For more information, please see full course syllabus of Multivariable Calculus

For more information, please see full course syllabus of Multivariable Calculus

### Maps, Curves & Parameterizations

i) f(0)

- Since f is a function, to compute f(0) we let t = 0 so that we have f(0) = [1/4](0) − 4.

ii) f(0) − f(3)

- We compute f(0) and f(3) and apply their difference. Now, f(3) = [1/4](3) − 4 = [3/4] − 4 = − [13/4].

iii) [ f(0) ]

^{2}

- Note that [ f(0) ]
^{2}= f(0)f(0).

^{2}= ( − 4)( − 4) = 16.

^{2}, g(t) = (t − 1,t + 1) compute the following:

i) g(0)

- Since g is a function, to compute g(0) we let t = 0 so that we have g(0) = (0 − 1,0 + 1).

^{2}, g(t) = (t − 1,t + 1) compute the following:

ii) g(0) − g(1)

- We compute g(0) and g(1) and apply their difference. Now, g(1) = (1 − 1,1 + 1) = (0,2).

^{2}, g(t) = (t − 1,t + 1) compute the following:

iii) g(0) ×g(0)

^{3}, h(t) = (t,[t/2],t

^{2}) compute the following:

i) h(0)

^{2}) = (0,0,0).

^{3}, h(t) = (t,[t/2],t

^{2}) compute the following:

ii) h(0) − h(1)

- We compute h(0) and h(1) and apply their difference. Now, h(1) = ( 1,[1/2],1
^{2}) = ( 1,[1/2],1 ).

^{3}, h(t) = (t,[t/2],t

^{2}) compute the following:

iii) 4h(1)

^{2}graph f(t) = (0,t − 1)

- Plotting a few values for t will give us an idea of the two dimensional graph.
- t = 0 yields f(0) = (0,0 − 1) = (0, − 1)
- t = 1 yields f(1) = (0,1 − 1) = (0,0)
- t = − 1 yields f( − 1) = (0, − 1 − 1) = (0, − 2)
- t = 2 yields f(2) = (0,2 − 1) = (0,1)
- t = − 2 yields f( − 2) = (0, − 2 − 1) = (0, − 3)

^{2}graph g(t) = (t,t

^{3})

- Plotting a few values for t will give us an idea of the two dimensional graph.
- t = 0 yields g(0) = (0,0
^{3}) = (0,0) - t = 1 yields g(1) = (1,1
^{3}) = (1,1) - t = − 1 yields g( − 1) = ( − 1,( − 1)
^{3}) = ( − 1, − 1) - t = 2 yields g(2) = (2,2
^{3}) = (2,8) - t = − 2 yields g( − 2) = ( − 2,( − 2)
^{3}) = ( − 2, − 8)

^{3}:

^{2}graph h(t) = (cos(t),sin(t))

- Plotting a few values for t will give us an idea of the two dimensional graph.
- t = 0 yields h(0) = (cos0,sin0) = (1,0)
- t = p yields h(p) = (cosp,sinp) = ( − 1,0)
- t = [p/2] yields h( [p/2] ) = (cos[p/2],sin[p/2]) = (0,1)
- t = [3p/2] yields h( [3p/2] ) = (cos[3p/2],sin[3p/2]) = (0, − 1)

^{2}graph f(t) = (2,1)

^{3}graph f(t) = (t,t + 1,t)

- Plotting a few values for t will give us an idea of the three dimensional graph.
- t = 0 yields f(0) = (0,0 + 1,0) = (0,1,0)
- t = 1 yields f(1) = (1,1 + 1,1) = (1,2,1)
- t = − 1 yields f( − 1) = ( − 1, − 1 + 1, − 1) = ( − 1,0, − 1)
- t = 2 yields f(2) = (2,2 + 1,2) = (2,3,2)
- t = − 2 yields f( − 2) = ( − 2, − 2 + 1, − 2) = ( − 2, − 1, − 2)

^{2}, f(t) = (t

^{2}+ 1,t) find a function of x and y which represents the same graph.

- Let each component of the vector (t
^{2}+ 1,t) equate the x and y coordinates, that is x = t^{2}+ 1 and y = t.

^{2}+ 1.

^{3}, f(t) = (2t,t

^{2}− 1,0) find a function of x, y and z which represents the same graph.

- Let each component of the vector (2t,t
^{2}− 1,0) equate the x, y and z coordinates, that is x = 2t, y = t^{2}− 1 and z = 0.

^{2}− 1 = [(x

^{2})/4] − 1.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Maps, Curves & Parameterizations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Maps, Curves and Parameterizations 1:10
- Recall
- Looking at y = x2 or f(x) = x2
- Departure Space & Arrival Space
- Looking at a 'Function' from ℝ to ℝ2
- Example 1
- Definition 1: Parameterized Curve
- Example 2
- Example 3

### Multivariable Calculus

### Transcription: Maps, Curves & Parameterizations

*Hello and welcome back to educator.com and welcome back to Multivariable Calculus.*0000

*We just finished our discussion of vectors and lines and planes and things like that.*0005

*Now we are going to actually start getting into the Multivariable Calculus.*0009

*Today's lesson, I am going to discuss a very important topic.*0013

*There are not going to be too many examples, but it is going to be mostly discussion.*0015

*It is going to be a global discussion about what we are going to be talking about when we talk about functions from one space to another.*0020

*So we are going to introduce this notion of a map.*0029

*Now you remember that we talked about parameterizing a line and the parameterization of a plane, now we are going to start parameterizing every single function that we deal with.*0032

*As it turns out, parameterization is a very powerful technique, much more powerful than the representations that you have been used to all of these years.*0042

*For example, if y = x ^{2}, y = x^{3}, where you have the 1 independent variable and the dependent variable.*0050

*Now, we are going to take a little bit of a twist and have a different way of looking at that, a much broader, a much more general way of looking at that.*0058

*So, with that, let us go ahead and recall a few things and we will go ahead and jump right on in.*0065

*Let us recall a few things, in terms of notation.*0072

*What we are talking about is 1 space, basically just a real number line, that is what the R stands for.*0077

*This is 1 space.*0087

*Now it is my habit to actually notate R slightly differently.*0090

*I usually... in books you are going to see the real numbers with sort of these double lines, for the real numbers, the rational numbers, there is a specific notation, a letter representing these things.*0096

*So if I do it sometimes this way, or I do it sometimes that way, they mean the same thing.*0108

*It just means the real number line.*0113

*So we will do R2, that is 2 space.*0115

*In other words, what they are saying is you have 2 number lines.*0120

*R3, that is 3 space, you need 3 numbers to represent a point in 3 space, and of course RN that is the general.*0125

*That is just n space, so that is the most general dimensional space we can talk about.*0137

*Let us start by looking at the simple function y = x ^{2}, but let us look at it in a slightly different way.*0144

*Let us look at y = x ^{2}, or f(x) = x^{2}, a slightly more general than this, but again this is how you are used to seeing it all of these years, this is how we have dealt with it.*0150

*We take a number, we do something with that number, and then we spit out a number that ends up being the f(x).*0170

*It ends up being the y value, and then we end up graphing this.*0179

*We are going to look at this in a slightly different way, I am going to draw two separate number lines.*0181

*I am going to draw one number line over here, then I am going to draw another number line over there.*0187

*Let me label this as 0, then we will do 1, then we will do 2 and we will do 3 and of course it goes on in both directions.*0196

*Over here let me go ahead and put 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.*0201

*Now, what f does, what this y = x ^{2} does, we are going to take a value and some number, a real number, and we are going to do something to it.*0215

*We are going to square it and spit out another number.*0226

*Notice, we started in the real number line, the 1 space, and we have done something to it and then the number that we get back is still another number.*0230

*We end up back in the same space, or if you want to think of this as two separate spaces, it is probably the best way to do it.*0240

*In other words, this function takes a number... for example if I put 0 in there... it takes 0, it does something to it, and it ends up at 0 on another number line.*0247

*Consider this just another space but it happens to be the same space as the number from which you chose x.*0260

*If I take 1, and I put it in there, I square it and I end up getting, spitting out, 1.*0265

*If I take 2, I end up doing something to it, I spit out 4, and 3, I spit out 9, and so on in both directions negative and positive.*0273

*What we are doing is we are associating to each number in one space, in the real number line, we are associating another number in a copy of that space.*0285

*Again, the real number line. What we say is that f maps x to f(x).*0298

*In other words, we are mapping 0 to 0, 1 to 1, 2 to 4, 3 to 9, whatever the function happens to be.*0307

*This idea of a map, 2 separate spaces, in this case they happen to be the same, but they do not have to be.*0314

*That is what we are going to generalize in a minute, you can actually take a number in 1 space, do something to it, do a bunch of things to it, and actually end up in a completely different space all together.*0321

*You could end up in R2, R3, that is what we are going to do in a minute.*0331

*But we want to be able to see these functions that we have been dealing with in a different way.*0336

*We do not want to think of them as being y = x ^{2}, we want to think of it as taking a number x, doing something to it, and getting a number in a different space.*0342

*Again, in this particular example, the two spaces, the departure space, the arrival space, happen to be the same space.*0352

*Now, there is a notation for this.*0362

*This notation is actually going to be very important, it is going to be a notation we are going to be using often.*0366

*It is going to be ubiquitous throughout mathematics.*0371

*It is a very simple notation, but it is actually quite powerful when you speak generally about functions.*0374

*We say that f is a mapping from R to R.*0382

*Defined by x going to x ^{2}.*0388

*This is the general notion, what we are telling you is that f is a function that takes a number in the real number space, does something to it, and then spits out a number that also happens to be in the real number line.*0394

*A number in other words, not a vector. We actually give the definition of that, it means take x, and square it.*0405

*This is a pretty standard notation for a mapping, that is what it is.*0411

*You want to think of a function as a mapping, that is the most important thing.*0417

*Another way of looking at this, which I think is a little bit better, a little bit more pictorial way of looking at this is let us just say we have this space, and let us say we have this space.*0422

*They can be the same, they can not be the same.*0433

*We call the numbers that you choose, in other words, the independent variables, the x's that you choose randomly the departure space.*0438

*Once you do something to that particular number, whatever it is you are doing to it, however it is you are transforming it, we call it the arrival space.*0446

*So, this is... whoops, we are not going to have these random stray lines, we definitely do not want that especially when we are dealing with pictures here... let us close this off here... you know what, let me go ahead and do this on the next page, because I want you guys to see this really clearly.*0457

*It is the best way to consider this, so let me go ahead and erase that.*0483

*Let me move on to... there we go... so let me draw a little something like this, and a little something like this.*0488

*This is the departure space, it is the number that we choose.*0498

*This is the arrival space, so let us just pick a couple of random points.*0504

*So this is x1, x2, x3, so this is x1, x2, x3, and what we do, is when we take x1, do something to it, operate on the function, or use the function to operate on this... let us say this maps to here and this maps to here... we end up in certain places in the arrival space.*0510

*That place is in the arrival space, so this is a pictorial version of what it is that we are doing when we think of a map.*0532

*That is what is important, we are taking a number, we are doing something to it and we are ending up with another number, or vector, or could be anything.*0537

*It turns out in this particular case that the y = x ^{2}, this space happened to be the real number line, and this space also happened to be the real number line.*0546

*These are the functions we have been dealing with all of these years, ever since we were kids, and up through calculus.*0556

*But now, in multivariable Calculus, we want to become a little bit more sophisticated.*0560

*We want our spaces to become more sophisticated.*0565

*Now, what Descartes did was as far as what you know and what you have been dealing with as far as graphs are concerned, for example if you saw the y = x ^{2} function, you are accustomed to seeing something like this.*0569

*What Descartes did was he took these two real number lines, this departure space and this arrival space and he set them perpendicular to each other, that is all he did.*0584

*So this is one space, and this is another space.*0594

*If you juxtapose them, because you can do that on a 2 dimensional sheet of paper, then if you map, you know... like... 1,1,2,4,3,9,4,16, the particular function, you end up getting this thing called a graph.*0599

*You get a curve in 2 space, that is what he did, that is where the notion of a graph came from.*0612

*What is really happening is you are mapping from one space to another space.*0618

*You are taking numbers from one space, doing something to them, and ending up in another space.*0624

*In this particular case, they both happen to be the same space, the real number line, R1.*0628

*Now, let us try looking at a function where now we are going to pick where our departure space is going to be the real number line.*0636

*Our arrival space is going to be 2-space, it is going to be 2-dimensional, let us see what that looks like.*0644

*Let us try looking at a function, and I will put function from R to R2, defined by the following: f(t) = t and t ^{2}.*0655

*Notice what I have done here.*0686

*This is vector notation, this is the x component, this is the y component.*0690

*If we are speaking in terms of x's and y's, so basically we have... we are taking a number, any number from the real number line and we are saying that what we are going to produce is a vector, a 2-vector, and the first component of that vector is just going to be the number t itself.*0697

*But, the second component is going to be t ^{2}.*0716

*This is a map, so I have taken a number, a real number from the number line and created a vector in R2.*0720

*Well a vector in R2 is a directed line segment, but it is also just a point, and the coordinates of that point are t and t ^{2}.*0727

*So what we are doing here is we are taking the real number line and we are taking numbers from the real number line and we are mapping them onto 2 space.*0736

*That is what we are doing.*0750

*So we might take the number 1, map it to 1 and 1, take the number 2, map it to 2 and 4, take the number 3 and map it to 3 and 9, and that is what we are going to do.*0752

*In fact let us do that for a couple of these so that we do a pictorial representation.*0765

*We are just taking numbers from the real number line, so here the departure space is the real numbers, and the arrival space is going to be R2.*0773

*We consider them 2 spaces, that is the idea.*0782

*So now, if I actually do this map, let me rewrite the map again, f(t) = t and t ^{2}.*0787

*Now let me go ahead and map a couple of these things.*0800

*Well if t =0, we will get 0,0.*0802

*If we do t=1, we will get 1 and 1.*0808

*If I map a 2, so it is going to be 2 and 4, so I will go to 2, go to 4.*0813

*If I take t=3, the point is going to be 3,9.*0821

*Notice what you get. You still end up with this parabola y = x ^{2}, but now it is not written as y = x^{2}.*0826

*It is written as a map with a given parameter, that parameter t, you take that 1 parameter t and then the x... you actually map it to 2-space, well 2-space you need 2 numbers to represent a point, a vector in that space.*0835

*The first point is t, the second point is t ^{2}.*0851

*This is the power of parameterization, so this is a parameterization of the function y = x ^{2}, and this is a much more powerful way of looking at it.*0856

*What you have done here is this -- this is a vector, so (0,0) that is that vector, (1,1) that is that vector, (2,4) that is that vector, (3,9) that is that vector.*0864

*When you connect all of the endpoints of the vectors, you end up getting your curve in 2 space.*0877

*That is what we are doing.*0885

*Let us do an example and then we will give a formal definition of what we mean by a parameterized curve in n space.*0890

*So, example 1.*0898

*Let us say that f(t) = ... oh, let me use my notation so that you get used to seeing this... so f is a map from the real number line to 2-space.*0903

*Defined by f(t) = (cos(t),sin(t)), where t... and this particular case I am going to specify the interval t, I do not have to, I can just let it go, but what the heck, let us just go ahead and specify... so t goes from 0 all the way to 2pi.*0920

*If I take t = 0, and if I do... so that becomes, so t = 0, that is the point that I am pulling from the real number line, I am going to do something to it.*0955

*I am going to do this to it, and I am going to end up at (cos(0),sin(0)), which is (1,0).*0968

*So, I end up over here. That is my vector, that is my point in 2 space.*0971

*I am going to go through some other points.*0976

*Let us just take t = pi/4, a 45 degree angle, yeah pi/4.*0979

*Again, a radian measure is a real number, so pi/4, if I do (cos(t),sin(t)), I will get (1/sqrt(2), 1/sqrt(2)).*0986

*That will end up putting me over here.*0999

*If I take t = pi/2, now I am going to get (cos(pi/2),sin(pi/2)), that will put me at (0,1), so that will put me over there.*1001

*If I keep going like this, go all the way up to 2pi, these vectors that I get, that vector, that vector, that vector, that vector, that vector, if I connect all of the dots of these vectors, I am going to get the circle on the plane.*1009

*So this thing right here, this function is the parameterization of the unit circle in R2.*1026

*That is what we are doing. We are taking the parameter and deciding... we are creating curves in a space of a given dimension.*1038

*In this case, we ended up creating a curve in 2-space.*1049

*Now let us go ahead and give a formal definition of what it is we mean by a parameterized curve.*1055

*This one I am going to go ahead and move on to the next page.*1059

*It is going to be a slightly long definition, but hopefully we have notated it properly.*1064

*So, a parameterized curve... I am going to write it a little bit better here... a parameterized curve in RN, or n space, is a map... let us call it c... from R to RN, which to each point in R associates a vector, which is also a point, in RN.*1070

*When these vectors, or points, are connected, we get a curve in RN.*1143

*So as we saw in the previous example, we got a curve in 2-space.*1167

*So we are not thinking that it is a function anymore, the way we used to as far as y = x ^{2}.*1173

*You can do that, I mean you can certainly turn this parameterization into a y = x ^{2}, but now we want to sort of break away from that.*1178

*We want something a little more free, a little more general, a little more sophisticated, and you will see just how sophisticated in a bit when we start differentiating.*1183

*We want to think of it as, now let us say I do it in 3 space.*1193

*Let us say I take some value, some t, something from the real number line -- it is always going to be a number, the map is always going to be like this, the departure space for a curve, in n space, the departure space is always going to be the real number.*1197

*It is always going to be like this, for a curve in n space the departure space is always going to be the real number line.*1212

*The arrival space can be any dimensional space, R1, R2, R3.*1218

*In R3 you might have some random curve that goes all over the place. There is a way to parameterize that.*1223

*In other words there is some map, and of course for R2 and R3 we can visualize it, but again this is an algebraic definition.*1229

*Because it is an algebraic definition, it is very powerful.*1236

*We can think of a curve in 15 dimensional space, it is perfectly valid and we can work with it mathematically.*1241

*We may not be able to see it or visualize it, but it does exist, it is real. We can work with it.*1247

*Curve in RN and, I will say one more thing about that.*1255

*The coordinates of x -- of c, I am sorry -- the coordinates of c are functions of t.*1261

*You saw that already, so c(t) = some one function from... actually let me use f instead of x... It is going to be some function of t and then another function of t, and so on, depending on how many dimensions you are dealing with.*1280

*In the previous example, we had 2 functions of t, we had cos(t) and we had sin(t).*1301

*Again, a vector is made of components, those components are actual functions of t. This is the power of parameterization.*1308

*Let us look at another example here. Let us do a 3 space example.*1318

*So, example number 2. Let us let x(t) and we are going to be several different letters, x, y, sometimes c, occasionally I am going to be writing it with capital letters where I am not going to put the vector sign on top of that.*1325

*The notation is important, however the notation I am not going to... there are going to be different types of notation, but it will be consistent within the notation itself, in other words you will see the same arrangement, but we are just going to be using different things.*1344

*So, x(t) is a map from R to R3, so this time we are doing the curve in 3 space, defined by x(t) = (cos(t),sin(t),t).*1358

*Now, we have a curve in 3 space.*1390

*Let us see if we can find out what this curve looks like.*1395

*Let me go ahead and draw that, and that, and this.*1398

*This is the general representation of a 3 dimensional space, this is the x-axis, this is the y-axis, this is the z-axis.*1403

*We already did the cos(t),sin(t), we know that that is a circle, so in the x,y plane that is the circle.*1412

*In the x,y plane there is no z component, but now we have added a z-component.*1420

*For every value of t we actually not only move in a ... we do not just move in a circle... every time we go t, we actually go up a little bit, sort of like a ramp, a parking ramp... have you been to those parking ramps?*1425

*You are still moving in a circle as far as the x,y, coordinate is concerned, but every time you move, you are also going up.*1441

*What you will end up getting, the curve you are going to get, is a spiral.*1447

*That is exactly what it is, so this curve is going to end up being a spiral.*1453

*It still moves in a circle as far as the x,y plane is concerned, if you are looking down the z-axis it still looks like a circle, but now you have this third component t, the z-axis.*1461

*So, it is actually moving up as it turns. I hope that makes sense. It is going to look like this.*1474

*This t is changing, in other words t is also the z axis, there is a third component here.*1480

*This is a curve, so this is the parameterization of the unit spiral in R3.*1485

*That is the idea. If you are given a particular curve, you can parameterize it with some map that looks like this.*1494

*Then, when you have this map you can begin to play with it algebraically.*1502

*As you will see in the next lesson we will begin to start differentiating it and... well, yeah, other things too.*1507

*So, now, let us go one more up here, I will just do one more example.*1515

*Example number 3.*1527

*Let f be a map from R to R2, in other words a curve in 2-space, defined by f(t) = t and t ^{2}.*1532

*So this is the x-coordinate, right? This is the y-coordinate, it is a curve in the 2 space, this happens to be the parameterization of the parabola in the plane.*1550

*Now, let us go ahead and recover a function, let me see here, 1, 2... you know what, no, this is... well, that is okay we have already started it so let us go ahead and finish it.*1560

*So recover a function y in terms of x.*1583

*Now I have given you this parameterization, and we want to be able to recover some function the way we are used to seeing it, y as a function of x.*1596

*Dependent is something function of independent.*1605

*We are given the parameter t, the t is what is in common, so essentially all you have to do is eliminate the t from them and set y equal to some function of x.*1608

*So, x = t, and y = t ^{2}.*1619

*Well if x = t, let us go ahead and just put this x wherever we see a t.*1624

*So, = x ^{2}, there you go, that is it.*1630

*So you eliminate the t, the parameter, and you end up recovering some function of y = x ^{2}.*1635

*It becomes a lot more difficult... in fact I am not even sure, well... in 3 space and 4 space, but this is the power of parameterization.*1641

*We have been dealing with simple functions all these years simply because we are accustomed to dealing with 2 space and 3 space.*1652

*Our natural intuitive surroundings, you know, we have a sense of paper, we have a sense of flat paper.*1656

*We have a sense of 3-dimensional space, however things can obviously become a lot more complicated even when you are dealing in 3 space.*1663

*4-space, 5-space, 6-space, it becomes virtually impossible to deal with.*1670

*This is why we do parameterization, it is a much more sophisticated way of dealing with functions.*1674

*It also happens to work for lower dimensional cases, in other words R2.*1680

*Here is how you want to start thinking about functions.*1687

*Now, let us go ahead and recap here.*1693

*What we have done, as far as all of these years, what you have been working with are maps, or general functions, y = some function of x.*1698

*Those are maps from R to R, in other words, you take some value x, you do something to it, and you spit out another number. Numbers to numbers.*1707

*Today we have introduced the notion of curves, which are maps from the real number line, you take a number, and you create a vector to RN.*1717

*Well, we are also going to be going the other direction.*1728

*We are going to begin talking about functions in which you will take a vector, and we will do something to that vector and what we will spit out is a number.*1731

*To top it all off, we are eventually the most general is RN to RP, we are going to take a vector in n space, we are going to do something to it, and we are going to end up spitting out a vector in p space.*1740

*We are going to work from one certain dimensional space to another completely dimensional space.*1758

*This is the power of multivariable calculus, that you can actually do this, that you can actually do the calculus you know.*1764

*Everything that you learned in single variable calculus still applies and yet now it is going to become a little bit more sophisticated, more powerful.*1770

*It is really quite extraordinary.*1779

*So, next lesson we are going to start with the actual calculus in n space.*1781

*Until then, thank you for joining us at educator.com and take care, bye-bye.*1785

1 answer

Last reply by: Professor Hovasapian

Wed Oct 30, 2013 2:43 AM

Post by Christian Fischer on October 26, 2013

Amazing lecture! Thank you!!

3 answers

Last reply by: Professor Hovasapian

Sat Dec 29, 2012 5:06 PM

Post by Riley Argue on December 2, 2012

Great Lecture.