Multivariable Calculus 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 = x², y = x³, 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 = x², but let us look at it in a slightly different way.0144

Let us look at y = x², or f(x) = x², 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 = x² 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 = x², 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 x².0388

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 = x², 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 = x² 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 t².0655

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 t².0716

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 t².0727

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 t².0787

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 = x², but now it is not written as y = x².0826

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 t².0851

This is the power of parameterization, so this is a parameterization of the function y = x², 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 = x².1173

You can do that, I mean you can certainly turn this parameterization into a y = x², 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 t².1532

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 = t².1619

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

So, = x², there you go, that is it.1630

So you eliminate the t, the parameter, and you end up recovering some function of y = x².1635

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