 Question? Sorry? Oh no, I'm working out, but I'm not working out. I will be here on Thursday. So in as much as I sympathize with the organizers and I think that they really have very valid points, they've raised very valid points, I also feel that my job is to be here, so I will be here on Thursday. Any other questions? Perhaps even more important, next Thursday we'll have our first midterm exam. Okay? Yeah, everybody's excited. So I already put an announcement on the B-Space page, which by the way, you should check B-Space site because there's all kinds of interesting stuff in there and useful information also. So homework solutions, something I have already told you about, but also there will be some information about the exam. I'm going to, this week I'm going to post a mock midterm so that you will have a chance to practice for the midterm and we'll have a review lecture next Tuesday, a week from today. And the exam is on Thursday exactly during the class hour, one hour, 20 minutes more precisely. We'll start at 3.40 and finish at five. Because this room is not big enough, because everybody be packed, so I have requested another room, so we'll have to separate into two groups. I will post information about this and I will tell you about this and everything about this on Thursday or next Tuesday. Okay? And now I go back to what we talked about last week, which is limits, I just wanna say a few words about limits, and then we will move on to the next subject, which is partial derivatives. So about the limits, too many late arrivals today. I'm going to charge a late arrival tax, if this continues. So let's just quiet a little bit, because it will be good for everybody if we are more focused, okay? So limits, I illustrated limits by way of an example, and I looked at a particular function into variables, namely x squared divided by x squared plus y squared near xy equals zero, zero. And the reason is that at this point, the denominator becomes equal to zero, and so this expression becomes problematic. It may or may not have a limit. If we did not have that, if for example, we looked at this function near a point xy equals one zero or one one, a point where the denominator is not equal to zero, that would be an easy question to handle. But precisely when the denominator becomes equal to zero, we have to be careful, and we have to analyze it more precisely. And so in this particular case, I explained that this function doesn't have a limit. This function does not have a limit at this point. And I explained this, I explained the reason. The reason is that we could find, the reason I gave was that I showed two different paths, two different paths on the xy plane, which approach zero. One was the path when x is equal to zero. So it's a green arrow. And the other one was the path when y is equal to zero, and that's a red arrow. And we have seen that along the first path, when x is equal to zero, when we look at this function on this path, very close to zero, but not quite equal to zero. What we get is zero divided by zero plus y squared. But as I said, we really look at it not at this point, but just near this point. So near this point, y is not zero. So this is not zero near the point. Because it's near the point, we can actually evaluate this. And we see that for all values of y near this point, this is actually just plain zero, because you divide zero by something non-zero. And so this means that along this path, this path, there is a limit and it is equal to zero. On the other hand, if we look at this path, we get x squared divided by x squared plus zero. And again, we will assume that x is very close to zero, but not quite zero yet. So this is actually non-zero expression. And therefore we can cancel them out. So this again is non-zero. And this actually gives us one. So that means that along this path, again, there is a limit along this path, there exists a limit, there is a limit along this path, namely one. So we have found two paths along which we achieve a different limit. We attain a different limit. That means that the function itself does not have a limit. You see, it does not have a limit. Because to say that the function has a limit, function has a limit, if along any path approaching the point, the limit exists and all of them are the same. In this particular case, clearly, they are not the same because there are two paths, at least two paths along which the limits are different. So how can it be that a function has different limits along different paths? In other words, what is the meaning of this? What is the geometric representation of this? Well, to explain that, we can already see an analog of this phenomenon for functions in one variable, which I explained last time. In the case of one variable, we look at the graph of a function as a geometric representation and we can have the following situation where the graph is discontinuous and grab the same color. So graph is discontinuous, for example, like this. So there's some point at zero and if you approach from the left, you're going to end up at this point and if you approach from the right, you end up at this value. So that means that function does not have a limit at this point because along different paths, you get different limits. They are both finite. So in some sense, you can argue that it's a more benign situation than, let's say, a situation of a hyperbola where actually it goes to infinity. Here, by the way, on the right, it goes to plus infinity and the left goes to minus infinity. So in some sense, it's even worse than just going to infinity. It actually goes to two different infinities in some sense. But first of all, if it goes to infinity along any path, it's already, we already say it doesn't have a limit. But even if it's finite, this situation, which is in some sense more benign, it still doesn't have a limit. It has a limit from the left, it has a limit from the right. In the case of one variable, there are only two possible paths which converge to this point. Well, you can go sort of at different, with different speed, different velocity, but it doesn't matter. I mean, geometrically, it's the same path. Either this one or this one, only two. And already that creates trouble. In the case of functions in two variables, there are many more paths. I have drawn two paths here. Here is one, here is another. But you have many more, right? You also have a path like this. Finally, a path doesn't have to be a straight line. It could be a spiral. So there are many, many different paths, and that's the difference between two variables and one variable. To say that the function has a limit for a function in two variables is a very strong statement. It's a statement that along any of those paths, you are going to achieve the same limit. So this actually shows that in a way, it's much easier to disprove something here to show that the function does not have a limit, than to show that the function does have a limit. Indeed, to show that it doesn't have a limit, it's sufficient to just exhibit two different paths along which you have different limits. And usually it's pretty clear which ones you should take. For example, in this case, you just look at x equals zero, y equals zero. Sometimes you might need to look at a linear path like this one where x is equal to y. Again, to convince yourself that indeed the function doesn't have a limit. But to prove that actually the function does have a limit is much more difficult because it wouldn't be enough, for example, to say that the limit along this path and along this path is the same. You would also need to show that the limit along this path is the same and along infinitely many other paths, right? So to show that it has a limit is more difficult and this is not a very efficient way to show it. It is very efficient. This way of argument is very efficient to show that it does not have a limit because then it's enough to show just two for which the limits are different. But to show that it has a limit, it wouldn't be enough to show that along two paths you get the same answer. You have to show that the same holds for all paths, okay? So for practical purposes, what you need to know is the argument showing that it doesn't exist. You need to show, you need to know this way of argument. When the limit doesn't exist, you should be able to demonstrate that there exist two paths along which the limits are different, okay? Yes? Yes. The distance, right? The distance. What really matters is the distance. What should matter? So the question is what really matters? The shape of the spiral, the shape of the curve or the angle? It depends on the situation, right? The point is that to have a limit it means that as soon as you get close, say within one over 100 of an inch of the origin, the answer is going to be within some small neighborhood of the value that you claim is the limit. It doesn't matter how you approach it, it should be within that limit. If it's one over 1,000 of an inch, it should be even closer, it's one over a million, it should be even closer and so on, right? So the notion of a limit should not rely on the way you approach. It should be about, it should be uniform with respect to all directions and all points in the neighborhood of the point, of the point zero, zero, you see? So in a sense, this argument that I gave you is kind of misleading. It's a very nice argument to disprove the existence of limit, to show that limit does not exist. But it is misleading if you try to think in this way about the existence of limits, okay? So for existence of limits, you have to use a different kind of argument. And now I'm not going to require that on the exams. But I'm going to just give you, just to give you an idea of how it works. I'm going to explain it in the following case. Suppose you have a function which is just slightly different from this at first glance. Namely, instead of x square, I will take x cube. You see, so what happened? So the problem here, the problem with this function was that both numerator and denominator had degree two. So both numerator and denominator in some sense are going to zero at roughly the same speed. But not exactly. It depends along which direction you go. If you go along this one, it will be zero, along this one is one. Along this one, for example, if x is equal to y, you can see it's going to be one half, you see? But because the powers are the same, that's why you end up with different answers. What happens now is that I put the numerator, I choose as the numerator x to the third power and the third power goes to zero much faster than second power. So that's why this will dominate and this will kill this guy. So it will become zero. You see, so that's, it's hardly kill because they both go to zero. So in some sense, it goes faster to zero. So it doesn't really kill it, but anyway. It depends on your point of view. So how would I show that this actually has a limit? So I claim, I claim that this function does have a limit at zero, zero. Namely, namely the limit is equal to zero. How would I show that? Well, for that, I would actually have to estimate the value of this function. I would have to estimate the value of this function when I approach zero. So I will say, let's suppose, so let me explain this as a proof. Suppose that X, Y are within, belong to the small disc of some radius R. And I purposefully don't want to use delta and epsilon, not because I don't like Greek alphabet, but because I know that people immediately feel violated when I try to talk about epsilon and delta. So I might as well, I think if I use a different letter, you will feel much more comfortable, some of you will feel more comfortable. There is a certain, just the way it sounds. Actually, one of the students told me that it reminds him of going to dentist's office at epsilon delta. So let's call it R. And let's look at the disc of radius R. I'm drawing it as a big disc, but actually you should think of it as very small. I just magnified it. So this is DR. And I want to look at all the points here. And I want to estimate the value of this function for all points within this disc. So the function, as I explained already many times, is a rule which assigns to each of these points a certain number, which is the value of the function, right? So that's the function F. Oh, that's why. And I want to estimate the value of the function. And what I want to show is that the closer I get here, the closer I'm going to get to the neighborhood, to the neighborhood of the zero value on this line. So how can I estimate this? Well, what does it mean that it belongs to this? It means that X squared, square root of X squared plus Y squared is less than or equal to R, right? That's what it means, because that's the distance. That's how we measure the distance to zero. To belong to a disc of radius R means that for your coordinates X and Y, the square root of X squared plus Y squared is less than or equal to R. So that means that X squared plus Y squared is less than or equal to R squared. But you see both of these are positive. So if the X squared plus Y squared together are less than R squared, this also implies that X squared is less than R squared. And so X is actually less than or equal to R. This is actually also clear geometrically because if you have all points in the disc, you will see that all of them will have the X coordinate less than R, except for the two sort of points which lie on the intersection of the circle with the X axis. So I know this as soon as my point is in DR. And I also know this, right? So now what I would like to do is I would like to write the following. I would like to write X cubed divided by X squared plus Y squared. And I want to measure the absolute value. What matters is the absolute value. When I say it's close to zero, it doesn't have to be positive or negative. It should be very close to zero on either side. So that's why to estimate it's better to take the absolute value rather than just the value. And so now I want to write it like this. I want to write as X times X squared divided by X squared plus Y squared. I've done nothing. I just pulled one X out of this fraction. And then I want to write it as X times X squared divided by X squared plus Y squared. So now I want to look at this and clearly this is less than or equal to one because in the numerator I have X squared but in denominator I have X squared plus Y squared. And this guy is always positive or zero. So the largest value this can attain is one when Y squared is zero. But if Y squared is non-zero, the numerator will be strictly less than the denominator. So this fraction is less than or equal to one. And X as I have just explained is less than R. Less than or equal to R. So that means that the whole thing is less than or equal to R. So you see, I have been able to effectively estimate the value of this function. I cannot find it for all X and Y. I mean, they're all different for different X and Y. But I can say for sure that as long as X and Y, as long as the point X, Y is within that disk of radius R, the value is going to be less than R. Less than or equal to, it doesn't matter. So that means that if I make this disk smaller and smaller, in other words, as I take R closer and closer to zero, this value of the function, or more precisely the absolute value of the function, will also tend to zero, right? Because for all points within that disk, all points, this value is going to be less than or equal to R. And I have control over R because I can take my points in a smaller and smaller disk. You see, nowhere in this argument am I talking about particular paths approaching zero. I'm talking about the entire disk of radius R. And then I'm squeezing that disk and taking smaller and smaller and smaller. And by being able to use this estimate, I can control the value as a function of the radius. I can say that if I am within the radius one inch, the value of the function is going to be less than one. If I'm within the radius one over 100 over an inch, it's going to be less than 100, one over a million. In other words, I can get as close as I want to zero for the value by choosing sufficiently small disk. And that's the argument. That's the proof that the function has a limit and the limit is equal to zero. And that's what traditionally mathematicians explain in the epsilon delta language. But that's all it is. That's all it is. It's just saying that if you take a disk of radius R, then the value of the function for all points within that disk is going to be less than R or less than something which becomes smaller with R. So that's the argument. Are there any questions about this? So yes, that's right. In general, it will not necessarily be R. It could be, let's suppose I proved, let's suppose I had a different function. For example, I had two times x cubed. Then I would prove it's less than two R. That would still be okay because I have to get something which will become smaller and smaller as R becomes smaller. Or if I get R to the one half, the square root of R, I would also be okay, or R squared. It will not be okay if I just say that it's less than one, that the whole thing is less than one. That wouldn't help me. That would just tell me that I get within a certain range, but that range doesn't get smaller as the domain gets smaller. I have to make sure that the range will get smaller as the domain gets smaller. And that's the perfect situation where I didn't have to make any adjustments. It's just I get R on the nose as the estimate, as the largest possible value for this. Okay, another question? Would that only work? Yeah, well, this is a simpler, the question is whether this is a general argument. This is not a general case because in general, you're going to have maybe some polynomial and x and y with additional terms. I have really looked at the simplest case. But in general, the argument is going to be very similar. And if you like in the book, there are more examples of this type which are analyzed. But as I said, I'm not going to require you to know this. So my point here is just to give you an idea of how this kind of proof works. And I think that even though this is a simplest example, it already illustrates this idea. Okay, yes? L'Hopital's rule. Well, L'Hopital's rule is really specific to functions in one variable because you have to differentiate, right? So L'Hopital's question was about L'Hopital's rule, which was one of the powerful methods of finding limits in four functions in one variable. And so the way it works is that if you have a function which is say p of x divided by q of x, and both of this tend to zero, say, and you don't know what the limit is, you can estimate the limit by taking derivatives. So now the situation is different because now we have, this is one variable case. And now we have two variables. And in two variables we have functions in two, say p of x, y and q of x, y. So let's say I wanted to generalize this. I would have to take some sort of derivative here. And this actually brings up the question which we need to study next, which is what kind of derivatives can we do for functions in two and three variables? So clearly there isn't a single derivative because derivative is about the rate of change, right? The rate of change. In a one-dimensional case, there is only one direction in which you can change your variables. You can increase that, you can just go away from x. Apart from the fact that you can change it going left and right, there is essentially only one way to change it. More precisely, there is only one degree of freedom that you can change. You can only change in one direction. But in two variables, there are more directions in which you can change and estimate the rate of change. And therefore there are many more derivatives that are possible. So in fact, this is very close to what we discussed up to now. There are so many different ways to approach zero that we have to be able to take care of all of them, right? So what L'Opital's rule in two dimensions would give us at best is a way to approximate the limit along a particular direction, right? So let's say if I were to take the derivative with respect to the x direction, then I would be estimating what happens when I approach along this line. And if I were to take the derivative with respect to y, I would be estimating along the green line, along the y axis, right? But neither would give me the full picture. The full picture I can only get as I have argued by sort of looking at all paths, right? Looking at all possible directions. So in that sense, the L'Opital's rule doesn't help us. Now sometimes, sometimes it happens that you can convert this function that you have into variables, into a function one variable. One of the exercises, I think it's the last one on homework, is about this. Where you can use polar coordinates to realize that the function you have is something like, it involves x squared plus y squared. I think it's something like logarithm of x squared, y squared times x squared plus y squared. Something like this. So you realize that actually, it only looks like it's a function two variables, but actually it's a function one variable, namely this. And then you are back to one variable case, and then it becomes a fair game to use all the methods that you know in one variable case. A general function is not going to be like this, right? For example, this one is not like this. So I cannot directly use, I cannot directly use L'Hopital's rule for two variables. And there isn't any obvious way to use it because there is more than one possible derivative. So which actually brings up the question as to what are possible derivatives for functions in two variable? And that's our next subject. So I have already kind of alluded to the answer because there are two variables, we can actually differentiate with respect either one of them and we get a meaningful derivative. And these are called partial derivatives. Partial derivatives. So what are partial derivatives? So we have a function, let's say we have a function f of xy in two variables. And when we talk about derivatives, first of all, we should fix the point, we should fix the point at which we are taking the derivative, right? Because for a different point, you'll have a different derivative. It's the same thing happens same thing happens for functions in one variable. So let's say we have a point which has coordinates a and b. What we can do is we can convert this function into a function in one variable, but by freezing one of the variables. Freeze one of the variables. So for example, we can say y is equal to b. Freeze the second variable and say that y, the variable y is equal to b, which is that the second coordinate of this point. So what we get then is a f of x and b. And let me indicate the fact that we have frozen, we have frozen this by red. So red would be a fixed value. So here also would be, I'll put them in red to indicate that these are numbers like one or five or 27 over 11, whatever you want, right? But x is a variable. So x, we can plug in any number you want and you'll get an answer. So you want to view it still as a function, but because you have frozen one of the two variables, there's only one variable that's remaining. Therefore, what you get is a function in one variable only. Function in one variable. And once we get a function in one variable, we can then differentiate it just in the usual way, how we differentiate functions in one variable. Differentiate it. So we'll get say g prime. And then we can substitute the value that we wanted, the value a. So then finally we get a number. So in other words, we had a function in two variables. First, we freeze one of the two variables and then we take the derivative with respect to the second variable at the particular value of that variable, namely a. So the result of this is what's called the first partial derivative, or partial derivative with respect to x. Partial derivative with respect to x. At this point, at the point a, b. And the notation for this is f sub x. Likewise, I could freeze the second variable, I mean the first variable x, I could say x is equal to a. Then I get a function again in one variable where the first variable is frozen, but the second one is free. So I get a function in one variable, let's call it h. And then I can differentiate it. So what I get is this h prime of b. And that's called the partial derivative. Get partial derivative with respect to y. For which notation, I'm just abbreviating the same sentences I have at the top of this board. The notation for this is obviously f sub y, f sub y of a, b. So we got two derivatives for functions in two variables. First and second. Now, so let's look at an example of what this looks like. Let's say f of xy is x to the five plus x times yq plus cosine x times e to the y. And we would like to find the partial derivatives. Now, when I define them, when I define them, I was insisting that the value of, that the derivative has to be evaluated for particular values of a and b. Sorry, particular values of x and y, which I denoted by a and b. Just like in the case of function in one variable, let's say if you have a function, let's say for function in one variable, say f of x is equal to x cube. I could say that f prime for any value a will be a, well, will be three a squared, right? That's the rule, because I know the rule. The rule is that the derivative of x cube is three times x squared. And then if I substitute x equals a, then I get this. So usually we don't write it like this. Usually we just write f prime of x is x squared. In other words, we would like to look at not just one value of the derivative for particular value of x, namely a, but for all, at all of them. For all possible values of x, we would like to know what the value of the derivative is. And then we can substitute, substitute x equals a, for example, one half, then you will get, oh, I'm sorry, I forgot three x squared. And no one corrected me, or at least I didn't get three x squared, of course, for this function. And we substitute x equals a, and we get three a squared. But it's too pedantic to go this long way each time and say, well, what if I ask, what is the derivative of this function? You say, well, for a given value a, the derivative at the point a is going to be three a squared. Instead, we just write f prime of x is three x squared. And that what is understood is that if I want the value at a particular, for particular a, I'll just plug a into this formula and I'll get the answer, right? So we will use the same shorthand for partial derivatives. In other words, I will not be writing each time that fx of a, b, I will just write fx of x, y. I'll just write, or just fx sometimes. And that will be a function of x and y so that if I substitute a instead of x, b instead of y, I will get the value of the derivative of this particular partial derivative at that point. So let's see how it works in this case. In fact, nothing could be easier. You just look at this function in order to, and in order to calculate the partial derivative with respect to x, you just view, view y as a parameter, but not a variable, not a variable, not a variable. This is exactly what I meant when I said that we freeze the value, freeze y. It just means that we view y as a parameter, okay? And then you just differentiate what you see. What do you see? You see x to the five. So you get five x to the fourth plus, differentiate this. It's y cubed plus differentiate this. You get negative sine x times e to the y. That's it. That's the answer. That's the answer, that's the way you write the answer. Now, if you want, if you are given some x and y, some values for x and y, like a and b, you can substitute them and you'll get a number. But in fact, you can view this first partial derivative with respect to x as a function of x and y, which is just obtained in this way. Likewise, we view x as a parameter and then take the derivative with respect to y. So if x is a parameter, then from this point of view, this is just a constant, right? It's independent of y. Therefore, its derivative is zero, right? So it's going to be zero plus, here it's also a parameter. So we just differentiate y cubed, so we get three x y squared. Cosine x is also constant and the derivative of e to the y is e to the y. So that's the answer for the second partial derivative. Is that clear? Yeah, so this is really straightforward. You only need to know how to differentiate functions in one variable, really. Yes, y doesn't cosine go away. In which one? In this one? Well, let's suppose instead of this, you had five times e to the y. Then the derivative would still be five e to the y, right? Or any other constant would just show up as a overall factor. So in the event the constant is cosine x, right? That's what I mean when I say we treat x as a parameter. If we treat x as a parameter, it's treated as a number. And so any expression involving x, like cosine x, is a fixed number. So it just shows up as an extra factor. Any other questions? Okay. So next I would like to explain the geometric meaning of this, because as you see in this course, algebra and geometry go hand in hand and all of the concepts that we discuss algebraically, they have geometric interpretation, which is very important. So for functions in one variable, the derivative of the function has to do with the slope, right, of the tangent line. In one variable, derivative gives the slope of the tangent line to the graph. And so the way we draw it is like this. We have xy plane. We have a function f of x. We have y equals f of x is a graph. Note again that y here has a totally different meaning than y in here. Y in here is a second variable. So it's an equal footing with x. X and y are two independent variables. But now I'm talking about functions in one variable. So there's only one variable, x. And y is not a variable. It's actually, it denotes the value of the function. I already talked about this. It's an unfortunate choice of notation, but that's how it is. So I'm not going to change it. So we pick a point, let's say x equals a, and we draw a tangent line. We draw a tangent line, and we know that the derivative, let's say the angle is theta, that the tangent of theta is a derivative f prime of a. That's the geometric meaning of the derivative for functions in one variable. So then it's natural to ask what is the meaning for functions in two variables. To understand that, we have to look at the graph of the function in two variables. What does that look like? Well, for a function in one variable, a graph is a curve on the plane. And I already talked about many times why do we need a plane? Because to represent a graph, you have to have your variables, and you have to throw in one additional variable which will represent the value of the function. For a function in two variables, there are already two variables to begin with. To draw a graph, we have to throw in one more, one more variable, which will represent the value of the function. So as a result, the graph of a function in two variables is going to live in three-dimensional space. So it will have coordinates x, y, and z. It will have coordinates x, y, and z. And we will have a graph of this function which will be a surface. So I would like to just draw part of it which lives in the first octant. On the plane, this coordinate system breaks the plane into four quadrants, four corners which are called quadrants, because there are four of them. In space, the coordinate planes break the entire three-dimensional space into eight pieces which are called octants. So this is one octant. It's looking at us like this. And so the graph actually lives everywhere, but I have just drawn the intersection of the graph with each of the coordinate planes. And so you should think of this as something like a dome, like a sphere, like part of a sphere. It's not necessarily a sphere. I'm just like this is not a circle. I mean, I'm just drawing a sample graph. And to emphasize this, I want to show a particular point on this. So let's say I take this point. And so this point has coordinates. To find the coordinates, I have to drop perpendicular on the xy plane. So that's going to look like this. And then that's the z coordinate, maybe a little higher. So this point is, what is this point? So this point has coordinates a and b. And the third coordinate is the value of the function because it lives on this yellow surface. I don't want to shade it because otherwise it will not be clear. What am I shading? Am I shading this or am I shading the plane and so on? So I'm just trying to just try to imagine that there is something here which looks like a part of a sphere. And that's the point which belongs to it. And that's the graph of a function, f of xy. So it's defined by the equation z equals f of xy. And this is a particular point, let's call it p, which has coordinates a, b. These are given. These are given. These are just the values of x and y. And what about the z coordinate? Well, since it's a graph, the z coordinate has to be f of the x and y coordinates. So that means that I have f of a, b to be consistent, I should put it in the red. So that's what this point is. So this is f of a, b. Is it clear so far? Okay. So now, what is the slope? What is the slope of the graph? Well, first of all, it's not really a slope of the graph. It's a slope of the tangent line. So here, actually, it doesn't make sense to talk about the tangent line to the graph because the line is one-dimensional and the graph is two-dimensional. How can it not be two-dimensional if we have function in two variables? Function one variables will have a graph which is a curve but function in two variables has a graph which is a surface. So it's two-dimensional. So it doesn't make sense to talk about a tangent line unless we make some choices, give some additional information. So in fact, the proper notion here is a tangent plane. And this is something we'll talk about on Thursday. Okay? So that's really, ultimately, what we would like to understand is the analog of this picture in two dimensions and to get the full analog of this picture, we should really talk about the tangent plane. But for now, I have a more limited goal. I want to illustrate the concept of partial derivatives. And when I talked about partial derivatives, I said that I freeze one of the two variables and then I basically go back to the one-dimensional case, to the case of function one variable. So that's what I would like to do. I don't want to talk immediately about the tangent plane, the entire tangent plane. I want to see what I get when I freeze one of the variables. So in my algebraic calculation on that board, I first froze the second variable, y. So what happens if I freeze y? If I freeze y, it means that I look at the part of the graph which has a fixed y coordinate, namely b. So it means that I cut this graph, I cut this graph with the plane which is y equals b. So the result is going to look something like this. So the blue, this blue is the intersection with the plane y equals b. That's what I get. So now, instead of a surface, I get a curve. This intersection is actually a curve because now y is frozen, y is equal to b. And so it's out of the game. So the game now is between x and z. And it's the same game as a game for functions in one variable, right? So in fact, I can draw this curve as a graph for the function in one variable x which I get by substituting y equals b. This is by the way the function which I called g of x on that board. So let me draw this. So now, as I said, I only have x and z variables remaining and this blue curve is going to look like this. And of course it continues somewhere but since I didn't draw it on the big picture, I'm not going to draw it much beyond the first quadrant. It would be tempting to draw it here. I know you might be wondering why am I drawing it like this and not like this. But the point is you have to look at it not from this angle but from the back of the blackboard where x and z become the oriented coordinate system. You see what I mean? You have to turn this. You have to turn this coordinate system like this, 90 degrees in this way. So to make the x to go to the right and z go vertically up, go up, right? If I look like this, it would be x will go here. So I don't want this. I want to look like this. And that's what I will see. If I turn it, this is what I will see. So this is in fact the graph. This is a graph of what I called g of x which is obtained by taking f of xb. It is part of the surface which is a graph of the entire function but I have frozen one of the variables so I actually was able to reduce my problem to the problem of function one variable. I get a graph of function one variable namely z equals g of x. And now I can calculate for the value of x equal to a, I can calculate the slope of the tangent line. Let me draw this tangent line white. So there is this tangent line and it has a slope. And the tangent of the slope is the derivative g prime at the point a which is what we call the first partial derivative of the function f at the point ab. You see what I mean? I have, I'm doing geometrically here precisely what I did algebraically on this board. Algebraically, I freeze one of the variables. I get a function one variable called g of x and I differentiate it at x equal a. Now I'm doing the same geometrically. Freezing the second variable means intersecting the graph with the plane y equals b. Then I'm down to two variables. I can look at it as a graph of function one variable and then I look at the tangent line to this graph at this point and I measure the slope of this tangent line. The slope is that derivative which we were looking for namely the partial derivative with respect to x. Any questions? So let me draw it now on this board. So this tangent line that I drew over there is going to look like this, right? That's this tangent line. It's not the entire tangent plane. It's one line on that tangent plane. If you think of the tangent plane as this, it doesn't want to turn anymore. I didn't know that it had some knobs and some things to play with. If this is a tangent plane, then I have drawn just one line on it and that's the line of intersection with the xy plane. Maybe it's better like this. If you think of the, not the xy plane, sorry. It's a plane y equals b. If you think of the y equal b plane as this vertical kind of vertical plane, then that's the tangent line that I got. So this green plane is not yet on the picture. I have not drawn it yet. I have only drawn this. And so now I'm going to draw the second one. So that's my point. That's my point, yellow. And now I will talk about the second tangent line which corresponds to the other, freezing the other variable. So this corresponds to y equals b and this corresponds to x equals a. So I intersect now with the plane x equals a. And I'm going to use a different color for this. So it's going to be something like this. And now, so the red curve is an intersection with the plane x equals a. With the intersection with the plane x equals a which is what I drew on that. So x equals a would be like this. So that's, so what's the tangent line to this? Well, this already looks like tangent line but I want to erase this part so we don't get confused. The second tangent line is going to look like this. And that's the second white line which I drew on that board. So if you want the tangent plane, it looks like this. This is a tangent plane. The tangent plane is spanned by both of these lines. Both of these tangent lines. So the tangent plane is the green board, right? But I cannot put it there. So think of the graph as being a kind of a part of a sphere which is just below this plane so that this plane just touches it. Just a tangent one, tangent plane to this graph. But on this tangent plane, I can clearly distinguish two lines. One of them responds to the intersection with y equals b plane and the other one with x equal a plane. When I intersect with those planes, I get pictures just like this. This is the first one. And here's the second one. In the second one, I have two variables left also but those variables are y and z because I have fixed a x now. x is equal to a but y remains a free variable. So I'm talking about this red curve. And this red curve looks like this. Well, the way I have drawn it, it looks almost identical, blue and red but I just did it to simplify the picture. Of course, in general, they're going to be totally different. And on this curve, I pick the value of y equals b. So this is my yellow, maybe emphasize. I forgot to put yellow in that place but anyway, I'm sure you understand this. And then the tangent line is here and then maybe let's call this theta prime. The tangent of this theta prime. Ah, maybe I should say that this is a graph. This is z equals h over h of y where, which is f of a y. And the tangent is the derivative h prime of b, which is f y of a b. So this line, this tangent line is, I have drawn here the tangent line to the red curve on the graph. So that's the picture. So to summarize this, in the case of one variable, you only have one derivative and that one derivative corresponds to the slope of the tangent line to the point, to a given point. In two variables, you have a tangent plane and what partial derivatives give you, they give you the slopes of two tangent lines which belong to this plane, namely the lines which are obtained, like these two lines. The two lines which are obtained by intersecting the tangent plane with the plane y equal b or the plane x equal a. That's the idea. So we just kind of look at, from two different angles. We look at this tangent plane from two different angles and what we get is two different lines and once we get lines, we can talk about slopes. You cannot talk about the slope of a whole plane. The plane doesn't have a slope, lines have slopes and so there are sort of two independent slopes that we can talk about. One with respect to x and one with respect to y and they correspond to the two partial derivatives, one with respect to x and one with respect to y. Okay? Any questions? Yes? Yeah, not a good notation because prime is for derivative, let's go with tilde. I just wanted to distinguish from the other theta. I wanted to make sure it's not the same as that theta but putting prime is like the worst possible notation because it looks like I'm taking derivative of theta which I'm not. Or even better to call it something else. Epsilon, alpha, that's a good compromise. It's Greek but not epsilon or delta, which are taboo. Okay, so what else can we do? In the case of a function, in the case of a function in one variable, we can also take further derivatives we don't have to take just one derivative, we can take the second derivative, third derivative and so on, right? So it's natural to ask whether we can do something similar for functions in two variables. And the answer is yes. We can also take, for example, second derivative. So in other words, we start with a function f of x and y and then we can take the first derivative with respect to x and we can take the respect to y. So we got two new functions, which I'll give you an example of this for a particular case of f. So both of them are also functions in two variables. So we can again apply the same procedure and do partial derivatives for these functions. Then if we go this way, we obtain fxx of xy. What do I mean by this? I mean that I take f sub x, this function, and I take the derivative with respect to x one more time. That means again, freezing y and then taking derivative with respect to x. Okay, if I go this way, I get fyy of xy, which means I take fy, the derivative of f with respect to y and then I take derivative with respect to y one more time. But of course, I can also do mixed derivatives. For example, here I can take this and I can take the derivative of this with respect to y. So that I will denote fxy of xy. That means taking first this derivative with respect to x and then with respect to y. But I can also do fyx, which is first with respect to y and then with respect to x. And then of course the natural question is whether I get the same answer if I apply these derivatives in two different order. Question is whether these are actually equal. And in my example, in my example, let's see if I remember. I think it was fxy x to the five. What was it? xy cube plus cosine x into the y. So what will be? So let me write f of xy. So then derivative with respect to x was five x to the four plus plus y cube minus sine x e to the y. If I do one more derivative, I get 20 x cube. Now y cube is a constant as a function of x. I view y as a parameter. So it doesn't depend on the x, therefore its derivative vanishes or disappears. Then I take one more derivative of sine, I get cosine, cosine x times e to the y. On the other hand, I can take the respect to y, I get three xy squared plus cosine x e to the y. One more derivative, six xy plus cosine x e to the y. And now the most interesting thing, I can, so let me do it like this, so that we don't lose track of where we are. So first we take derivative of this with respect to y, this disappears, this becomes three y squared minus sine x e to the y, so that's this way. And if I go this way, oh sorry, not this way, this way. I get the same, right? I get three y squared and I take this with respect to x, so minus sine x e to the y. So clearly I get the same answer, right? So that's actually a general result, which is called Clérot, or I guess if we pronounce it with a French accent, it will be Clérot, so Clérot theorem, which says that under favorable conditions, which is essentially the condition that in a small neighborhood of a given point, you have all partial derivatives, which are continuous functions up to the second order. Under these favorable conditions, the two mixed derivatives are the same. So this is in fact Clérot's theorem under some conditions of continuity, which will in all our examples, this will be satisfied. So this is actually great because what it means is that if you think of a way of doing partial derivatives for function two variables as this plane, where if you go this way, you differentiate with respect to x, and if you go this way, you differentiate with respect to y, right? You could do that. We could continue this picture. I can go one more step will be like f x x x, or I could go this way, and it will be f x x y, always the last one. The new one is the last one. And then if I go more, it will be for example, x y x, but the point is that it doesn't matter in which order you take. What matters is how many times you differentiate at x, and how many times you differentiate at y. So for instance, f y x is equal to f x y, but likewise, f x y x is the same as f x x y, the same as f y x x. Again, under favorable conditions, when functions in question are continuous and differentiable. So all that matters is not the order, but the number of times you differentiate with respect to x and y, which is kind of nice. So it has the same commutative structure as the structure that you have for the variables x and y themselves. In fact, differentiation is in some sense, the process which is opposite to the process of multiplication by x or y. So you have two operations of multiplication by x and y, but you also have two operations of differentiation by x and by y. And multiplication by x and y commute, two multiplications commute, and two derivatives also commute. Which by the way actually begs a kind of a better notation for this operation because for now the operation is denoted by inscribing this additional subscript next to the function. But there is another notation. So I go back to this, this is our notation, but the other notation is df dx, like this. Also if you wish at AB, but it doesn't have to be. And likewise, the notation for the second derivative is df dy. So this is a particular notation. This should not be confused with the straight D with just a straight letter D. It's not the same. In fact, this actually makes sense, which I will explain as I will explain on Thursday. I will finally explain what dx means. But this by itself doesn't make any sense. This makes sense, d dx. D dx is a procedure which you can apply to a function. You apply it to a function and it gives you its first partial derivative. This is the operation which in the case of one variable we just denote by prime. Also in the case of one variable we write, in the case of one variable we write, if you have f of x, you write f prime of x, or you write df dx. But now we cannot write like this as I will explain in more detail on Thursday. You have to differentiate, you have to specify in which direction you differentiate. And this is one way to do it. Say you choose to differentiate with respect to the x direction and then you get this. But this is not to say, the numerator by itself doesn't make any sense as an notation and the denominator also doesn't make any sense. Only these two things together makes sense. This is a notion of partial derivative. And likewise you have the notion of partial derivative with respect to y, which makes any function into its derivative with respect to y. This is f sub x, this is f sub y, okay? This on the other hand, df is an entire different object, the differential. This is an entirely different object, which is called a differential. And it's not the same, likewise it's not df. So this is not even, it's not even a letter if you think about it. It's not even a letter of any reasonable alphabet. It's just a mathematical notation for partial derivative. So this of course begs the question as to what is the differential? What is a differential and what on earth does this mean? Because this is something we've been using quite a lot, but never really spelled out what we mean by this. But actually it has a very precise meaning, differential and dx and dy. And this is what we're going to discuss next. So in fact, so I have about five minutes left, so I'll give you a little preview of what's coming on Thursday. And it's really very important, that's a very important subject, which unfortunately has been made really, really obscure by a very unfortunate choice of notation. It's a very bad notation, makes it very obscure and very difficult to understand. So I remember when I was learning this for the first time, it was impossible to understand. So it took me a long time to figure it out. But I'm happy to share it with you. I'm happy to tell you. Because it's actually very simple. And we already know everything that we need to know about this. So what I want to do is just to tell you just a little bit, just a couple of things about it. And I will, as always, I will start with a function in one variable, because that's already a very good example where you can understand what the differential is and what all this notation means. So in fact, I should have erased it because I'm going to draw it again, but I just wanted to draw it in a slightly different way. And the more, the way I usually draw, which is kind of a, this is an optimist view of realities, as it goes up, you know. The other one's down, so that's why I erased it. So we talk about tangent lines. And we talk about the importance of tangent lines. And the importance of tangent line really is that it gives you a very useful approximation to a complicated function on a very small scale. Okay? So the differential, the differential really is the function whose graph the tangent line is. So the funny thing is that we talk about a function in one variable. So in this case, let's say you have a function f of x. In this yellow curve represents the graph of this function. That is the set of solutions of the equation y equals f of x. So for this function, we have two objects. We have the function, namely f, and we have the graph, which is the yellow curve. We have two objects. And then we talk about the tangent line. And tangent line, of course, we understand geometrically very clearly. We choose the particular point, so let's say x is zero. But we never talk about the function which gives us this tangent line as a graph. But somehow we don't, we kind of ignore, we usually ignore this question. So the yellow curve, the yellow curve is the graph of this function. But the tangent line is also graph of a function. A much simpler function, which is actually a linear function. And what this function is, it is a differential. This one is a graph, also graph of a function, function, namely df. This is what we mean by df, let me write it in words. Namely the differential of f at this point. That's what differential is. The only subtle point is that for the differential, we shift the coordinates, we choose a new coordinate system where the origin is at our point. That's all. In other words, with you now, this line is a graph with respect to a new coordinate system where the origin is not here. It's not at some arbitrary point, but actually at the point which we're analyzing. And if you write down the function whose graph, this tangent line is, you will have precisely the differential. That's how it works for functions in one variable. And now it's absolutely clear what will happen for functions in two variable. For functions in two variable, we are going to look at the tangent plane to this graph which is represented here. Or if you wish, that's the tangent plane which I was talking about. And then you will think of this tangent plane also as a graph of a function, of a linear function. And that linear function is a differential of a function in two variables we started with. So that's the short version of this, and I will give you more details on Thursday.