 I don't want to go into what makes it like this for me today. Alphons do that during recitation. Any time here. Are you Jeff? Yes. That's I think 40. Thank you. About the, for example, suppose, so the low score I think was 12, 15, something like that. If you were that person, that doesn't mean you will automatically fail this class. It does mean you automatically fail this exam. So I posted the, let me back up. I posted the solutions and the grading scale on the web. Basically similar to the other exam. So 60 and up was A or A minus. I think 65 was the A minus line. Either 45 or 50. I think 50 to 60 is a B. All kinds of B's. And 40 was the kinds of C's. Not counting C minus. And if you've got below 40, then you need to make some changes to your life. If you want to have this class. So I guess I should say a few words about what I do when I pray. It is possible for someone to have a very bad day. I understand that. If you had a very bad day or a very bad week, that does not mean, so suppose you got a zero on this exam. That does not mean there is no hope of you passing the class. It means that I will look at your performance in the other grades on this test. I'm in this class and I will possibly discount this score accordingly. That doesn't mean I throw out the worst test score. What it means is if you did much, much, much worse than your average, then I conclude something was wrong. If of course the standard deviation of your grades are all over the place, that is you get a zero on this exam, a perfect score on the next exam, and right in the middle on the final, then I will average them all. But if it is clear that your grades do something like, then this one doesn't count so much. Signing a grade is to assess the material in the course. So we have four samples of your knowledge, the first midterm, the second midterm, the final, and the homework. And I view those as four samples. If you know any statistics, I'll view those as random variables. Well, I hope they're not totally random variables, but I'll view them as with, you know, there's some noise here. So you're using some noise range, and this one maybe really has a big variation because that was a bad day. And so I'm looking for a grade somewhere here. So that means that if you did very badly, don't give up all hope, but take that as a sign that you need to do better, and if you manage it, then your average decides. If you did really much better, well, much before you. So, right, if your grades do this, then you still get the average, which will be somewhere here. So, you know, even though I shouldn't file, I'll expel that one out, too. But I won't do that. Okay, so a couple more comments. Please check of your grade. Let's correct her. 15, 15, 15, and 10, you should tell me. If your grades got a perfect score, it would be nice if you would tell me, but I am not going to hold you to that. So, tell me if your grade is not, and that's enough. So I don't want to go over the problems on the exam because I post about the solutions. Call them something else. Armando, recitation, either administrative or any kind of questions. Yeah? In fact, there are other grades you just use the percentage you've got. Yeah. So this counts for 25% of your final grade. So, if you've got an A on this exam, and you get all zeros on everything else, and you've probably got at least a D in this class. But, yeah. You can add it to the letter. What matters? What do you calculate? I calculate the number, okay. But the number is not the percentage, right? Yeah. So here's what I actually do. I apply a piecewise linear mapping to your grade to put it on a standard scale where a 50 is the middle C. Then I average those. So the number matters, but it's not the percentage, right? I guess it's piecewise. Good. Okay, so other things? Okay. So now let's go back to doing some math. So we'll do some number, let's call it A, which is the way they teach limits in most calculus classes. That means if you look at A here and look at the graph here, it means as you zoom in here, wishy definition, let's call this a definition, definition. This means that for other little number delta, so that here, so that the epsilon is there, sorry, f of x minus l is less than epsilon in the domain minus A is less than delta and bigger than zero. So that's another choose. So let's choose that epsilon. So here's l, here's l minus epsilon, l plus epsilon. So you choose some distance around the point l, then there is some width here. It's here, well, let's just use that. There's some width here, A plus delta, A minus delta, that traps the graph inside this box. If you want to get within epsilon, you have to be, there's some number delta, some amount of control that you can apply to the function to get it in there by being within some, the real definition of the limit. And the reason that I'm pointing to calculus one classes, that's glossed over a lot, because it's, I don't know, too confusing or something, but it's important to realize now we want to generalize this to functions of more than one variable. So I want to generalize this to some function f from Rn to Rn now, so this is a vector function f. And I claim that everything is just the same as long as we interpret all the symbols in the appropriate way. So I'm going to say that the limit x, the vector, goes to some point, actually, let me call it x0. Some vector x0 of my vector value function f of the vector x, I'm going to say that that limit is some vector l. This vector is in Rn. This is in Rm. This is in Rm. So everything's the same except I put arrows over stuff. This means how small I want it. So for any, this can still be, this is my distance. Any distance I choose, there is some other input distance. I don't know. There is some input distance. So that, let me write it with two lines just to emphasize, the vector, the distance between the vectors f of x and l is small whenever the distance between the input vectors is small enough, but not necessarily equal. x is in the domain. It's exactly the same except that, and so I'm writing double lines too. So here, I make a mistake. Should be a zero vector. This is, no. No, it's a zero zero. It's a zero vector in R, which is also the scaling. So here is the length. Exactly the same as that guy as long as I interpret it. So it's saying if I want to get, if I want to say that f is getting close to some vector value l, I have to ensure that I take x close to, and if this is true, then we say the limit exists. Is this thing in the way? Should I put it lower? So let me actually draw that picture of some function. So let's say my vector value function f takes R2, R2. So a function from the plane to the plane. Let me just draw this same picture in that way. So let's say the domain, here's the domain of f. It's defined in that blob over here, and I have my point x naught, which is inside the domain of f over here somewhere. Let's f match this blob, I don't know, some way over here. And if I want to ensure that some epsilon maps onto that disk, there's some little disk here of distance delta around x naught so that everything in this disk goes inside here. Goes to some blob, and maybe it's got a long finger, goes to some blob here around l, but it fits inside every little disk of epsilon that I can make. And if that's the case, then we say that the limit, as we move in here, is the same as that. Now, this is saying, no matter how, not just along the coordinate axes, but no matter how you get there, here it has to get there, too. So the functions, which were not continuous because they moved, they were fine if you come in this way and if you come in this way, but if you come in that way, you get a different answer. So we had functions, so we have another way beyond this, you know, in one variable. We take a line, I'm just going to draw this analogous picture, and it goes to a line. This is not the graph of the function of this line, and if I shrink this in, then this has to shrink in. This is continuous, but something that takes this is not continuous. So we have to be a little bit careful that stuff doesn't fold in your way. We have that. And I guess one other x is in the domain of f. Maybe I'd have a function which is defined on this blob, this big blob, which also sends this point, x2, to this point, to this point, isolated. So a point in the domain, if it's in the domain, is x1. Then that's called isolate. If it's defined in an isolated point, it's automatically continuous. If this just holds automatically going x1, then we just say the limit might want to define a function at a point. So I'm going to take 141 for me. She's not even listening. Okay, so... But I won't even talk to her. Fine. Even though the sum vector f1 comes x vector f2, so we have that. It's not hard to show f of x, sum vector l, which is l1, l2. These are all now scalars. It's something exactly when the limit of each of the coordinate functions is the coordinate bit. So this is true if and only if x, as opposed to x, the coordinate function fi of x1. The limit exists if we just look at the coordinate functions that eat vectors and give numbers out. And that makes our life much easier. And let me not do the proof of this, but let me just tell you why it works. So can somebody give me a hint why this would work? Or even what this is saying? What it's saying. Okay. It means that you can take the limit of the individual components, so I can take the limit of f1 of x vector. It means that it'll equal l1 of the scaling. Yes. So we just can take the limit on each piece. And why does this work? Well, this works. I mean if you just look at this picture, here we say we take something going in, we look at what comes out, and we see that here it's trapped in this bit, and here it's trapped in this bit. And so as this shrinks to zero, these both shrink to zero. And as these both shrink to zero, this shrinks to zero. So we can just look on each of the coordinate axes and see how they shrink. Yeah. Why does it go towards x0? Do I have to have a limit going to something? Well, why does it go to zero? Would you rather I call the gym? Well, shouldn't it go up from zero? Feel cool. So it's not, I mean x0 would just give it some place to go. In my notes actually I called it A, but then when I get further along I want to call something else A. Yeah, no, it's just a gym now. So as things tend towards gym, yeah. It's a gym, so it's... No, that's fine. I didn't say anything about continuity. I didn't say f of gym is l. I'm just saying x, this is just saying we're here in the plane, here's some x that comes in some way and goes that way. And squeeze this, that forces the circle to squeeze, and as we force the circle to squeeze, it forces these two sweeps. In other words, what we're trying to vector minus l vector is just the square root as one of l2 squared plus blah blah blah plus fm of x vector minus lm squared. Just the distance formula. All of these things, if all of these go to 0, and that squeezes this to 0, if one of these doesn't go to 0, then this doesn't go to 0. So this observation is just another way of saying that picture. You said, thought I heard you say a minute ago writing the limit in that situation is not the same as it being continuous. Well, I didn't talk about what continuous means. So maybe that limit exists, but it's not what you want. It's a condition. Certainly, if I want it to be continuous, I have to have a limit. But you have a limit, it's not necessary. Not necessarily the value. Just like in one variable, if I have, just like in one variable, if I define a function like f of x equals the cosine of x or x not equal to 0 and 5 and 3 quarters for x equals 0, the limit exists everywhere. x goes to 0 of f of x is 0. This definition doesn't affect the limit. This is well defined everywhere and it's saying, you know, the graph of this function is the cosine, except here it's 5 and 3 quarters for all my scales that go off. It's not a continuous function. Now I didn't define continuous yet, but sure. Okay, thanks. So, I mean, limits are closely related to continuity, of course. Yeah? Because it's 1. And then also, for all i, for all i, between, what did I start with? 1 and m. This is just a hint of how the proof goes. You want to read the proof, open the book to the appropriate page and there's proof. But it just does that. It just says this can't shrink. So, I mean, I can do the proof if you want, but, you know, I don't want to spend. Why is it 1? Because the cosine in 0 is 1. But the limit as x goes to 0 of this function doesn't even see this because we're only looking for x near 0, not equal to 0. So as x squeezes down to 0, it looks just like the cosine. And cosine is continuous, so it's just the value. But the value is 5 and 3 quarters of this function. I didn't say anything about derivatives. I'm only talking about limits. I'm sort of starting over, but not really. This is defined everywhere. Here, I can say, how about x bigger than 0? How about x bigger than negative 1 and x not equal to 0? And it's 7 for x equal to negative 5. Negative 5 is isolated because this is only defined from negative 1 and up. So this is isolated. I was just throwing the isolated in there in the case that we have an isolated one. And I was talking about you but you weren't listening, so it's okay. Okay. I guess another thing, another piece of terminology that I want to mention, a name which is that we have correction functions. The book says that this is the function I'm going to use the book's notation for a minute. pj which takes, I don't know, rn into r of x some vector is just let's say x is the vector x1 up to xn and the jth projection function the jth projection function just picks out the jth coordinate. It's just a way of saying forget everything except the jth coordinate. Now, the more common notation at least in more advanced math use the letter pi because people get confused if you start calling a function pi. But it's fairly standard notation in a lot of more advanced mathematics to call these projection functions pi, pi, the Greek letter p so I may actually I probably wouldn't use it in this class. Yeah. And then this book uses capital p but it just says pick out the jth coordinate. Yeah. It's rn in if you r out. So always right so p2, 7, 8 pi means something else in the fundamental group but yeah, we need more letters. Yeah. So it's really confusing when you want to project the fundamental group on this whole thing. Pi, pi. So you only use either letters mostly for fundamental. Yeah. This is p2. 1, 7, 8 is 1. And p4 doesn't make sense because it's only in 3 seconds. So this is sort of a standard function from rn to rm and sometimes it's handy to use these projection functions to mess with the prior things but it's sort of so some function some function f in rn oops, sorry I need something first is in the interior. The same is plugging in. If that's true, then the function is continuous. It's exactly the same definition as in one variable. When you plug in it's the same as taking the limit, the function is continuous. So respectively f is continuous at x0 and x is continuous at g at the appropriate place and also where it's appropriately defined the composition. Composing, continuity, continuity provided that you're not divided by certain continuity, all that stuff is good and let's call it a matrix. And it's continuous and scaling them by some scale. projection functions, you bang down on each of the projections looks like a1x1 plus a2x2 plus a2 11 12 11 plus a2mxm n1x1 plus amm all linear transformations are continuous Does that say a2mx1? Yeah, I need two I need two indices because if you prefer I could put a b here and a z here. I'm going too fast, is that too confusing? Why do we care? I'll show you in a second. We're almost there. Let's respond to r for a minute. So this is a real value function on a vector input which is in the interior F of gym I'm writing a difference quotient here by the distance between x and gym Well, it is a number but it's not just any number but this is where this a comes from So, instead of writing it on this side I'm going to write it here, it's the dot product where you're going. What do you talk about? Do you take each of its components and simultaneously increase them first? We're careful than that Let me draw a picture If your function is nice sure that's good enough So let me draw a picture here and I take any vector x and it comes in in any old way Just look on the coordinate axes The function might be nice on the coordinate axes but terrible if I do something like this it will be generally Is it up to you? This is like the map that Sarah Kaylin put up You have to show it works from all ways So really what I'm doing is I'm zooming in my scope because you can have one component of the vector go really far off as long as it comes back So in the end in the circle You shrink this entire circle and the things that are trapped in that circle have to approximate minus sign A equals zero is that the limit as x goes to gem of f of x vector minus f of gem vector divided by the length of x minus gem vector vector times the vector x minus gem but we scale it that's why I can't do it See, I can't do it I can't do it So what this is really saying is if I look at this thing more and more and more it looks like looks more and more and more like a this is a unit vector more and more like some vector or projected onto this unit vector It's saying no matter how I come in here eventually I'm going to come in sort of along a direction and on that direction there's some vector A that works like just measuring the dot product I mean the dot product is measuring the size This is yeah So the top is equal to zero and the tops only equal to zero if the two sides are equal to each other So why do we need the bottom part of the limit at all? So think about the one variable case In the one variable case I have f of x minus f b over x minus b and I take the limit as x goes to b This is f prime of x Let me try and rewrite the one variable limit in this kind of language Well that's the same as saying that minus that is zero and then I can scale this up by x minus b over x minus b except I don't want this to be like that So here we usually put an absolute value here and an absolute value here but I don't want the absolute value compensated for by making sure that they come from the same side But it seems like this whole thing is just complicated like only holds true in a way way simpler less case holds true Like the really simple case being the limit of f of x minus f of j equals A dot x minus j There is some A so that they will be equal You do But it doesn't seem like you do because the only time that that limit is ever zero is when that top part is zero Most of it You're losing information It's saying if you scale things up so this is a linear function and we're saying as we zoom in here we're scaling it up so we're zooming in and scale up the corresponding linear function They look more and more the same Is there any time that the limit is zero but the top is equal to zero? Well it's more than just the top being zero But is it? This is tending I mean this limit so this is never zero x is never general We're only getting close to general Right But it's never equal to general Let's say as you get closer and closer I mean it's saying as you scale it up almost I think there's a counter example but I'm not sure Okay so let me So this vector A, I called it A here but it has another name and some of you may already know what the name of that vector is Does anyone know what the name of that vector is? This special vector A So the definition says it's not derivative This is how we write Robert This is how we write Robert Mine was pretty close What's up with this? Oh you can't see It's a gradient I should tell Robert That doesn't like that That's the definition That is all stuff It's the gradient of f It's the gradient of g Yes the gradient of g And that's what A is And that's what A is So A This particular A If it exists It's called the gradient vector Now Many of you know Really The following theorem We've done any multivariable calculus before or seen any multivariable calculus before We have some function It's continuous I'm going to stop thinking of Jim Continuous Differential It's differentiable At x0 Then The gradient of f Is just the vector Partial f x1 At x0 Partial f x2 At x0 Blah blah blah Partial f xm And that's unfortunate So suppose that xy No ze to the xy So that's a function From r3 to r So the gradient This function is continuous everywhere The domain is everything So at any point x, y, z The gradient Is going to take the partial Of this with respect to x So that gives me y yz e to the xy That's the first x bit And then x to the y gives me x plus xz e to the xy That's 1, that's 0 And that's e to the x This is not a D This is a nap line You type in macro-western It's easy, very easy to calculate You might also write it as y plus yz e to the xy Yeah, this is in there Because I don't have m kind of like A kind of derivative vector It's not Well, it's a vector that's Related to the derivative It tells us about it And in particular It gives us a good sense of Where the tangent plane is But here It's even a little bit hard To think of this tangent plane Because this is a vector in R3 This is where the C later Work geometrically So right now It's just that And Maybe So this is not a definition This is a theorem This is a definition We find some vector That when we dot it with the function It works nice It gives us The limit as we zoom in So let me say a few words So the proof actually Is pretty easy The thing that when you zoom in It looks more and more like It has to be that The hypothesis is it's differentiable So that A exists So that's what it has to be So What do these things mean? These things mean I come in on the various coordinate axes And so I can write A So really what I Here's A In terms of the coordinate pieces Somehow I lost Jim Jim became A Yeah, that's because I switched from Oh well, it's okay So if I write it that way Then notice that if I just Look at I can now look at this piece by piece So I can just look at this on each piece And I see that it converges Well those are exactly the partials It can't be anything else If it's there Let me point out that if Partials are continuous Then f is automatically differentiable Without the partials If it was functions Then it's easy to see that the function is Differentiable Even if the function can be differentiable Even if the partials are not continuous Let's just an easy example An example of this Is the function From r1 To r1 Which is say x squared Sine 1 over x If x is not 0 And 0 Then the derivative at 0 is 0 But It's the same as the derivative that's not continuous So there's a name That has two things happen Also partial Is continuously differentiable The derivative that we get Is continuous there So if you remember I mentioned this That said the mixed partials are equal When the function Has continuous partials This is exactly when Claro's theorem holds Given this The second derivative The second partials Fxy is Fyx We have to go to the third For three dimensions Say you're doing it What I wrote here works for 17 Dimensions if you want So it wouldn't necessarily be the second No, just the second So The function is continuously differentiable That f of xz Is the same as f F Or y Or q Just the mixed partials The second derivatives Doesn't matter which way you go So this is really the same as saying To tangent plane There's a tangent plane And it moves around in a nice way You can find a tangent plane