 Okay. Next Thursday we'll have the first midterm exam. Okay? I know it might come as a surprise to some of you. I hope not. All the information is now available on B-Space and also on my home page, but there are links between them anyway, so you can find all the information at either place. And on Tuesday we'll have a review lecture, so you are welcome to ask me questions on Tuesday about the material. The material for the midterm exam is everything up to this week. Okay? Everything we've studied up to this week. So next week we'll have review, both in sections, and in my Tuesday lecture, and then on Thursday we'll have the exam right here. Now, I requested another room so that we have a little bit more space, but unfortunately it looks like no rooms are available, so we'll just have to use this one, utilize this as much as we can. So in a way, it makes things easier, so we don't have to split into two groups. We'll all be here. So the exam will be just an usual class time. We'll start sharp at 3.40 and we'll finish at 5. Right here. Any questions? Yes? No. All questions like that are addressed. So I don't want to waste too much time describing this because I believe that all the information is available. All right. And in the meantime, I want to go back to the topic which we started discussing last time, the topic which we started discussing last time, namely the differentials. And already last time I talked about the differentials in the case of functions in one variable. I would like to repeat that now and then talk about differentials of functions in two variables. First one variable case. So in the case of one variable, we'll have a function f of x and we'll have a graph of this function. Graph of this function lives on the plane where in addition to our variable x, we introduce one more variable which is responsible for the values of the function. And usually we denote this second variable by y. But I would like to depart from this tradition today and use a different labeling for the second coordinate because after this I will talk about functions in two variables where x and y will be two independent arguments, two independent variables of the function. And so I don't want to have any confusion between this variable here in the case of function one variable and the second variable for functions in two variables. So that's why what I want to do is I want to call it z. Then it will be more close to what will happen for functions in two variables because for functions in two variables we'll have x and y two independent variables and then z will be responsible for the values, right? So x and z. And as I said already many times, we don't have to use x, y or x, z. In principle, each time you can use any letters you like. It's just a tradition to use x and y. But might as well today use x and z because then we'll be able to appreciate the analogy between the one-dimensional case and the two-dimensional case more. So the graph of this function then will be given by the equation z equals f of x. This will be the graph of this function f of x on the zx plane. And so let's say this is a graph. Now we pick a point x zero. This is a particular value of x. It could be any number, any real number you like. And what we are interested in is we are interested in the tangent line. Interested in the tangent line to this graph at this point. That's a tangent line, right? So what's so special about the tangent line? We've talked about this many times before. The special thing about tangent line is, well there are two special things. First of all it's a line. And a line is a simplest curve, right? It's given by the simplest possible equations amongst all curves. And just geometrically it is a simplest curve that you can draw. That's the first thing. And the second thing is that among all lines which pass through this point, which pass through this particular point, this red line approximates the graph in the best possible way in a small neighborhood of this point. In other words, I'm not claiming that this line is a good approximation to the graph everywhere. Certainly it's not. They diverge. The farther away we get from this point. They diverge more. The farther away we get from this point. But just in a small neighborhood of this point, it is actually a very good approximation. In fact when I draw it, it's kind of difficult to draw in such a way that actually to insist that they are actually different because they are so close to each other. That already gives you the feeling that indeed it is a very good approximation. So that's what tangent lines are good for. They give you a linear approximation to your function. In other words, they capture the essentials of the behavior of this function in a small neighborhood of this point to the first order, as we say, which already contains a lot of information. So therefore it is useful to write down the equation for this tangent line or more precisely to think of this tangent line as a graph of a linear function. So what is this linear function? Well, we know that since the studies of one variable calculus, we know that the slope of this tangent line is given by the derivative of this function. So the tangent of this angle theta right here is actually f prime of x0. And so the equation of the tangent line, and I'll just write it in words, of the tangent line x equal 0 is the following. I see I'm writing y because I'm used to writing y, but like I said I want to use z. z is equal to f prime of x0 times x minus x0 plus z0. What is z0? z0 is the value of the function at this point. And so z0 of course is just f of x0. But I could actually write here f of x0 also. But I prefer to write z0 to make notation a little bit less heavy. And I want to rewrite this also as follows. z minus z0 is equal to f prime of x0 times x minus x0. I want to emphasize that this is a particular number. This is a particular number, namely the slope. And it appears in this formula as a coefficient of proportionality between the increment in z along this line and the increment in x along this line. So the increments in z and x are proportional. That's what this formula expresses. Proportional to each other. And the coefficient of proportionality is nothing but the derivative of f at this point. So it's important to realize that all of this, everything that I've done so far is relative to a particular point x0. It's relative to x0. If I choose a different point, let's say some x1. So this point will live here. I mean, the point with such x coordinate will live here on the graph. And so the tangent line to this point will be this blue line passing through this point. And surely this blue line has nothing to do with this red line. And for a good reason, tangent line is useful as much as we want to understand the behavior of the function in the neighborhood of that point to which we draw the tangent line. So the tangent line becomes irrelevant when we start talking about a point which lies sufficiently far away. So when you talk about tangent lines, you have to specify the reference point, the initial point. Tangent line at what point? There is no such thing as tangent line, the tangent line to a graph. There is a tangent line. There is tangent line for each point on the graph. That's the first important thing that you have to remember. And then once you fix x0, your reference point, or more precisely, your reference point, if you look at it on the plane, it's x0 and z0, where z0 is the value of the function at x0. Once you have that, then you have the equation for this tangent line. And sure enough, it involves the equation x0 in a very essential way. It involves it in two places. First of all, the increment in x is counted relative to x0. And second of all, the coefficient of proportionality is the derivative of f at this point x0. So surely this coefficient, the slope, will be different for a different point. And you can see that the slope of the blue line is different from the slope of the red line. So this formula really refers to the red line, this one. Okay. So now, the concept of differential is just the concept of this equation that we have written, of this equation which expresses the proportionality of the increments in the value, in the argument of the function, the value of the function under linear approximation. So the notation, the important thing to understand about differential is that it's a particular, it revolves around a particular choice of notation for the increments, which I'm now going to explain. So all the difficulties in understanding the differential, in my view, really boil down to understanding the notation. That's why I'm going to be very careful when I define this notation. So we'll introduce the following notation. I would like to denote x minus x0, I would like to denote x minus x0 as dx. This is just a notation. And so I want to emphasize, right definition. So I define this to be just the difference between x and x0. Now, if I write it like this, you already see that this notation is deficient. Because on the right, this expression depends on the choice of x0. When I write x minus x0, it depends on the choice of x0. If x0 is equal to 0, this will be just x. If x0 is 1, this will be x minus 1. If x0 is 2, it will be x minus 2, and so on. If it's pi, it will be x minus pi. In other words, on the right-hand side, I'm not talking about a single function in x. But it's a function which you get, it's a linear function, it's very simple. But it's a function which you only specify once you specify x0. But my notation on the left does not carry x0. So that is the first major problem in the notation which we use. So let me cure that by actually introducing it explicitly on the left-hand side. So I will keep track of x0 by putting it as an index in this notation. To emphasize that this is something that we are doing for now, and we will actually discard later on, I will put it in yellow. So let me just introduce notation dx relative to x0. I will just denote the following linear function x minus x0. I have the right to do this. I can introduce any notation I like. So this is notation I want to introduce for whatever reason. Likewise, for z, I want to do the same. dz also relative to x0. I would like to write z minus z0. Maybe it's better to say relative to z0. But z0, of course, is determined, since we are talking about a particular function f, z0 is determined by x0. So in fact, I could write as the parameter zx0 or z0, whatever I like. But let's write z0 to make it a little bit more consistent. It's actually determined by x0. And that's just going to be z minus z0. So far it's totally thought logical. There is nothing in this. It's just a choice of notation. But if I do that, then I can rewrite this formula in a following way. Just this formula on the left-hand side, I recognize what I now call dz relative to z0. And on the right-hand side, I recognize dx relative to x0. So the same formula will look like this. dz relative to z0 is equal to f prime of x0 times dx. So I've done nothing. I've just used a certain notation for, just introduced some new notation for the increment in x and z. But now the formula starts looking more familiar, because you've seen this formula before in one variable calculus. But what's usually done is usually we drop the indices. So then it looks like this. dz is equal to f prime of x0 dx. And also we replace, often replace, dz by df. Oh, we get df is equal to f prime. You see? So you recognize this formula, right? Because you can also use this to write f prime is equal to dx df over dx. But now you understand, I hope you understand, now I appreciate this formula more. And understand what it means. Usually in the textbook they don't really explain what we mean by, what is meant by dx, what is meant by df? Or dz, which we use here interchangeably because we are talking about a particular function f for which z serves as the value. The point is that dx and dz are nothing but the increments in the coordinate x and in the value z along the tangent line. So the equation of the tangent line, which we know is given by this formula, just becomes the old formula that you knew, that dz is equal to f prime dx. Or if you wish, df equals f prime times dx. You see? The only problem in understanding this formula is the fact that usually we, what we can call abuse notation, abuse notation in a sense that we drop some essential information from the formula. This would be to me a much more consistent way of expressing the fact that what we are doing is just writing down the formula, the equation for the tangent line. It is important, at least from the outset, to indicate the fact that dx is not an absolute notion, like x. x is an absolute notion, it's a coordinate. So x makes sense without any reference to anything else. It's a particular coordinate. dx is not an absolute notion, it's a relative notion. dx is defined once you choose the reference point. Once you choose the reference point x0, then dx is defined. So, and then if the reference point is 0, dx is just x. If the reference point is 1, it's x minus 1 and so on and so forth. It's just x minus x0. Likewise, dz is not an absolute notion. It is really relative to the reference point. Once you choose the reference point, it's just an increment. Once you realize that, then you see that this formula is nothing but the expression of proportionality of the two increments along the tangent line, which is just the equation of the tangent line. Do you see what I mean by this? Do you have any questions about this? I wouldn't, in a way, you can say, why am I talking about all of this? Because we've learned nothing new. The only piece of essential information is already available here. That's the equation of the tangent line. Something we've known all along, well, all along since one variable calculus, right? Since studying one variable calculus. But the reason I explain this is because we are going to use this notation dx and df and dz and I would like to explain what it really means. So, now I have explained this, explained what this notation means and I've explained what the formula, the old formula that we've known, df equals f prime dx means, and it's just the equation of the tangent line. All right, now, this expression f prime of x dx is called a differential x0 dx and I would like to keep insisting on putting this x0 just to make sure that we understand that actually this is something which is a relative notion. It's called the differential of the function f at the point x equals x0. So, let me give you an example. Let's say f of x is x squared minus 3x. What is the differential of this function say for x equals differential for x equals 1? Well, we simply have to take the derivative of this function. What is the derivative of this function? It is just 2x minus 3, right? Just have to take the derivative of this function at the point x equals 1. So, this 1 is what I indicate in other formulas by x0. This is x0. So, I simply substitute this. I have to substitute x0 into this expression like here. So, I get 2 times 1 minus 3. So, it's negative 1, right? And then I multiply it by dx relative to the point x0, which is in our case is 1. So, in fact, I could also write it as minus 1, negative 1, times x minus 1. So, the differential of this function is really this linear function negative 1 times x minus 1, which is the same as if you want. It's just 1 minus x. Let's just leave it like this. Minus x minus 1. That's the differential of this function. But usually what we're going to do is we're going to abuse the notation. I mean, for now on, once we understand what the meaning is, we will actually abuse the notation and we will drop all the indices, right? So, when we drop all the indices, we will actually just write minus dx and say this is the differential of this function at the point x equals 1. We have to realize that actually in this formula dx is referring to this particular point. And if I change the point, I will get a different answer. So, change the point. So, let's find the differential for x equals 2. So, now it's 2, which is x 0. So, I have to substitute this value into the derivative. So, I substitute 2 here. I get 2 times 2 minus 3. So, I get 1, right? So, now, now f prime of 2 is equal to 1, whereas here it was negative 1, right? And so, I'm going to get 1 times dx at x 0 equals 2, which is just, I can just erase 1, of course, just dx. And then, just in this case, we're going to just write dx dropping the indices. So, suppose it's this yellow chalk becomes invisible. And so, the answer just becomes dx, right? But you have to realize that this is a differential at the point x equal 2. And it's different from the differential at the point x equal 1, which was negative dx. So, what I'm saying is what I drew here on this diagram is that, well, the differential is the function whose graph is the tangent line. The tangent line depends on the choice of the point. So, therefore, the differential also depends on the choice of the point. The coefficient in front of dx is going to be just the slope, which is the derivative here in negative 1, here 1, right? But also, dx itself actually has a different meaning. Here, dx is x minus 1. And here, maybe I should write, here it's just x minus 2, really, okay? Yes, there's a difference between the two. So, the differential is a function of x, which is equal to the f, which is equal to this. This is a differential. It's a function of x, which is equal to the derivative of your original function f at the point x 0 times dx, which is understood as x minus x 0. That's the differential. It's a particular linear function. It is precisely the function whose graph is the tangent line. It's the equation of the tangent line. It's a function which gives the tangent line. You cannot say tangent line is the function, right? You can say tangent line is the graph of a function. You see what I mean? I'm trying to be precise. It's not because I want to be pedantic, but there is too much confusion as it is. So, I try to separate different notions. There's a notion of a function. There's a notion of a graph of a function, right? So, that's right. So, a line, you cannot say tangent line is a function. Function is a rule which assigns to each number, you know, we represent a function by its graph. This tangent line is the graph of a function. Which function, the differential? Exactly. There is only one function given, which is f of x, but there is more than one tangent line. There is a tangent line for each point, which I have illustrated. I've drawn two of them. I've drawn the red one and the blue one, right? Yes? That's right. That's right. So, you're right. So, this is, that's right. So, to be absolutely precise here, the, would be df for x0 equals 1 is equal to minus dx at x0 equals 1. df at x0 equals 2 is dx at x0 equals 2. And if you want, I can also do df at x0 equals 3, for example. I just need to calculate the derivative at the point 3. The derivative at the point 3 is 2 times 3 minus 3, which is 3, right? So, then the answer will be 3 times dx at x0 equals 3 and so on. But of course now, you can kind of guess what this coefficient is. This coefficient is just 2x minus 3. So, instead of writing an infinite list of answers for different values of x0, I just write, I just write a formula in one stroke. I just write one formula which is responsible for all of this. I just write df is equal to f prime of x dx. Yeah. Well, it's like, you know, when you're, certain things are allowed when you become adults, right? I mean, so that's sort of the, it's the same, same reason. It's just to simplify things. But it, it leads to incredible confusion in my opinion. So that's why I'm trying to unravel the precise definition and then explain how do we actually arrive at the formula which you see in the book. And I, I agree. In some sense, we should not be allowed. Or more precisely, we should be mindful of this. So whenever we see this formula, that's what we should see. We should see not a single formula, but a bunch of formulas which depend on x0, on the choice of x0. That this one formula is not just one equation, but it's a collection of equations. For each value of x0, for each value of x0, there's a formula which says that df relative to that x0 is f prime at that x0 times dx at that x0. And this formula is, is nothing but just saying z minus z0 equals f prime of x0 times x minus x0, which is the equation of the tangent line. But we write in one formula, we write down all equations for all tangent lines. That's what we do. You see? So that's the point, that's the point I'm trying to convey here. Okay? And that's why I give you examples. So, ah, so maybe in this case, I would actually want to write more precisely, in this particular case, I will write f prime of x. I have found it, right? I have found that f prime of x is 2x minus 3. So in fact, I will write df is 2x minus, df is 2x minus 3 times dx. So more precisely it is this formula which is responsible for all of this. Because if you have this formula, you will get this one. If you substitute instead of x, you substitute value 1, right? Substitute 1, you get minus, like I put, at x equal 2, you will get 1, at x equal 3, you will get 3, and so on, right? So this one formula will give you all of them. In other words, I give you this formula, you should be able to substitute any value x0 and you will get the equation of the tangent line at that x0. Does it make sense? Any other questions? Okay. So there is nothing mysterious about the differential. The only mysterious thing is that it should actually be defined relative to a particular point x0, and what we have done, or people before us have done, is that at some point they decide to drop this from the notation. So the formula starts looking like this and starts looking very confusing because what does it mean? What is df? What is dx? It becomes very confusing. But if you remember that it actually refers to a particular value x0, and once you remember that, you know that dx corresponds to x minus x0 and df corresponds to z minus z0, then the mystery disappears. There is no mystery. It becomes extremely simple and mundane formula, just a formula for the tangent line, the equation for the tangent line. That's what it is. Okay. So, one more piece of notation to sort of finish with this. We can also write delta f. So that sort of, to make things even more confusing, there is also notation delta f. And delta f, and again to be precise, we also have to say that it is relative to some x0. This by definition is f of x plus x0 minus f of x0. So that's what we have done. So it is the increment of the actual function. It is the increment of the actual function, whereas df, df is the increment of the tangent line, of the linear function which approximates our function, you see. So that's the difference between this big delta and small d. So should not be, should not be confused, not be confused with df relative to the point x0. Which is f prime of x0 times x minus x. And let me draw again the picture. Let me magnify it. Let me not draw the coordinate system, but let me magnify, blow up the small neighborhood of this point. So here is my, here is my graph. And here is my tangent line. Here is my tangent line. So here is x0 somewhere. And here will be x, here will be, I'm sorry, I'm sorry, I wrote something correctly. x minus x0, sorry. Half of x minus x0. So there is x0 and there is x. Oh, so now I'm trying to confuse you. I'm sorry. I was going to draw it on the picture anyway. So look, this is delta f. You see what I mean? This is x0 and this is x. Next is very close to x0. I can measure the value of the function itself, which is the yellow curve. I can measure the difference between the values of the function, that's this. Or I can measure the difference between the values of the linear approximation, the linear function, which corresponds to the tangent line. That's df. You see what I mean? These are different because it doesn't quite go up to here. They start diverging. They start deviating from each other. But the punch line is that they are almost equal. The difference is negligible. The closer you get to this point, the closer x and x0 are, the smaller the difference is going to become. That's the punch line, which expresses the fact that linear approximation is useful in the following sense, that when you are very close to the point, your linear function whose graph is a tangent line is almost as good as the original function. In case, of course, your function is a nice one like this. It's kind of a smooth function. Such functions for which this linear approximation works are called differentiable functions. The punch line of all of this discussion is the following, that there is a large class of functions f for which the difference between delta f and df is negligible. It's much smaller than the difference between x and x0. We can actually ignore it when x is very close to x0. The notation is different. The point is that the two quantities for differentiable functions are very close, which is to say that a linear function, the function which gives you the tangent line, is a good approximation to the original function. Let me write this down. This f is called differentiable x equal x0, if delta f at x0 is equal to df relative to x0. Plus, I promised you not to use epsilon, but let me use it anyway. To indicate that it's something very small. I will not use delta, where epsilon of x goes to 0 as x goes to 0. In other words, this piece is negligibly small. You see, because this already goes to 0, and this certainly goes to 0 when x goes to x0. This goes faster than x minus x0. It goes to 0 faster than x minus x0. This is precisely what I indicated by that little difference. Let me point out this. This little piece, this is what is epsilon of x times x minus x. It's a very little one. This tiny little piece is something that goes to 0 faster than x minus x0, because this already goes to 0. This goes to 0 as a square. It roughly goes as a square of x minus x0. Let me unravel this for you. In other words, what we are saying that function is differentiable, that's the expression. Delta f is df plus this. What is delta f? I explained that delta f is just f of x minus f of f is 0. Let me just spell out what that formula means. Let me call this star. Formal star is just saying the following. Now I want to spell out what df at x0 is. df at x0 is this. It's f prime at 0 times x minus x0 plus epsilon of x. Now, if you remember Taylor series, this gives you a good perspective on this formula. Do you remember Taylor series? You don't want Taylor series? How about just the first two terms in Taylor series? I will not ask you for more than that. Taylor series, the idea was that f of x is equal to f of x0 plus f prime of x0 times x minus x0 plus 1 half f double prime x0 times x minus x0 squared plus and so on. I just want to explain to you what we are doing now. We are just looking at the first two terms of the Taylor series. We are just focusing on the first two terms of the Taylor series because this one term is this. Now it's on the left-hand side, but big deal. We can just rewrite it like this. Just take it to the other side. You see this matches this. This term matches this. In the Taylor series, we then have all the higher derivatives and higher powers of x minus x0 squared, second, third, and so on. Now, I have taken all of this stuff and just denoted it by this one expression. The main point is that the powers of x minus x0, which show up in this term and the next term, are 2, 3, 4, and so on. They are higher than 1. I can sort of chip off the first power x minus x0 and what will remain, for example, here is second power. I just split x minus x0 squared into this and one more x minus x0. In other words, this thing goes to 0 by itself. You see what I mean? What I am doing roughly is, let's take this term, 1 half f double prime of x0 times x minus x0 squared. What I am doing is just I am writing it like this. I mean, I have done nothing. I just wrote the square as a product. After this, I take this piece and call it epsilon. Then what I get is epsilon times x minus x0. The point is that this guy by itself goes to 0. It goes by itself to 0 because this is finite. This is just some expression. For example, for the function which I had, which was the first derivative was 2x minus 3. Second derivative would be 2. This is just some number, which is 2. It's 1 half of 2, so it's just 1. But then there is x minus x0. x minus x0 is something that goes to 0 as x approaches x0. But in addition, I have one more power x minus x0 because I started out with a second power. The only term for which this will not be the case is the first term. In the Taylor series, the first term will have x minus x0 to the power 1. That's the one which I retain. Then everything else is really negligibly small compared to this term. That's the idea. That's the main point of our calculation now. Before, we tried to write it as an infinite series which involves all powers. Sorry, all derivatives of the function. It involves no derivative at all, the first derivative, the second derivative, and so on. What we are saying now is that how about let's just keep the constant term, which is f of x0, and let's keep the first derivative term, which gives rise to the linear term in x. Let's just keep this. Everything else, we'll just put in a bag and call it epsilon of x times x minus x0. It's something very small times x minus x0. It's negligibly small compared to the first two terms. The special thing about the first two terms is that they give you linear function, which is the function whose graph is a tangent line. Writing down this formula means that you approximate your original function f of x by a linear function. This one plus some really, really small, negligibly small error term. That's what we've done. Questions? Yes? That's right. I'm talking about functions in one variable. I'm of the opinion that before you do complicated things, you should start first with a simple thing. This is a case of one variable. I wanted to explain everything in the screw shading detail for functions in one variable. Now we'll have to talk about functions in two variables. The point is that everything is going to work in exactly the same way. I believe that you first have to understand what happens for one variable. Only then you'll be able to fully understand what happens for function in two variables. Let me explain now functions in two variables. What happens for functions in two variables? What can I do? I want to keep this. I kind of want to keep everything. Let me erase this. Now we have a function in two variables, x and y. We have f of x, y. Aren't you glad I haven't used y in the previous calculations? We had x and z. Now we have x, y and z. Now we're going to have a graph of this function, but in space. That's going to be a surface. Let me try to draw this. I will draw it in a slightly different way than last time, because I think it will be a little bit more clear. You tell me. I want to draw it like this. Now we have a point. Now I want to say that there are two curves here, which actually last time I drew them with red and blue, but it's not worried about this. This is one curve, and that's another curve. Maybe I can do it like this. Because he's never satisfied with his drawings. I think this is better. You see what I mean? It's a surface like this, but it's concave. Last time I drew it convex like this, and now it's concave. I think it's a little bit easier on the eyes, so to speak. This, of course, all lives in three-dimensional space. There is a three-dimensional coordinate system, as usual, x, y, z. This corresponds to a particular point with coordinates x, 0 and y, 0. Now I want to draw the analog of the tangent line. This yellow thing is the analog of the curve, of the graph of the function. When I say yellow thing, I mean, of course, the whole thing, the surface. Now, before I had a tangent line because it was a curve, so for a curve it makes sense to talk about a tangent line. But for a surface like this, it makes more sense to talk about the tangent plane. It is two-dimensional, so in your linear approximation you should use a linear surface but not a curve, a linear surface, which is what we call a plane. In fact, I brought something to illustrate it even more. This should play the role. I tried to find a basketball in a math department, but finding a basketball in a math department is like, well, you complete the sentence. It was difficult. This is what I found. This is the surface, and I just want to explain what the tangent plane is. The tangent plane is what is a plane, let's say, if I pick a point. The tangent plane is the plane which is the closest plane to this surface. That's the first point I want to make. The second point I want to make is that tangent plane is going to change when you change the point of contact. If you are interested in a tangent plane at this particular point, that's the plane you have, but if you want at this one, that's going to be this one. It's exactly the same thing as we had for curves, for tangent lines. Tangent line depends on the choice of the point. Now, let me draw this tangent line for you. You see, to indicate that to give more shape and volume, well, not volume, but really the two-dimensional aspect to this, perspective to this, I drew these two curves. What are these curves? These are the curves, exactly the same curves that I drew last time. This is a curve of intersection. This is intersection with a plane which goes like this, which is parallel to the YZ plane. It's really the plane where we fix the X coordinate, X equal X0. You will see that what I'm doing now is very similar to what I did last time, but there are small differences. For example, the graph I was using was convex in a different way. Also, instead of X0, Y0, I use notation AB and so on, but otherwise it's very similar to what I did last on Tuesday. What I do is I can cut by a plane, a vertical plane like this. I can cut the graph by this vertical plane, vertical plane which is parallel to YZ. That's the curve I get. This is a curve which lives on the graph, on the surface, on the vase if you want. This is part of the vase. This lives on the vase. Then there is a perpendicular one which I get by cutting it by a plane which is parallel to the XZ plane. That's this one. The only thing is I confused them. What I was trying to say is that this is the curve you get by cutting with a vertical one parallel to ZY. Now this one is what I get by cutting with a plane parallel to Z. This curve is intersection with the plane Y equals Y0. The graph is complicated. The graph is the surface, but on this graph, for my point X0, Y0, I have drawn two curves which in some sense are perpendicular to each other. These are the curves of intersection of the graph with two natural planes, two vertical planes. One plane goes parallel to the blackboard. That's this one. The other one is parallel to this one which has an angle too. Since I'm just giving it a three-dimensional perspective, I'm drawing it like this. In fact, you should think you should realize that it's a plane which should be perpendicular to the blackboard, this XZ plane. I get two curves. Curves is something I can handle because we now know everything about curves. We know everything about tangent lines to curves. These two curves have tangent lines for sure, just like this curve has tangent line at our point. Let me draw two tangent lines. One will be the tangent line to this curve of intersection, and the other one will be the tangent line to this curve intersection. I'm trying to separate them a little bit. That's why you see as though there is some distance between them, but it shouldn't be. Just make it more visible. Now, these two lines, in fact, they span a plane which looks like this. That's the plane which is slightly underneath the graph. That's what I was illustrating when I put a vase here. That would be the tangent plane. On this tangent plane, I would have two lines, which are tangent to the intersection, to the curves on the graph which you get by intersecting with the vertical plane. You see what I mean? Is this clear? That's right. It's a tangent to that particular point. Exactly. If I take a different point, it will change. Imagine there is a sphere here. That would be the tangent line. If I take a different one, it will be like this. It will get tilted in a different way, not just like this, but also like this in all possible ways. Everything I say is relative to a point, relative to a particular point, which I now call x0, y0. What I need to do is I need to write down the equation for this tangent plane, just like I wrote down here the equation for this tangent line. Then I will say that this tangent line approximates the graph in a very nice way if the function is differentiable. The equation for this tangent plane will be called the differential of the function at this point. It will be exactly parallel to what we did in one variable case. The immediate task at hand is to write down the equation for the tangent plane. Fortunately, we know everything we need to know about equations for our planes. We are going to use it now because we have previously studied the question of writing down the equation of a plane when we know two vectors which belong to this plane. We know how to do it. First of all, we know that to write down the equation of a plane, we need to know a normal vector to the plane as well as a point. Surely, we have a point that's x0, y0. We need a normal vector. We can calculate the normal vector by taking cross product of these two vectors, which go along these two lines. What are those two vectors? Those two vectors add the tangent vectors to those curves which we can easily find from the one variable calculation. That's what we're going to do now. Let me explain what actually it would be more like this way because it will be this way and this way. It will be towards the direction of x and y rather than the opposite direction. We want to write down the equation for the tangent plane to the graph. We want to add the point x0, y0 and z0. By the way, z0, of course, is again f of x0, y0. That's the value of the function. What is the equation? We know that the equation is going to look like this, a times x minus x0 plus b y minus y0 plus c times z minus z0 equals 0, where a, b, c should be a normal vector. How do I find a normal vector? I take these two vectors, let's call them r1 and r2, and I take their cross product. Now you can apply the knowledge which we have acquired earlier about equations for the tangent plane. Now, let's appreciate why it was important to actually learn these techniques about planes and lines and so on, and cross product. We have to figure out what r1 and r2 are. Let me talk about r1. R1 is something which I find by using a curve. This curve is obtained by intersecting the graph with the plane y equals y0. In other words, it's what I used to call, what I called on Tuesday, freezing the second variable. Intersecting with this plane means that we freeze the second variable, y. y is fixed, it's y0. Effectively, the problem, which is three-dimensional, because we have x, y and z, becomes two-dimensional because only the unfrozen variables participate, namely x and z. That's why now I'm going to draw that curve on the x and z plane, and it's going to look like this. Where is my yellow? Here it is. Just for consistency, I'm going to draw it with yellow. It's going to look like this, because I'm looking at it now from that angle so that x goes this way and z goes this way. This is the curve I'm drawing. It is decreasing. That's why I draw it like this. This is a tangent vector I'm interested in. This is my r1. It's going to be a tangent vector. We have to find a formula for this r1, but we know how to find tangent vectors to parametric curves. How to do this? Well, first of all, what is this curve? This curve is z equals f of x, y0. We have frozen the second variable we have frozen y. We have just said y equals y0. This function, which, by the way, last time I used to indicate it by writing it in this red. The function effectively becomes a function in one variable only, namely x. Let me call this g of x, because the function in one variable only, so let's just call it g of x. This is a graph of the function z equals g of x. This, by the way, is the same notation I used on Tuesday. Now, I want to convert this into parametric form, because we've learned how to do tangent vectors when we have parametric form. Let's do parametric form. For graphs, it's very easy. Parametric form is x equals t and z equals g of t. I have this vector r of t, which is tg of t. That's the vector value function, which corresponds to this parametric curve. Now I know that this vector r, which is a tangent vector, r1, is just the derivative of this. It's just r prime at x0. My point corresponds to t equals x0 is our point. This is a formula we've learned before, which now comes very useful, namely the formula for the tangent vector to a parametric curve. I have converted my curve into parametric curve into variables with this parametrization. Now I can use this, and what do I find? The first derivative is 1. This is a prime. The prime means derivative with respect to t. I'm just recalling. It's derivative with respect to the t variable. The derivative of t is 1. The derivative of g of t is g prime of t. But t is x0, so it's g prime of x0. Now remember we agreed that the derivative of g at x0 is the first partial derivative of f, because that's how we define the first partial derivative. The first partial derivative was defined by taking that function g, which we get by freezing y, and taking the derivative. What this is is just 1, and f sub x at x0 y0. This is what I needed. I have found r1. Now, this is not exactly true, because r1 is a vector in three space, and now I have done the calculation in two space where I kind of ignored the y variable. But in fact the y variable should be here. It's not like this, but it's going into the blackboard to comply with the rule. What I found is this vector in two space on the plane. But if I want the corresponding vector in three space, I have to also remember the y coordinate, and the y coordinate will be 0. It will be in between these two. The true vector r1 in this formula is just 1, 0, and f sub x at 0, y0. Why like this? Because I found this guy's 1 and f sub x, but to have a true vector in three space, I also have to give it the third coordinate, which corresponds to y. This is the coordinate which corresponds to x, and this is the coordinate which corresponds to z, and I need one more which corresponds to y. But it is 0, because everything is happening on the plane which is perpendicular to the y axis. So that's r1. Now you need to calculate r2, and of course you can guess what the answer is. The point is that now 0 will migrate here, because now everything will be happening on a different plane, which is the yz plane. So the x coordinate will have no role at all. The y coordinate will now play the role of x, so this will be 1, and this will be f sub y. It's exactly the same calculation, except I should draw it on the plane yz, and I should call it r2, and I should do the same parameterization and the same calculation, exactly the same. So I'm not doing it just to save time. So that's the answer I get. So we are almost there, because now all we need to do is to find the normal vector to our plane, which as we already discussed is the cross product. So finally we get to use the cross product for something really important. So this is our application of the cross product. So r1, r2 is going to be, as we discussed, and no doubt some of you may have wondered why the hell are we doing all this stuff. Now you can appreciate that it is actually important. For short hand, I'm not writing, I'm dropping x0, y0, but I will restore them later. So what is this? This is i times this, which is minus fx times i, and minus j times fy, fy times j. And then k, I just get this matrix. So it's 1 plus k. Okay? Does everybody agree with this? We have found the normal vector to our plane. That's the one. The normal vector is perpendicular to this plane. The one like this. So say this is r2, this is r1, and this is a perpendicular vector, the normal vector. That's it. We're done, because now we can write down the formula for the tangent plane. You see this? So the formula for the tangent plane becomes minus f sub x, and you get minus f sub y plus z minus. And now I just want to rewrite this in a nicer way. z minus z0 is equal to f sub x of x0, y0 times x minus x0 plus f sub y times x0, y0 times y minus y0. That's the equation of the tangent plane. And of course, you should compare it to the equation of the... I have erased it, but I want to write it again. In one dimensional case, the equation was f prime of x0, x minus x0. So it's totally analogous. z minus z0 here was the derivative times x minus x0. There was only one variable, therefore only one derivative. There was no choice. Now there are two different derivatives, and both of them show up. This is the derivative with respect to x, or partial derivative with respect to x, times the increment in x. And this is a partial derivative with respect to y, times the increment in y. So this is a linear function. The right hand side is a linear function whose graph gives you that tangent plane. And this function is called the differential of f at this point. So more precisely, all right, this is a, b, c. This is what I got. Then I wrote the equation of the tangent plane over there by using these three coefficients. I mean, if you want, I put one here. We have two different choices, two notation for vectors. One is with i, j, k, and the other one with three components. When I have a vector, I call this a, b, c. If this is a normal vector, then the equation of the plane is a times x minus x0 plus b times y minus y0 plus c times z minus z0. This is what I wrote. Second formula from the bottom. Do you see that now? Yeah. The second formula from the bottom is the equation for the plane, which is perpendicular to this vector. All right? Okay. Any other questions? All right. So now we're almost there. And so the right hand side now is called the differential. It's called the differential, differential of f x, y at the point x0, y0. It is a linear function, it is a linear function which approximates well our function provided that our function is differentiable. In fact, f of x, y is called differentiable x0, y0, compared to the one-dimensional case. If delta f, which is, let's just write formula, f of x, y minus f of x0, y0 is equal to this linear function up to a small correction term. fx of x0, y0 times x minus x0 plus f sub y at x0, y0 times y minus y0. The correction term I will now write as sum of two terms. Before I had a correction term which was something small times x minus x0, but now I will write it something small times x minus x0 plus something small times y minus y0. So if you like, if you like to think in terms of Taylor series, you can imagine that I'm writing a Taylor series expansion for my function, where I first write the constant term, then I write the linear term, which now has two summands, one coming from the first partial derivative and the other coming from the second partial derivative. And then I will have quadratic terms and cubic terms which would involve all the mixed partial derivatives of higher order. But then actually I kind of, I don't want to specify them, other than to say that they all together combined have this shape, something which is negligible compared to these two terms, something that can be viewed as an error, as a negligible error term. So epsilon 1 and epsilon 2 have to go to 0 as x, y converges to x0, so that's the condition. Okay, so in fact I could stop here, but I want to explain the notation again, because I think all of this sounds great until you encounter a notation and then it becomes a little bit, it could become a little bit confusing. But now we can very easily unravel the notation as well, because we now have a very good example of that in one-dimensional case. So here's how the notation works for this. Just as in a one-dimensional case I will have dx, but to really do justice to dx, I have to keep track of the reference point, which now is x0, y0. And so dx is x minus x0. There is also dy, which is y minus y0. And there is also dz, or df, which is z minus z0. So this formula, which I put, which I have framed, can be written as df equals f sub x dx plus f sub y dy. More precisely, you have to put everywhere x0, y0, here x0, y0, here x0, y0, here x0, y0, here x0, y0, and here x0, y0. And if you do that and you remember what this means, this will become identical to this. It's nothing more, nothing less. But in other words, this formula makes sense for a given reference point x0, y0. And it's nothing but the equation for the tangent plane. But after this, just to simplify the formulas, you draw up all the indices. You say, okay, let's just forget about this. Let's forget. Not forget. Let's remember it, but let's not write it on the board, or let's not write it on a piece of paper. So the formula really becomes just df equals fx dx plus f y dy. So if on the homework, if on the homework, you're asked to write, to compute the differential of a function, that's what you're going to do. You're going to take the first partial derivative times dx plus the second partial derivative times dy. So just nothing could be easier than that. You just take two partial derivatives, but what time is it? Oh, it's five. Okay. But now I have explained to you what the meaning of this formula is. So we'll talk about it more during the review on Tuesday.