 A week from Wednesday is okay, and the next cover, a week from one to another, we'll cover basically chapters one, two, and four, which right now we're sort of partway into four. I can make it a week later if you want. A week from Wednesday just means two weeks. So a week from Wednesday is okay for everyone? Yes. Okay, then it will be here a week from Wednesday. Okay. This is a three class test. Yes. The exams in this class are during class five except the final. The final is during final five. Okay, so last time, looking at functions from R1 into Rn, which we can think of as vector value functions. So we looked at something like f of t is a vector. I'm just going to try it generally. f1 of t, f2 of t, fn, for example, f of t is sine t, cosine t, t. I don't care. Something like that, which describes, let's see, now the way I did it, since I started with sine, it must start here. And the sine increases, so it's absolutely like that. Something like that. Right? So we talked about that. Maybe I screwed it up a little bit. And then we talked about what the dt of such a thing. So this is a vector, which is just f prime, f2 prime, fn prime. So it's just the vector of derivatives of the components. And that corresponds to the tangent vector to this curve at time t. So I talked a little bit about that, how that works. This is also the limit, h goes to zero of the vector t plus h minus the vector t over h. Same definition, except that this is a vector, this is a vector. So we get a vector in, we push it by a real number, and we get a vector out. So that's what we did last time. What I neglected to do last time, so I want to take a few minutes to do, is so we know about derivatives, let's think about what an integral would be. So certainly, we can just define the integral of this vector value function t dt. Well, this, we can just define it to be the vector, which consists of integrating each of the components. So we can certainly define it to be that. So this is a vector. We'll get a constant of integration, so we get a constant here, which will be a vector. So the constant of integration is another vector. So this will be another vector value function, let me call it g plus some constant of integration like that, and we'll get the property derivative of this guy, the vector. Nothing hard here, you just integrate term by term and it gives you a thing. And that thing that it gives you, well, you see that since this is the velocity vector, then this guy could be a vector, could be a vector that takes velocities and turns them into a position. So it's a position value for a function whose velocity is, right, it's exactly, this is the anti-derivative. So if I tell you how a thing is moving and I give you an initial condition, then you know where it is. So I guess maybe a stupid example. So let's say, let me just do, I'll just use that same example. So suppose that the velocity vector is, let's call it v of t, which is given by something like, let's just put it into the equations, sine t, cosine t. Did I put it in three together? Yeah. So suppose that the velocity vector is sine t, cosine of t, and I want to find a function like this, well, this will not give me an answer. I would need, with the initial position being, I don't know, 1, 0. Or I could do 1, 1. It doesn't matter. So this is saying that I know that my thing starts here, and then that I know that the velocity, as it moves at each point, well, let's see, sine, cosine, that as it moves around, I tell you the vector, which says how it moves. So this is at time t to 0. It's here, that's the vector. Some later time is over here, that's the vector, and so on. So we're putting together these collection of vectors that tell you how the point moves. Just in terms of calculus, we just integrate sine t, cosine t, t. And that gives us, well, the integral of the sine is the cosine, except it's negative. And the integral of the cosine is the sine. And then there's some constant. And that constant I need, well, that constant seems to be the 0. So that was a stupid example. Let's make it homework. So now t was 0, so I was 0. OK? So I have that, and I need to use the fact that minus cosine of 0 plus c1 is 1. And that the sine of 0 plus c2 is 1. So that tells me that c1, cosine of 0 is minus 1, I guess is 2. You do that right? Yes. Yeah, I guess so. And that c2 is 1. So that means that my function isn't really what I drew because I was wrong. Well, not quite. Is the function f of t is minus cosine t plus 2, sine t plus 1. It's just doing the same anti-derivative problems that you do with one variable, except we have n of them. And that also tells me exactly where the point is and its future path. You can, I mean, those of you that do this sort of thing in physics all the time in a mechanics class, you know? Well, OK, if you're in that class that you do this, if you're not, you don't. It's OK, you know where you know the law. You know the force is acting on the thing and you want to figure out how the thing moves. It's exactly this, giving you how its speed goes and where it started and you figure out where it goes. The problem is exactly the same as the one-dimensional problem. It's just several one-dimensional problems. Another, so I don't want to do, you know, 37 of these. I think, is there anybody that doesn't understand what I'm doing? OK. Another thing, if you think back to what you do with integrals, the area doesn't really make sense here. There is a way to interpret things as areas, but we're not quite there yet. So we can't think of this as an area under a curve. But another thing that we do with integrals, which is very natural in this context, remember this is related to curves. So I'm telling you how a thing moves, both what direction it's going, how its direction changes and how its velocity changes. And then we figure out what the thing is. But another example would be arc length. So in calculus one, maybe remember we have some graph, y equals f of x, and we want to figure out, we want to figure out how long the curve is. This is a terrible hand. How long this piece of curve is. And the process that you go through in calculus one is, we just draw the same picture again, it's bigger, that you write it as a collection of little chunks of width delta x. And then you integrate the length of each of these chunks. And of course this chunk here is related to the slope. We have to use this diagram in theorem where the base is delta x and the height is something related to the derivative, that this is delta x f prime of x. So you do all of this stuff and you arrive at the formula that the arc length is the integral of the square root of 1 plus the derivative squared gx from x to x1. People sort of vaguely remember this formula, maybe not. But it comes down to doing this kind of thing and then letting the size of the strip to zero and so on. But this is actually much more natural here in the context of a parametric equation. This is actually easier. So again I'm thinking I have some function which parameterizes my curve and I'm going to draw it in two variables but it doesn't matter if it's two or five variables. I have my function here. And if I just go a little bit on some small range, this is f of one t and here's f of another t and this vector between them, well actually let's think about it a different way. Let's do it a different way. I have a vector that tells me how far I move in a unit time. The derivative vector here is how far f goes in a unit time even if it has a point. And if I were to just add up all of these speeds then that should give me the length. So the length is just going to be some of the speeds over a small time period times how far you went and then I take delta t going to zero so this converges to the integral of the speed. So that formula actually becomes very simple. It's the same formula. So what is the speed? The speed is the length of the velocity vector. Just how long the vector is. I don't care which way it goes. I just care how long it is. So that means that the arc length is just going to be the integral of the velocity vector dt. That's it. Let's see that it's the same as the original thing. So if I have, say, y equals f of x, I want some other thing. y equals g of x, that means that I can think of this as the parametric equation as vector is x g of x. This gives me the graph. So let's just see that that's the same. So here, if I take the derivative of this, so the derivative vector is 1 g prime of x. This I have to get back and I want to go up on the space here. It's like a simple there. And so that means that the integral is the integral from my given, starting to my given ending of the length of that vector. Well, the length of that vector is once where plus g prime of x where vx. Which is exactly the same formula that you hated in calculus too. Or calculus b or bc or whatever name you call it. When you were doing applications of integration of the formula that you got, you're looking at it. So they're the same. But this one is easy to understand. And this one involves some messing around with the diagram and theorem and slopes and stuff. This is sort of a very natural representation of this one. It's just what it works out to be. This is a formula you can remember. This is a formula you have to figure out every time you want to do it. So I guess, let's just use the same example. Just moving backwards today. So the arc length of the piece of the helix h of t is sine t cosine t This one's parameterized by arc length. Excuse me. But t from p equals 0 to I don't know, 2. Well, vector is cosine t minus sine t 1. It's length is always square root of 2. And so it's length is 2 square root of 2. So the length is the integral from 0 to 2 to t. Of course, if I make this harder and make this a t square then this becomes a 2t and then this becomes the square root of 2 plus 2t and then it's not as easy but it's still doable. Any questions on this stuff? So this was leftovers from last time that, oh, it works again. So now I'm kind of done with so if you remember our goal my goal, whoever's goal the goal is to understand calculus in functions from n variables into m variables and then so last time in case of n equals 1 and m was n because n's are easier to write than m's. So now so last year n equals 1 last time and a little bit today this is easy, we're done we'll come back to it a little bit but we're kind of done here and so the next natural thing is m equals 1. So we want to now turn to functions I'm still going to use f because f is a function it's not a vector anymore it's a function that eats n variables and gives me one variable back so as an example from x, y is x where plus let's do something like that, x squared plus y cubed plus 1 so here I have two independent variables that I combine and out comes another variable another example would be we just square everything and add it up so we can obviously use various powers and various things we can put it together okay so this guy is a function I put in four numbers out comes one number here I put in two numbers out comes one number this case you can draw this case we have to think a little more about so let's mostly focus on a graph mostly we'll focus on this case a lot because we can draw it but we can also think about other cases so one tool which is really useful in calculus of one variable the graph is really having the graph is a handy thing now in this context how many of you don't know what the graph is okay at least there's one, two, three semi-honest people or a few more honest people it's good some of you do I know so let's think about what the graph would be in the case where I only have two variables well let's think about the graph in one variable we draw a line this is my input and we draw another line which we call x so this is y equals f of x and then we just plot all of the points all of the pairs x, y so that they're really x f of x and that gives us the graph so the height here f of x and that gives us the graph now if we want to generalize these two more input more input variables but still one output variable could do a similar thing only now we should think we have lots of inputs I have a whole in this case of playing but you could think of your plane as being a six-dimensional hyperplane if you want and then I have one output which gives me a height so I'm thinking that you know going in let me draw it red I have some inputs and I'm going to turn well let's just be conventional so that's my x's and that's my y's so you turn that's my x's and my y's going in and then for each guy who goes in so here's some typical point x naught y naught I go up to the height of f of x naught and I do that everywhere so my inputs are like a sheet of paper and I raise the sheet of paper and bend it around by the amount given by the function so here I should have something sheet of papery like as my outputs made a mess where the height of each thing is given by my function so this is the graph of the function in two variables it's a little harder to think about what the graph of the function in three variables would be I would take a space and I would move every point in that space up a certain amount but up has to be in the fourth dimension it's a little hard to draw that right so in the fourth dimension I'll take a cube going in and in the fourth dimension it will become something else going out that's a terrible picture wow thank you so I could draw some let me see if I can get this computer to draw some because that sort of better provided that I can figure out I don't want the audio okay come on okay it's alright so I've got nothing here so where's the so let me let's think about the easy one I'm waiting for the computer to do whatever it is it's supposed to do so let's think about z equals I don't know x squared plus y squared plus 1 so what would that look like so think about what that would look like so how can you figure out what that would look like well one way to figure it out think about what slices look like so this is f of x, y let's think about what f of x at some constant like 0 would be this would be a function x squared plus 1 okay so on here there's a 3D graphing applet and I'm just going to pop out this window don't tell where's run thank you should I turn the lights down does it matter I accept yes of course I accept you so let's well there there we go so this one here let me make it 2 instead of plus 1 so should I turn the lights is it good you don't care either way so this looks like a paraboloid fine so but let's think about it if you notice each of these slices that we're seeing here of course I can't ever seem to get them spraying each of these slices if we look can first spin these things straight you see a bunch of parabolas bunch of parabolas going this way and a bunch of parabolas well if I spin it going the other way going the other way there they are of course it fell over but that's okay and that's exactly what's happening if we take a grid in the xy plane we don't pay attention to whose x and whose y these lines this way well their graphs if I think of putting a standard xz this way the graph of this line is going to be a parabola whose vertex is at height negative 2 and y is 0 as y increases the vertex goes up so I get a bunch of things and as y increases in the negative direction the vertex goes up so I get a bunch of things like that if I make the slices in the other direction I get the same sort of thing here when x is 0 that's supposed to be a parabola when x is 0 I get the parabolas that look like that as I increase x from 0 they move up and as I decrease x they move up that's exactly what you're seeing in this picture well I turned off the wrong lines that's exactly what we're seeing in this picture here parabolas which move as I move x and y this the best way to get used to these things is just to play around with them one thing that this will do click a point and drag it around this is my input here is my input which gets wrapped up to the surface and you can't really see it because of the color let me take just another function this function instead which is some complicated thing it has some bumps and stuff which is what is that 7xy times e to the x squared plus y squared whatever it's a function got a couple of bumps on it as I move this point around you can see the point well I can see the point I don't know if you can moving around on the surface you should imagine that this base point here is sitting under here and it gets lifted up to the surface by that okay what I'm going to do since I want to come back to this I'm going to turn on this one one of them should have gone off no neither one of them went off I want to make it go away I turned that one off I meant to turn the other one off but okay I'm going to leave this up because I'm going to have blue it's supposed to go off alright so talk a little more you don't need a computer to figure these things out but it does make it easier there are various things we can do which I sort of mentioned suppose that I have a function let's just use that one let's use the one I had before x squared plus y cubed plus one suppose I have a function and I don't have my computer I mean I do have my computer there but let's assume I don't and let's figure out what this is going to look like does anyone have a feeling for what this would look like that's part of it yeah I'm sort of happy to do this thought process but if you do what you did before and set y equals zero and x equals zero along the y axis it would sort of shape to a parabola and along the x axis or maybe I'm reversing it but it would shape into sort of like an x cubed looking function and I just I can't really visualize that am I right or am I not so let's work on so this is true f of x and a constant is a parabola oh yeah yeah f of constant y is a cubic let's think about what these parabolas would look like this parabola is x squared plus something that depends on a constant which changes in y so here looks like x squared plus c where c is y cubed plus y so that means that y increases c increases and as y decreases in other words y y negative c decreases so the slices fixing x and allowing y to vary are going to be curves so this is x z and y constant are going to be curves that look like this and they move up and down they move up as y goes up and they move down as y goes down if we look at this part oops did I just do the wrong thing yeah but I'm sorry these are parabolas sorry I just read the wrong line these are parabolas and they move up and down as y moves up and down so let's not think about how they move up and down so if I look in the x y plane and I cut this thing every cross section is going to give me some kind of a parabola like this and these cross sections are going to go down actually they go through one zero the origin is at one but I get something that looks sort of like this but it's not as flat as I drew it it looks like a piece of pipe cut in half but it moves around now let's look at the other cross section the other cross section so if we did an easier one where this is like one with this just x squared plus y we would just get a nice half pipe right if I did g of x y is x squared plus y I would get a nice half pipe kind of shape but that's not what I have okay let's look at the other slice if I do an x equals constant then here sorry I'm drawing the nature and x is constant so in yz land I get a cubic that looks something like this and it moves up or down but here this cubic I have f of x some constant k is sorry wrong one x of some constant y is y cubed plus 1 plus k squared now the thing to notice here is this is always bigger than 1 and as k is positive or negative it does the same thing so it will come down and go back up well that corresponds to these things right so here k equals 0 means that this is at 1 and then k bigger than 0 will be up higher and also k less than 0 so that means that this pipe shape will actually have the vertex along this pipe will move down flat now yeah yeah yeah I could yeah just say let's finish this one so x squared plus 1 plus y cubed okay and I don't want that oh you should go away so see it's sort of like this parabola thing in this direction it looks like parabolas but in this direction it looks like cubics so one slice one way I never managed to keep these things straight graph no I lost you graph so I get cubics in one direction and in the other direction I get parabolas so we should turn that so you can sort of sit in yeah so that graph should that graph should well I didn't tell you where the axes were you didn't really draw the axes I have it backwards you're right I'm looking from the wrong side okay so I mean you know we can do this for hours on end if you want I think what's important is that you get a feel for how to combine these things and what these graphs are telling you but this is not you know so you can just think about this but the important thing to realize is we get these things we understand already from calculus one in certain directions now another useful piece of information to figure out what these graphs might look like is yet another way of drawing these things is this notion of level curves z equals f of x, y and one way that I can sort of slice it up one thing I can do is look at the constant look at y constant that's what I did here but there's no reason that would also equals constant so that gives me something called a level curve or a contour plot and this is actually what you see if you like look at a weather map or many other kinds of maps so here what I want to do is plot so here I want to plot should just start predicting the bad ones I want to plot f of x, y equals constant two various values that's all right so that is in this case it's a little harder let's do the x squared plus y squared one again well so I want to look at so I have that and this gives me implicit curves this doesn't give me graphs so I get x squared plus y squared let's start with zero equal to zero so here I'm going to just look at the x y-land what x is in y satisfy x squared plus y squared equals zero there's exactly one that one now I look at x squared plus y squared equals I don't know a half maybe a quarter what's that this is a circle of radius a half that's this one now I look at x squared plus y squared how about I just do it in general equals r squared gives me a circle of radius r so I see that all of my sizes in the horizontal direction are going to be circles and I get nothing I get nothing for less than zero telling me that in terms of my graph there's nothing below the x y plane it's all up here and the cross sections that I get are a bunch of circles they grow in height and the only question is how do they grow do they grow like a cone or do they grow like a parabola or do they grow in some other way we already know that they grow like a parabola circles are not parallel they're probably not possible but those are not the things that I was solving remember what I'm drawing here is height equals x squared plus y squared so I have this surface here and I'm just taking out my chainsaw and I'm cutting it off at a certain height or maybe my lightsaber I cut it off at a certain height and what I'm left with is a circle and if I cut it off at some other height I get another circle so when we're holding z that's like a topographical map it's exactly like a topographical map or a weather map I mean here let's see I have it or I could have it here I think it's already in here there it is that one is two the other one is two so there's the function plot it there it is and I can say show me the contours oops except I drew the wrong contours crap that's wrong I want these contours sorry try that again draw the contours and I have to make sure I select the right function so there's the contours which is this collection of circles at constant height and this thing if I click on it it shows me it shows me the surface there they are spinning around falling over these contours labeled it's exactly like a topographical map if you look from the top you see the topographical map if you look from the side you see the thing ruled in that way if I take a more complicated one like let's just use this first one again so this bumpy guy and if I draw the contours for the first guy there they are and if I look at them on the surface stop spinning around you see them there it is from the top but there it is from the sides it's exactly like a topographical map so these are if you want to walk on this surface without going up or down these are the paths you will follow excuse me this athlete is that one of the links on your page it's from the page it's this graphing absolute link but you can do this with a lot of software you can do this in maple you can do this in mathematics basically anything that needs to be three-dimensional plotting on the limits you just have to say it's okay to run job okay so we have these things here any questions it takes a little while to get used to visualizing these things but these kinds of maps you see all the time you see topographical maps you see weather maps which show you bands of equal pressure or temperature all of those kinds of things are contour maps they are describing we're looking at a map of Long Island and we want to see how the temperature well it's not going to vary enough on Long Island we're looking at a map of the United States and we see that it's really hot in Texas and it's cold in Minnesota and you can see these lines of constant temperature okay so we have a little calculus here this is just why do we need to do this? well because you're only used to graphing in one variable being able to graph in more than one variable is another way to display more information in fact if I wanted to graph with two input variables and two output variables I could choose one to be for example the height and the other one to be the color and that would be a way of displaying a function from R2 to R2 it's a little hard for you to think of it that way but there are ways to display this stuff so why do we need to do this? why do you draw graphs in the first place? you tell yourself about functions but everything you do with the graphs you can also do just by calculating but having the graph helps you have intuition of how this calculation goes I suppose I should say something about standard like the quadric surfaces let me push that up I'll come back should I do that? let's do some calculus calculus but I'll say about quadric surfaces I can go off too that one doesn't know how to go off I love it let's play right shoe leg instructor it doesn't matter that one just won't go off nor will it come on I love it I don't need it anyway I don't need it anyway we'll just have areas of light and dark so what can we do with this? well if you think about if you think about having some surface I'll draw a contour of it there's some surface doesn't stick that's we want to use what you learned in one variable calculus but we want to adapt it here now actually I don't want that I have already there's my surface in goes a grid like that out comes a function like that so this is the graph of z equals f of x y and it is actually quite easy we know that if I fix a specific x value let's fix a y value I'll fix something here then I can get z equals let's call it f1 f1 of x I know how to do calculus on this I know how to take its derivative I understand this this is going to be some graph looking like that this is x this is c I know how to take derivatives I know how to do all that stuff already so if I now consider what happens as I move along this line with a fixed y value I can understand how the function for the fixed y value will change so this function f1 I can understand this by taking a derivative and I'll get f prime of x which tells me the slope as I move along the y equals constant that's fine and I can do that for any y so really maybe instead of y that should be not f1 but fy except that's the bad notation because the other one is fx so what you do you call it right? without introducing this intermediate function just write it again I have c equals f of xy and I want to do this thing where I hold y constant and I take the derivative so I want to look at the limit as h goes to 0 f of x plus hy minus f of xy h that's what this function f1 of x is as I move the y around this has a name this is called partial derivatives of f with respect to so this is the slope along y equals constant line in x and y that tells me the slope if I just move along parallel to the x axis so this is called partial of f with respect to x I can do the same thing in the other direction too to get the partial of f with respect to y this is a Russian it's a round it's a really deep so I have these two partial derivatives these are the slopes in two orthogonal directions do you have a question? I see a hand so all we do here is just like x would be just like f of x is derivative of x how about y? so shall I give you an example? yes so let's do f of xy is I don't know x square time xy there's some complicated function so if I compute the partial of f with respect to x I'm thinking this is a constant so it's derivative I have to use the product rule and the chain rule it's going to be 2x sine xy plus y is a constant here so x square times the cosine of xy but since y is a constant I get a contribution from it okay is that what you were asking? okay the partial with respect to y x is a constant so this will just be x square sine xy sorry cosine xy times the derivative of xy where x is a constant give me another x so this is x cubed cosine I'm using this as a differentiation if we want to get everything in the same time I don't know what everything in the same time if you want to instead of doing partial derivatives of what is x and then y if you want to do that at the same time yeah yeah I don't know what that means I mean I do kind of know what that means but please do not explain where you have d, x, d, y and those derivative parameters but if you think about it so what does this represent? this is saying I mean yes but not really this is saying that how much does the function change if I increase x a little bit how much does the function change if I increase y a little bit now if we want to combine them you don't just add them together we have a plane and we have to think about how we move in that plane so I can put them together but this is a function from r2 to r2 right? it's a plane if that's what you're saying then yes but I wasn't quite to the tangent plane here yeah because I keep accidentally writing a comma when I don't want to all of those commas are multiplication signs they just sort of slip and one of the dots yeah there's no commas anywhere so I mean computing partial derivatives is really easy you just have to remember everything except this variable is held fixed it's a constant it's easy but you need to keep in mind what it needs another notation for this sometimes written as f subscript x now that is a comma of x and y so another notation is the variable that you take the derivative with goes as a subscript so this would be f sub y do you have a preferred notation? no I mean this one, I mean just like f prime and df dx which one well depends it's more useful when you're doing complicated formula and this is sometimes more useful when you want to think of it as an operator that is a function yeah it's okay okay so let's do another example give me some function involving x's and y's don't make it too nasty what? go ahead okay so that's g of xy so now I want to compute g sub x which is partial with respect to x so that means x varies y in the constant so this is just a number so the derivative of a number times 2x is 2 times that number now we do the other way around g sub y this is just a number so the derivative of 2 number sine y so it's just like the thing that you know in calculus one some students really mad when you say d dx of a cosine 3x and they get mad what's that a? I don't know what to do with the a or you know the nasty trick that you do on calculus one exams this where people say oh yeah that's 3x where? where? well this is not a vector yet making a vector this is a number this is a slope it's easy to say well let's make it make it a vector well okay this is just a number this is a number which measures a slope in a given direction if you take those two slopes then you take the cross product that gives you that well you can't take the cross product you can't scale it so yes you can look at these things and you can get a plane and yeah but we're not there yet so let's just think of these as numbers but that's some stuff going on there so so I mean this is kind of like the thing you do when implicit differentiation went on white it's kind of like some other stuff but not quite now if we think back to our surface so now I want to I want to I want to think about so I have my surface here and I have some point on the surface and what we have just calculated is the slope in the x direction of course I just drew it in the y direction but the slope in the y direction and the slope in the x direction x now I have these two slopes in orthogonal direction so this does actually tell me something about the tangent plane if I cut this plane in one way if I cut this plane if I cut fixing y I get the line of course I have to switch text that's supposed to be this black line and if I cut fixing x then I get something in another direction so I can put these together and figure out the tangent plane as a combination of these two slopes and I know the slope this way and I know the slope this way so I know the plane between which is maybe what you were asking all those time disorder so in general what do you think? look so what would that mean? that would mean that I'm almost out of time that would mean that my tangent plane z equals f of x, y and let's say at a, b certainly this is my point and then my slope well I say that if I move in away from a a certain amount in the x direction I get the x derivative and if I move away from a in a certain in the y direction I get the y derivative that's it so I start at the point and if I keep y constant I keep y, b, this is 0 and the amount that the x increases is this derivative and if I move in the other direction keeping x constant the y increases is this amount this actually gives me the tangent plane so the normal vector here is fx, fy negative 1 but sure you can read off the normal vector that way because really I should think of this yeah so maybe maybe I should stop there it's a reasonable place to stop I'm meant to talk about second partials and other things like that but let's do that next time