 So I'll remind you that the exam is a week from today, it's here, I put on the web page somewhere here, link to stuff, there's things about the exam, this seems terrible. Anyway, there is the exam I gave in 2010 when I taught this class at the exam. Anderson gave it to us. Well, this is class A. Okay, so let's look at this exam for a minute. Over 90 is a meaningless concept because they added to 74. So it's hard to get over and on how hard the exam is. So this particular exam was pretty easy, I thought. So maybe somewhere around, I would guess somewhere, if you get either all right except for one, then you probably get an A or an A minus. Or if you get most of the questions mostly right, you get an A or an A minus. But it depends on how hard the exam is. So if I write a hard exam, it could be that 50% and up, isn't it? So usually I don't write an exam that is that hard. Generally, I try to write an exam, although this one was too easy. I try to write an exam where nobody will get a perfect score. Because if I write an exam where several people get a perfect score, the exam wasn't hard enough because those people were not challenged. Usually I try to write one problem that is difficult and some problems that are easy and some problems that are in the middle. I look at the exam and I decide how hard it is. And generally that works out that maybe 10 or 15% of the students get an A. But sometimes it works out that only one student gets an A and sometimes it works out that 60% of the students get an A. So I don't have rigid criteria that 90% it up as an A nor that only 10% of the students get an A. If you know the material well, you get an A. If you know the material badly, you do not. Generally in this class, most of the people that make it to the end of the class pass get an C or better. A lot of people usually drop out. Yeah. So last time I taught this class I had about this enrollment, I had about this time in the semester. And at the end of the move down date, I mean a few people did the PNC business. But I would say like five or six people decided this was not really for them. Which is fine. Okay so there's that. What? No, but you can. So if you have questions, I will not put up the answers until probably Monday. Maybe before class, maybe after class. I'd like you to try it, see what you think. I can talk about it a little bit on Monday. I don't want to spend Monday's class doing the sample exams. I'd like you to do that, but if you have questions that you see from it, then ask. Yeah. So what I wanted to do, where is the rest of this page? Oh, that's the exam page. So I wanted to continue roughly with where we were last time. I'm going to continue with stuff. I guess I'll start with some graph and these kind of stuff. Just like last time. So we talked about standard planes and partial derivatives and stuff like that. I want to continue with that. Okay, so talking about some function which we can think of, especially, and the idea here is the X and the Y with that height and you get a surface that is completely illegible on the screen. But anyway, you get something like that. I guess that Y is coming out of the screen in this picture the way I had it. And let me spin it around. Oh, okay. Oh, I screwed that. Let's start over. So there's X, there's Y, there's my function, draw it. So X is coming out this way. It's a little hard to see. There's the lights, but that's all right. And so it does something. If we fix a particular X value, we get a curve. So I'm going to fix a particular X value and think of cutting the thing open like that. So I have a curve here in the XZ plane and there's a well-defined derivative there. So we have a well-defined tangent line. I can do that here, tangent line at a point. Can you see that? Probably not. So this red line, let's put my point up here. Well, there. I'm trying to get it at the edge. Come on. I'll go past it. Come on. Why can't I? Okay, I'm having a little trouble here. Why is this going? The mouse was upside down. Okay. Let's go. Alright, so there's off the surface. So that red line is what I get. If I cut the thing open, I get a curve like this. Again, remember positive X is this way. So you have to look from the other side. There's the tangent line. And this is the partial, which is just taking the usual derivative. Let me write it this way. At a point which is, in this case, let's say X goes to A minus F of A over X minus A. And so that gives us the partial, which is just restrict to this slice. Oops, except I forgot to write B. And that gives us that line. Gives us this plane, this curve there. And similarly, if we slice it the other way. So let me go up to the corner here. And instead I look at the Fy tangent. So now I'm slicing it the other way. And you can see there's a tangent line there, which moves around as I move around on the surface. Right? So that gives us the partial. The other way, which is that slope, which put them together will give us a plane. One slope, and here another slope, and together they define a plane. Think of this. So this tangent line that I have here gives me a slope, which gives me, think of this slope as giving me a vector. So I can think of this plane either in coordinates, where the coordinates are saying the Z value is going to be, from A to X and from B to Y, well I have to adjust. If I just move along, so that's a number. Let's write it this way. So I'm going to gain X direction on the Z. In this case it's negative. And if I move just in the Y direction, I'm going to gain whatever distance I have on the point B adjusted by the derivative. So this is towards the end of the class. We can also, yeah. So I'm at the point AB, which in this picture is right here. Maybe we'll be tilted a little bit more. That one's kind of flat. Let's move it over there. So I'm at this point here, this point here. This height is Z of AB. Where AB is that point. And then now I'm going to move on this plane a little bit. Okay, so you're not changing. You're not moving on the point. I'm moving on the point. Which is exactly the analogy in one dimension. If I know the point here, that's A. And I have the tangent line here. I go to this height, which is half of A. And then I move some distance here to a point X. Then I take this slope. So the slope is F prime of A. And I move this distance, X minus A, in that direction. So I have to adjust to get this height. What I started with, plus the slope, which in this case is negative, times however far away I moved. It's the exact analogy. It's just now I'm in a plane. And I can move in the X direction or I can move in the Y direction or I can move some combination of both. And as long as there's a nice plane here, this will work. But this function has to be nice for there to be a nice plane there. I can also write this, I can also think of this in terms of vectors. Where I have this vector here in my point, which is on the surface that I'm not drawing. And then I have a vector. Well, in this case, this is Y. So this is some slope times J, times the J vector. Gives me that. And in this direction, which is the X derivative of AB in the I direction. So that means to get to some point here on the plane, out here on the plane, I move some amount in the X direction and some amount in the Y direction. So my plane can be written parametrically as my vector value function. So that's my position vector of the base point. Plus, I moved in the X direction. But I have to, I want to describe the whole plane so there's some parameter T. Because it's parametric. This is a vector. This whole thing is a vector. I mean this distance here in the Y direction. We need to back this off. Okay, forget about this parametric representation because I've just embarrassed myself terribly. We'll come back to that. Otherwise, I'm going to spend the whole day figuring out what I wrote down wrong with this garbage. So let's do an example, let's take this plane here and let's just write the equation of the tangent plane to minus 4 X squared plus Y squared equals Z. I don't know, 1, 1. Okay? Does everyone know how to do that? We'll use the word Y. We'll just treat X squared as the parallelism constant, right? Yeah? So, yes. Can everyone do this? Does anyone need me to do this? Yes, you need me to do it. Okay, so we need to compute the partial derivatives and evaluate them at 1, 1 and then look at the plane. So, partial of F. Let's write it in the other notation. Partial of F with respect to X at 1, 1. Well, with this function I think of Y as a constant and I don't like the quotient rule so I'm actually going to think of this as minus 4 X 1 plus X squared plus Y squared to the minus 1 power. So that gives me a minus 4 1 plus X squared plus Y squared to the minus 1 power plus minus 4 X times this derivative of plus. So I get a 4 X 1 plus X squared plus Y squared to the minus 2 power but then I need to multiply by the derivative of 2 X, right? Because this is a constant so when I take the derivative of this with respect to X I'm just left with 2 X. I might have made a mistake so if I made a mistake, please tell me. Again, now I'm thinking so I get minus 4 X 1 plus X squared plus Y squared to the minus 2 times 2 Y that's right except the minus power changes that to a plus and then I want to add the point minus 4 over 3 plus 4 over 9 times 2 which is what, 4 nines? Negative? No, positive, right? This is 8 nines This is definitely a bit This is definitely a bit Oh, that's 12, yeah, that's minus minus 2 nines and then here FY at 1 1 is 4 over 3 4 over 9 times 2 which is positive 8 nines so that's telling me of course it's not at the point that I'm drawing here at the point 1 1 which in this picture is there if I move in the X direction I decrease by a factor of minus 4 nines and if I move in the Y direction my slope increases by a factor of 8 nines so for example that plane that plane is the plane Z is the function at 1 1 which I guess they didn't compute minus 4 over 3 plus, well it's minus minus 4 over 9 times X minus 1 that's how far I moved in the X direction plus 8 over 9 times Y minus 1 so now it's easy to convert this into parameters 1 if you want if you want me to do that alright how about X equals 1 minus T X equals T and Y equals T, S and Z is well minus 4 thirds minus let's make X minus 1 T so that's plus 1 plus 1 then it's easier 4 nines T plus 8 nines alright this is the same thing what? what do you say it's a factor of 7? well okay so 4 T plus 1 plus 1 minus 4 thirds minus 4 ninths T plus 8 ninths S 4 T plus 1 I plus S plus 1 J that junk is plus no it isn't 9s plus 8 T 8 9s S so is it clear that these are all the same thing? there's nothing canonical about this T plus 1 you could choose just T here but then this would need to be a T minus 1 right just deciding where 0 is deciding where the origin on the plane is the way that I put it here is right here at the center of the plane but if I made T if I chose a different parameterization where this component was I don't know let's call it U then the center of the plane would be back here on the axis yeah before we do this do we need to say where's the origin? no the origin is explicitly obvious here okay right the origin is 0 0 0 the origin is always 0 0 0 yeah so the thing that was screwing me up when I was trying to write it is I was thinking of the plane as sitting at the origin the center of the plane as being the origin that my point was 0 0 0 and then I was getting lost so here we're putting in some sense in our parameterization T equals 0 S equals 0 is sitting over the point 1 1 not sitting over the origin which is sort of a more natural place for the center of the tangent plane to be is sitting right on the graph so we have to adjust so the coordinates are the usual coordinates okay any questions about that? you should all certainly be able to do that problem within a week or one like it this is something I view as a mid-level maybe easy problem somewhere between easy and mid-level if you can't do this don't expect to be able to get a C on the midterm can't do this or other things like it okay so when does this break? I said if everything's nice this works times things aren't nice what? there's going to be more than a tangent plane what does that mean? a functional sharpener okay so then there's not a tangent plane right so if the graph can I start erasing I don't know where the problem is putting these things down so if the well maybe there's one here that I can just use should be 0 0 should be one that's like a 1 over draw that something like 1 over square plus y that one has a problem at the origin it kind of blows up there right so there's no tangent plane at the origin well the function is not even defined at the origin so that's kind of a problem I could you know change this well let me just draw it rather than figuring out one so if I have in my analogy with the one dimensional situation that's something that comes to a peak looks like it would be hard to sit on um maybe so I want this to be so maybe if I take the arc tan of that I should have no that was too much oh that flattened so if I have something that comes to a point which I didn't have one in mind at the moment that would be a problem is that the only thing that can go wrong okay what else can go wrong okay I could certainly have this is bad I could have something like that right so this is this is supposed to remind you of a surface like this one it's not defined okay so I can have things like this we also have even worse things where stuff wiggles around like crazy and piles up in different ways um so something that is important is the idea of continuity can happen in two in one variable that is we can maybe calculate any derivative we want but they don't hook together nicely um I'll give you an example of that in a second but let's just think about what continuity means so in one variable continuity means that the limit this is not smoothness, not so much we want something similar just restrict to one and to one I want vector goes to equal I depend on how you get there something could happen that it's okay in some directions and not okay in other directions if I'm trying to get to this point it's perfectly well, no not so much here uh this point this point is perfectly fine if I come in this way all of those always go here but if I come in along the there so there could be bad directions this can have which are bad and the same sort of thing can go on the same sort of problem can happen with the one variable and in several variables that I get a different value depending on so suppose I take squared minus squared minus y squared it's weird at zero so it's basically I mean so there's sort of a crease along there's a straight line I mean if you look there if I look at it just in the x direction it looks fine and if I look at it just in the y direction it also looks fine but they don't match up the function in one direction is down here and another direction is up here and somehow there's a crease going on I mean there's a crease right in here where things are weird so I have a problem at zero zero now the partial derivatives of this thing are just fine it's a perfectly nice function the y squared is zero one equals zero zero y is y squared minus over y squared which is minus one so along one line this function comes in this way and here's my axis y axis if I come in over the y axis the function has a negative value if I come in over the x axis the function is one and if I come in in some combination then the function does something in between right if I come in along the line x equals y I get zero but if you didn't notice this and you took the partial derivatives you get a nice formula so it's a little more one has to be a little more careful it just gets zero when you take the partial derivatives at the origin so one has to be a little more careful with these functions than in one variable it's not obvious right away what's going on I just want to point out that there's some subtleties and you need to think about what's going on with these things we will spend more time on this so I want to point out something which so I guess I'm now done with these things so any questions on this kind of a surface I mean I can up I can obviously do many more examples of this kind of thing but we have finite times everybody understand the issue here if there's a subtle issue which I'll come back to later clicks and then it will go on function is nice if we have nice means in a second so assuming it's nice that is we have a tangent plane there's an important theorem called if it's nice here is continuous at some point let's call it a vector just say it in the state let me just say it in the case of 2 and also I'll do it in 2 it works in higher dimensions but so and all are are continuous partials are continuous if I take and I take the x derivative of f y this is part of y I can never remember which means what so this says function second partials i.e. partial of f with respect to x and then with respect to y so it's a second partial is the same as the second partial of f with respect to y with respect to x so the order doesn't matter I want the second derivative there's really three there's the f x x which is the same as f y a simple example I could do the proof but then we wouldn't get anywhere let me just tell you the idea of the proof without really doing it but first let's do an example cosine x y function if I compute f x this will be so this I'm thinking of y as a constant so that will be 2x cosine x y plus x squared should be minus sine it's much easier x squared sine and it's next it goes right taking the derivative of this it's a product rule the first is x squared times the second and then the first times the derivative of the second and then this one doesn't need a product rule because the only y is inside the cosine so that gives me an x cubed sine x y so now if I take let me write it this way just to emphasize d dy x which is also I will forget which one goes first I think this is f x y that's the right notation so I take the derivative of this thing but with respect to y this term gives me a 2x y sine x y gives me something messier so the derivative of this is 2x y sine now it's minus I've lost it 2x y sine x y minus x squared y cosine x y but I think of it as y I think that's right and if I take the x derivative of f y I should get the same thing so if I take the x derivative of this if I take the derivative of this I get 3x squared minus x cubed cosine x y okay I really screwed up somewhere I should take y derivative wasn't I how stupid this is just garbage sorry I'm taking the y derivative of this so this is 2x squared cosine x y and the y derivative of this cosine x y where? did I do it wrong? it's not y squared it's x cubed that's the last term here? for the next letter I need 2x squared sine x no you're taking in terms of y you gotta take the y derivative of this jeez anyway it works it's 2x squared cosine x y sine x sine and it's negative thank you and then this term I probably have right the y derivative of this is minus x squared sine x y plus negative plus negative x squared y x y pick up another y pick up an x pick up an x and that's what those are equal right almost okay it's not my game let's try again this guy minus a taking derivative with respect to x minus 3x squared sine x y is this part and then when I take this part I get minus x squared cosine x y with respect to x so I get another y minus x cubed it's x cubed x cubed there they match now good I get a c minus on the test so okay so why does Clairot's theorem work you can read the proof in the book but it basically just comes down to the mean value theorem twice so you look at so the mean value theorem says that mean value theorem if you remember says that if I have some function I look at it the slope of the tangent point and I take some integral around it I take a function of one variable and I connect two dots on it but somewhere in between there's a point where the tangent line is parallel what you do is you cleverly use this fact that you look at an appropriate function which says that I'm going to take my function of two variables which I can't really draw there's my function of two variables and I'm going to go this way to get a tangent plane and then I'm going to move that way to get the second derivative and everything is going to work out so I take the use the mean value theorem for x this way and then y that way and that's the same as doing y first and then x but this geometrically is this is saying the rate of change of the tangent plane doesn't matter if you go like this and like this then if you go like this and like this if I move around if I move around on the surface and I want to go the surface I'm at this point here's a tangent plane and I want to get to this point here with a different tangent plane I can move my tangent plane this way and then this way and the way in which the tangent plane moves is the same as if I move this way and then this way so the net rate of change which is how is measuring how the tangent plane moves is the same if I go by two different paths as long as my function is nice alright so you can read that proof if you want it's just a bunch of algebraic calculus let me skip that okay let me mention some other functions of more than one variable first let me remind you quadratic functions these are the quadratic origin and the form of a parabola is y equals A x squared this one you should be very familiar with I could have a circle or more generally an ellipse so that's supposed to be an ellipse this is of the form x squared plus y squared y equals b we get a circle semi minor axes and then we can have a hyperbola so this should not be new to you guys is this new to anybody okay so those are the sort of the simple the most simple degree two functions where I'm allowing for I v degree two and if we move to functions same things we can put them together in different ways instead of being called conic sections by the way these are called conics you know I just lost the eraser this room only needs two erasers these things are called conics because if you take a cone and you slice it this way I can get an ellipse or if the cone is round I might get a circle if I slice it I'll get a hyperbola depending on how I slice it if I slice it parallel to this but over here I get a parabola and if I slice it a little tilted I can get a hyperbola so slicing this cone in three different ways gives me you can't see what I'm doing depending on how I tilt this slice I get one of these three things they're called conic sections because they're sections of a cone in R3 the analog quadric surface since I started with the circle there I started with the parabola there because I started with the parabola these don't match but anyway I get something like that this gives me something that's sort of down-facing parabola one way and an up-facing parabola the other way all of the slices in one direction are parabolas which open downward and all the slices in the other direction are parabolas which open upward do you see this surface? it's a saddle surface it's called an elliptic this is called a hyperbolic most normal people call it a saddle surface and for some reason I switched my notation let me put my notation to match so this guy has a minus sign so the difference here both terms are positive equals the sum of two positive squares gives me paraboloid like this or both the same sign and this guy gives me, well since one is positive and one is negative they open in opposite directions so the slices here are parabola which either open down or open up depending on which way I'm slicing but we have other variations there too so we can have something so this is a slice this way is an ellipse and a slice that way is a parabola this one, a slice this way is a parabola and a slice that way is a parabola this one is a parabola either way and the cross section here are ellipses it's not obvious that the cross sections here are straight lines they are another option is we can have an ellipsoid so that will be again looking like this I didn't even write it down, cool slices in one direction in any of the three directions there are ellipses slice the level curves are the ellipses the x cuts are ellipses the y cuts are ellipses we can also comb kind of things pull up kind of things so we can also something like let me make sure I get them all yeah I've got something wrong c square that's what I want the sum number k so if I do that then that's saying that if I fix a z value I get elliptical things if I fix z to be a constant then this is like an ellipse if I fix one of the y's I get something that looks like a hyperbola so this will be something that looks like a nuclear power plant so this is called a machine I get the thing in between which is a comb cross sections are ellipses if a equals b if a equals b has a circular cross section but if a gives you a difference you get an elliptical combinations of those things in higher dimensions and with humans left these are chronic sections are things that you should easily recognize when you see an equation like that these are original analogs that are sort of good examples to keep in mind these are also sections of a 4D object so if you think about sections are all slices of a yeah but they're cut by a three-dimensional slice so just think about these guys these guys all fit together in a very nice way pick an a, pick a b and let k move then you can pinch it down where k is a fourth dimension that sits inside a fourth dimension where the slices are the two-dimensional surfaces that cut up the three-dimensional space in the same way right so these are slices you can move them around and think of them as the same you can do the same thing with the ellipses these things can evolve one into the other which you can think of as slicing of a higher-dimensional object so we have just a few minutes left but I have yet another topic to cover that I didn't even mention so now let's sort of come back to the analog that they had with more than one variable now I want more than one variable in and more than one variable out but let's just let's just think about, let's take an example let's take a function here I'm going to put in two numbers and I'm going to get out three numbers for example do this one f of u, v it's shorter to write is what do I want there cosine u cosine v what does that describe what? well it's like a spiral but it's not really a spiral it's like a working round so if you just think about this let's just fix one of the variables let's think of then the v the only thing v is v which we've already seen gives us a helix like this it spirals around and goes up and it starts at when v is one sorry when v is zero it starts at one zero zero and then it goes up like this now if I move over a little bit let's instead of u equals one u equals a and I pick up an a here and a here and now a little bit get another spiral up here a bigger spiral if I move a back and forth I can think of this as giving me something like a ramp going up so that is let's see which of I fix I fix the x value something like that something like that so it gives me a ramp and moving around and of course I just picked a positive I picked here a between one and two but it gives me a thing like that and if we look at the other dimension let me just do that in red if I instead of fixing a I mean if I'm fixing u let's fix v so it gives me like this that keeps spinning around because v is a constant so I get u u constant so the other direction is just lying to the constant so this idea that you see by this example we have a surface now in the special case where we're taking two dimensions into three dimensions it's easy to think of these as surfaces but it works the same way let's consider if we're taking say r2 into r2 now I can't draw the picture because the picture lives in four dimensions but I can think of f of some input square is maybe some maybe I should call it a gene here it's not that easy is some output square this is in the plane in the plane I put some yeah we didn't talk about vector fields but sure also view this as a vector field but we're not getting the vector fields until like sometime in November but yes I can think of this as a vector field I'm useful to think of this as a transformation of a piece of rn into another piece of rn the vector field is just the difference between this point and its image so I can think of it as a vector field but at this point I don't want to think of it as a vector field I want to think of it as moving stuff around which is the same you move stuff around what you get after you move it or you have the directions for moving this is the analogy