functions in two and three variables, by a function in two and three variables, I simply mean a rule in which the domain is R2 or R3, if I want two variables it will be R2, if I want three variables it will be R3, and the ranges are. So maybe I will write it here. Let's just write it like this, R2 to R and R3 to R. So then how to express such a rule? So domain is now say R2, so it's a set of all pairs of points and the range is just R. So it means that each x, y has to go to some number. So we have to denote this number in some way and the way we denote it is say f of x, y. For example, I can write f of x, y is equal to x, is equal to x squared plus y squared. What do I mean by this? I mean that whenever you give me some values x, y, I can substitute them into this formula and I get a number. You give me two numbers, but I give you one number because I do something with them. I take a squared and y squared and I take the sum, so I get one number. So it is a rule which assigns to this one number. That is an element of R. That's why writing this formula can be interpreted as giving a function in two variables. Two variables because I point in R2 has two components x and y which we think of as two variables. So again, I could try to make a table, different values of x, y and the corresponding values of f. So for example, I start with zero, zero, I get zero, I start with one, zero, I get one. You know, I start with one, one, I get two. But of course, I cannot make the table for the entire R2 because there are infinitely many points in that table. So instead, I simply write this formula. So this kind of functions appear in real life all the time. And a typical example could be a map of temperatures or atmospheric pressure. In this case, you can think of some domain being, you can draw a map of a certain region, save the Bay Area. And then you plot it on the x, y plane so that each point in the Bay Area gets the coordinates x and y, and then you say, you know, like on television when you watch the news, they show you this and they write temperature like 85 degrees or whatever. Here and then maybe here it's like 90. And then they hear, they say something else, 83. And in San Francisco, it'll be like 55. No, I'm joking, maybe on a day like this, maybe 70. Okay, so what is it? You assign a value to each point. So you are defining a function. Each point on this map, each point in this region, x, y, corresponds to certain temperatures. So to each point, you assign a temperature. So that's an example of a function. Likewise, you can think of the landscape, you can think of the map of this area and you can look at the height of, like, each point, the height of this point as compared to the sea level. So then again, the height gives you a function. For each point, x, y, you get a particular height. And this actually gives a way to, leads to a nice idea of trying to interpret functions or visualize functions into variables in terms of graphs. Of course, for functions in one variable, this is a very old idea. Functions for one variable, we've been drawing graphs for a long time and this are almost a reflex. If I say y equals, if I say y equals f of x squared, let's just say f of x equals x squared. So you immediately think about this picture. So this is a picture which you draw on the plane by introducing an additional variable. So this is your x and there's an additional variable you introduce to accommodate the value of the function. And then for each value, you plot the point on this plane by taking as the x coordinate the value of x and as the y coordinate, you take f of x. Of course, here again, there is a clash of notation which you have to realize. I mean, it's sort of again very clear from the context but I would like to mention it anyway. When we talk about functions in one variable, we talk about there are graphs and the graph of a function in one variable lives on the plane. So even though in your original problem, the domain is one dimensional, it's just x. There's only one variable. You are forced to introduce a second variable just because you also want to accommodate the value, the range, the element in the range. And we usually call this coordinate y. So we write the graph as y equals f of xy. Y is equal to f of x. But this y has nothing to do with this y because when I talk about functions in two variables, I talk about functions in x and y. Why? Because x and y is like the two variables, they go next to each other in alphabets. So it's natural. Once I say x, I have to say y. You would be surprised if I say function of x and a or something like, what's wrong with him? Why is he talking like this? So once I say x, the next one is you say y. But now in this context, y has nothing to do with the domain. This is a domain, it only has one variable. And this y corresponds to range, to the range, you see. But now when we talk about functions in two variables, we use the same notation, x and y, to indicate the two variables in the domain. So now to draw a graph of this beast of this function, we have to go to the three dimensional space because we have to introduce one more variable, which corresponds to the range. The range in both cases is r. But here I denote it by y because I use just x. So the next letter that comes is y. Now I already use x and y for my domain. So I have to use a different coordinate, different name for the range. And usually I say z, just because it comes, it's the next letter in alphabet, right? So for instance, if I choose as my function in two variables, this function f of x, y equals x squared plus y squared, what kind of picture do I get? It will just be a surface given by the equation z equals x squared plus y squared. And fortunately, we have already discussed the surface. So we know what it looks like, right? This is an example of an elliptic paraboloid, which looks like this, right? And I kind of want to indicate some more curves on it just to give it sort of more shape, okay? So that's a paraboloid. So this is a graph of the function, which is a paraboloid, elliptic, right? Likewise, you can also talk about functions in three variables. So here I talk about function in two variables, but I can also talk about function in three variables. So in this case, domain is R3, and the range is R. R3, and the range is R, x, y, z, right? So of course, this is also meaningful because for example, if you talk about parametric pressure, if you talk about atmospheric pressure, well, for temperatures, we usually just measure the temperature of the surface. But for pressure, it makes sense to also look higher. So we're really talking about a point in space, and we assign to a point in space the pressure. So it's really a function in three variables. So really for each point in this classroom, there is a particular value. Temperatures also, for example, we could measure temperature at each point in this classroom, and there will be some small variations. If you wanted to draw a graph of this function, though, you would have to introduce one more variable. And now we've run out of letters because z is the last letter of the alphabet, right? So then we start from the beginning. So then we can say a is equal to f of x, y, z. So the graph is now going to be, graph is in four dimensional space. But since we are not, we cannot really visualize R4, we're not going to talk about this. What we can talk about, though, are the so-called level curves and level surfaces. So when I draw this graph, I almost feel the urge to draw this additional circles because when you look from far away, this kind of gives a certain depth to the picture and it makes you understand what I'm talking about. And in fact, it looks even better if I draw a few more. And so I am actually indicating, I'm kind of thinking that they are transparent. So I'm indicating also the backside of those circles, but they're not as visible as the front ones, okay? So what are these circles? This is something that we've learned when we talked about quadratic surfaces, of which, of course, elliptic paraboloid is just one example. When we talk about quadratic surfaces, we learned that a very efficient way to understand the structure of these quadratic surfaces is to cut them or slice them by planes parallel to a given coordinate plane. So in particular, we could slice it by a plane, z equals something, cut by the plane, z equals k. So when we cut this graph by this plane, we get a curve because it's intersection of two surfaces now, and the intersection is a curve. What is this curve? Well, look at this equation and look at this equation. Now, in this equation, this is a number, k is a number. So k, for example, could be one, two, three, four, whatever. So that's different from A, B, sorry, a different from x, y, z, which are variables, which can take arbitrary values here. Now, if you look at this equation and you couple it with this equation, what you get is, of course, x squared plus y squared equals k, and that's one of those circles. So this, when you slice it, the set of intersection will be a curve, and that curve will be given by this equation, which indicates that this is a circle of radius square root of k. And these circles, we get by looking at all the points on the graph which share the same value of z. So in other words, it's kind of like the same level. If you think of this as being the landscape, so there's sort of a hole in the ground, then this will be all the points which are at the same height, or sort of the same level below the surface. You see? So in other words, what this is, this will immediately leads to the terminology, level curve. This is all the points on the graph which are at the same level, and saying level just means fixing the value of z. Okay? Likewise, when you have a function in three variables, you can also fix this value a of the variable a. So you say a is equal to k. And what you get is an equation like this, f of x, y, z is equal to k. And so what it is now, it is a surface, and we call it the level surface for this function. In other words, for each value of k, for example, one, two, three, four, and so on, we can look at all points x, y, z which share the same value with respect to the function f. So that's now a surface, just like before it was a curve, but now we have an extra variable, so we get the surface, and that's called the level surface. And so you can look at various examples of this. So that's roughly what we need to know about functions in two and three variables from the sort of qualitative point of view. And now we would like to understand them from a more quantitative point of view. What can we do with this function? And of course, our goal really is to do derivatives and integrals of these functions. And that's really the crux of this course. It's really understanding derivatives of these functions and integrals of this function. And you see right away that it's going to be more difficult than before because for instance, if we think just in terms of two variables, for example, we can differentiate respect to x or we can differentiate respect to y. And in fact, there's more because we could even differentiate in some direction which is kind of in between, which is kind of a combination of x and y, you see. So there are many more choices. And we have to understand much better what derivative means in this context. Same with integrals, but first we'll talk about differentiation. Now, when we define differentiation and I kind of talked about it earlier, when I talked about the derivative of vector function, the derivative is defined in terms of a limit. When you take the sort of the change in position divided by the change in time, when you get the velocity, for example, you take the limit. So before we can talk about derivatives, before we can talk about derivatives, we have to talk about limits. We have to understand what limits mean in the context of functions in two and three variables. And this is a somewhat obscure subject because to really truly understand it, we have to work pretty hard because we have to give a precise definition of the limit and things like that. We're not going to do that. We're not going to prove theorems about limits. If you like, you can read about it in the book. But what I'd like you to understand is the kind of a qualitative picture. In other words, the main ideas as to what the intuitive idea of the limit is and what kind of examples, what are the possible scenarios, what are the possible situations when limit exists or doesn't exist. And that's what I'm going to explain. So let me start with functions in one variable. If you have a function in one variable, let's look at the graph of this function. So let's say this is a graph of function. You pick a particular point, x0. So you want to see what happens with the value of the function in the small neighborhood of this point as you approach this point. You see? You want to see what happened. Can you hear me? Yes, okay. Tell me if you'd stop hearing because I have the red light. So I hope it works for another 15 minutes. Otherwise we'll have to see if I can find a spare battery. So we would like to see how the value of this function behaves in the neighborhood of this point. In other words, how predictable the function is. As we approach this point, we can trace the values of this function and we can try to see whether they also tend to the value of the function at the point x0. Now, of course, on this picture, it's clear that yes, because it's kind of smooth. I drew it in one stroke, which is pretty much to say that it is predictable function or what we call more precisely continuous function. But that's not the only possibility. I could draw something like this. I could draw instead something like this. Where are you going like this? And then you go like this. So in other words, the function jumps here. So let's say the value to the left of this point the values kind of behave in a nice way. That if I start from the point below x0 and I start moving towards x0, the value will move towards the value of the function at x0. And that I want to indicate that this is not, this circle means that there is no point here. This branch only takes care of the points which are larger, greater than this one. Okay? So on this side, if I start approaching this point, I get values on this branch. But then when I get to this point, there is a jump. The value is here and not here. So in this case, we will say that the limit of this function does not exist because it exists on one side, but it doesn't exist on the other side, you see. In other words, the limit of the function exists but is not equal to the value given. So the function is not continuous. There is another option, there is another option. The value could go to infinity. In this case, we will also say that the limit doesn't exist. For example, our favorite example for this is the hyperbola y equals one over x. So the limit, no limit as x goes to zero. And that's not all because another thing that can happen is the function may not go to infinity but it can oscillate like crazy. For example, the function, a typical example is function sine one over x. Around zero, this function doesn't have a limit because as you get closer to zero, one over x goes to infinity and the function starts oscillating faster and faster and in sort of, you never know, you cannot say what is the limiting value because it somehow is everywhere between minus one and one. So again, no limit at x equals zero. In other words, you see that there is sort of the most favorable scenario in some sense, which is the most typical one, what we are going to use mostly in this course. But at the same time, you have to realize that that's not the only possible scenario. It's really a favorable one, but it's quite possible to have different situations. It's quite possible to have a situation of a jump that the limit from one side is not equal to the limit from the other side. There is a situation where the value goes to infinity so again, there is no limit or the function oscillates in a way that makes it impossible to say what is the limit when it approaches a particular point. And that's just the case of function one variable. So in the case of a function two and three variables, there are even more scenarios. And what I would like to do is to explain just a couple of them so that you'll get an idea sort of what these functions of the way those functions could behave. So first of all, the favorable scenario still remains. All the functions in two variables that we looked at up to now, for example, the quadratic functions, they are all well behaved in the sense that they are predictable. As you move, you take points x, y in a small neighborhood of a given point x, 0, y, 0 and you take the value of the function f of x, y. Then when x and y tends to x, 0, y, 0 x, 0, y, 0, the value of this function will also tend to the value at x, 0, y, 0. So that's, in this case, we will say that the function has a limit x, 0, y, 0 and is continuous. And functions like polynomial functions, for example, and all kinds of functions involving sine and cosine and the exponential function are all continuous. You get in trouble when you start dividing by something that could become zero, which of course you see already in the example in the one-dimensional case. When you divide by something which could become zero, you are in trouble. But in a case of two and three variables, there are sort of more possibilities. There are more possibilities to divide by zero. And the typical examples which you will see in the book and on the homework involve division by some expression, some polynomial which becomes zero at a given point. So a typical example is a function f of x, which is equal to say x squared divided by x squared plus y squared. So the question that you'll be asked is whether this function has a limit, does it have a limit as x, y tends to 0, 0, 0. You see, if for example you had x squared plus y squared plus one, you don't even have to think about this really because this is actually never, never zero, right? It's never zero because you add one. For sure, it's not zero in a small neighborhood of this point. So you can even plus any small number will do. So then there is no issue. This doesn't become zero. And therefore you can be assured by various theorems which can be proved in a fairly straightforward way that actually this has a limit and the function is continuous. The only reason why we have to worry about this is because we have an expression which actually becomes zero at this point. So we can't really divide. We can't literally divide by this. We cannot literally divide by this value. It is zero, so we cannot divide by zero. Now, if here you had something like x squared plus one, again, you don't have to think about it too much because or even for simplicity, let's just say one divided by a square plus one. This will be exactly like this. I mean, the thing just blows up because when x and y go to zero, this becomes very small and you take one divided by this so it becomes very large. So when x, y go to zero, zero, this goes to infinity. So clearly there is no limit. So again, that's an easy case. You see, that's an easy case. In other words, if the numerator doesn't go to zero and denominator goes to zero, the thing goes to infinity and I mean the value of the function goes to infinity so there is no limit. So the only subtlety is when both numerator and denominator go to zero, like in this case. So in this case, sometimes it could be that it has a limit and sometimes it could be that it doesn't. And so I have just enough time to tell you, right, I have just enough time to tell you what happens in this particular case. And then you will see in the book and on homework you will see other examples, very similar to this. So I'm going to show you that this function actually does not have a limit. So here's how I'm going to do that. I'm going to approach the point zero, zero, the origin in two different ways. So approach in two different ways. The first way I'm going to approach it is I'm going to say, I'm going to approach it along the y-axis, like this. So in other words, in the first way I will say that xy is zero y and then y goes to zero. So this way I eliminate one of the two variables, namely x, and I get a function in just one variable y. So I kind of go back to the one-dimensional case which I can analyze more easily. So let me see, what do I, ah, I'm sorry, I made a mistake. It's f of xy. I have a function in two variables, but now I convert it into a function in one variable. So I set x equals zero. So what do I get? I get zero squared divided by zero squared plus y squared. If y is non-zero, this is non-zero, which is zero divided by y squared. When I'm approaching zero, right, I'm approaching zero, but I'm not at zero. I'm looking outside of the point zero, but very, very close. So outside of the point zero, this is very small, but still non-zero. So that's why I'm allowed to take the ratio of this, because why is not zero? It's not zero, but it's approaching zero. So this is okay, and it gives me zero. So along this ray, along this ray, along this path, I get the value zero. So that's the first calculation. In the second calculation, I will go along this path. That's number two. And now I'll do the opposite. I will say xy is x zero. And now I will take x to zero, but not equal to zero, and see what I get. So I get f of x zero is equal to x squared divided by x squared plus zero squared. So that's x squared divided by x squared. Now, if I have such an expression, x squared divided by x squared, and I know that x is non-zero, I can take the ratio, and it's just one. So the upshot of this calculation is that the result, the value of the function depends on the path along which I'm approaching my point. You see, it is in some sense, it's very similar to what happens in this picture. If I approach from this side, I get this value. If I approach from this side, I get this value. But if my domain is just one dimensional, that's the worst in some sense that can happen because the only ambiguity I have when approaching the point is from this side or from this side. But when the domain is two dimensional, there are many more choices. I can approach it along this path, along this path, along this path. In fact, there are many other paths as well. And so what you need to see for the function to have a limit is that along each of this path, you are going to get the same result. I have now shown you that along one of them, you get zero, and along the other one, you get one. So you get different results. That means that the function does not have a limit at this point. Okay, so we are out of time, so I'll let you go and we'll continue next week.