Multivariable Calculus Functions of Several Variable

Hello and welcome back to educator.com and welcome back to multivariable calculus.0000

We have been discussing mapping from R to RN, curves in n-space.0004

Today we are going to define the length of a curve in n-space, and we are going to start talking about functions of several variables, which is a reverse.0009

It is a mapping from RN to R, so let us just jump right on in and start with a definition.0020

We will let c(t) be a curve in RN, in n-space.0027

The length of the curve, t = a and t = b, so let us say from t goes from 1 to 5, or something like that, is the integral from a to b of the velocity of that curve.0046

You remember the velocity is just the norm of the derivative, of that curve.0076

It is going to be the integral from a to b of... I will write c'(t)... that is the norm symbol, dt. That is it.0083

If I have a particular curve in space... you know if t goes from 0 to whatever number... if I want to know the length of a particular segment of that curve, I take the derivative vector and I just take the norm of it.0096

Which is the derivative dotted with itself, the square root, then I put it under the integral sign and I just solve whatever integral, if it is solvable.0111

That is it. Let us go ahead and write it in an extended form so that you see it a little bit more clearly.0120

In extended form, we can write... so c(t) is going to be some function x(t) and some function y(t), right?0126

Two functions, it is a vector function, it is an example of a mapping from R to R2, so there are 2 functions.0151

Now if I take c'(t), that is going to equal x' and y'.0160

I am going to go ahead and leave off the t's just to save some notation.0165

So, the norm is going to equal x'² + y'² all under the radical sign.0170

The length is going to equal the integral from a to b of the radical of x'² + y'², and that is going to be dt.0190

That is it. This is... I will go ahead and put it in blue... this is the length of a segment of a curve in n-space, defined parametrically.0207

This is it, nice and straight forward. Let us go ahead and do some examples.0220

Let us go ahead and go back to black here. Umm... that is fine, let us go ahead and start down here. I imagine we are going to get some of the stray lines but hopefully not.0223

Example 1. We will let x(t) = (cos(t),sin(t)), where t > or = 0, and < or = to 2.0231

We want to find the length of this curve from 0 to 2.0250

Well, let us go ahead and find what x' is, and then we will find the norm.0255

x'(t) = the derivative of cos is -sin, the derivative of sin is cos(t).0260

Now let us go ahead and find the norm, so the norm is x'(t) = this vector dotted by itself, the square root, so we end up with a -sin × -sin, is sin²(t) + this is cos²(t), all under the radical.0271

Of course sin² + cos², that is a fundamental identity, it is called the Pythagorean identity.0294

It is equal to 1, so the sqrt(1) = 1, therefore our length = the integral from 0 to 2 of dt.0299

Just 1 dt, because that is the definition. You find the norm, and that is 1 dt.0313

You all know that that is equal to 2-0, so the length is 2. That is it. That is the length of the curve.0318

Notice, I can interpret this curve. This curve cos(t),sin(t), this is the circle in the plane. The unit circle in the plane.0328

So, it... I mean we can certainly think about it that way and that is nice, we want to use our geometric intuition... but again, geometry is not mathematics.0338

Geometry helps us to see and make sense of what is going on, but once we have a good solid algebraic definition, it is the algebra that we want to work with, so this is the algebra right here.0347

Okay, let us do another example. In this particular case, we are actually able to solve the integral explicitly, simply because we had the integral of dt.0358

That is not always going to be the case. In fact in real life most of the time you are not going to be able to solve the integral explicitly.0367

So, you are probably going to have to use either a symbolic algebra program, or you are going to have to use a numerical technique to actually evaluate the integral, approximate it, but evaluate it numerically instead of symbolically.0373

Let us see what we get. So example number 2.0386

Now, this time we will let c(t) be equal to cos(t),e(2t),t.0398

Now we have a curve in 3 space, and we want to find the length of the curve as t goes from 1 to 3pi/2, some value.0407

So c'(t), c'(t) is equal to, the derivative of cos is -sin(t), and the derivative of e(2t) is 2 × e(2t), right? and the derivative of t is 1.0420

That is the derivative of that curve, that is the tangent vector.0438

Remember the derivative, that is what it was, we thought about it as a tangent vector at that particular point along the curve.0443

That is what we are integrating, is we are just integrating all of these tangent vectors along the curve and adding them up.0450

So, the norm, so c'(t), the norm of that is equal to sin²(t) + 4 × e(4t) + 1.0458

Now, our length is equal to the integral from 1 to 3pi/2 of... oh sorry, this is under the square root sign.0477

sin²(t) + 4e(4t) + 1, all under the radical sign, dt.0490

Clearly something like this is not going ot be solved in any easy fashion, I mean certainly you could probably try to look up a table and maybe work it out.0502

Your best bet, we have plenty of wonderful, powerful, symbolic algebra programs. Things like Maple, Mathematica, MathCad, all kinds of things.0510

I personally use Maple, it is the oldest, anything will do. It will do it for you symbolically, it will solve it for you numerically, whatever it is that you want.0520

But that is it, this is a perfectly valid number. It gives us the length of the curve, and that is the length of the curve.0528

Okay, so now we want to start our discussion of functions of several variables, since this is a multi variable calculus course this is sort of where it really begins.0536

So let us go ahead and begin to talk about what functions of several variables really are, and we will take it from there.0545

We will draw a line here, and we will do functions of several variables...0552

So, it is exactly what you think it is.0564

When you were doing all of the math and the functions that you have been dealing with for all of these years from middle school, high school, and in calculus, were functions of one variable, x.0566

x is the independent variable, y is the dependent variable, so that is single variable calculus.0575

Multi-variable calculus, now you have more than one independent variable. maybe 1, maybe 2... not 1... you have 2, 3, 4, however many, and then you do something to those variable and you end up spitting out a number.0579

What we have done so far, in the last few lessons, is we have gone... let me write this out... we just did functions that take a number and map them to a vector in n-space.0594

So, a mapping from R to RN is a curve in n-space.0612

Now, what we are going to do is just the opposite.0616

Now we do functions that take a vector, a point in n-space. 3-space, 4-space, 2 numbers, 3 numbers, 4 numbers.0623

We are going to map them to a number.0634

We are going to take a vector and do something to that vector, the individual components of that vector, and then we are going to spit out a number.0638

This is why we call them functions. The word function is used specifically when the arrival space, the number that you spit out, is actually a single number.0644

R, that is it. 1-space. We sort of reserve that for functions.0655

So, again we are taking a point in n-space, a point in 3-space, and we do something to that point, the components, and we are going to spit out a number.0662

Let us just do some quick examples of that. You have seen them before, it is not like you have not dealt with them. You just have not dealt with them systematically like we do in this class.0672

Let us say f is a mapping from R2 to R, defined by f(x,y) = x²+y².0680

Now instead of f(x), we have f(x,y). We are taking a vector, a 2 vector, a point in 2-space.0695

We are doing something to it, the components, and that is it. For example f(1,3) is going to be 1² + 3².0703

It is going to give me a value of 10. It is going to spit out 10, a specific number.0710

Let us say f is a function defined from R3 to R, and defined by the following.0715

f(x,y,z), generally for 2-space we use x,y, for 3-space we use x,y,z.0726

When we talk about 4 space and 5 space, we generally do not use x,y,z, we usually say x1,x2,x3, but for now we are not going to have a bunch of other variables.0731

What we are going to be doing is we are going to be in 2 and 3-space, we are fine with x,y,z,.0741

So equals say 1/x + sin(x,z)... excuse me y,z... that is it.0746

I take some vector x,y,z, 5,7,9, whatever, and I put them into this thing, in other words, I operate, you know, on that vector, I do something to it, and I spit out the number.0756

That is all the function of several variables is, more than 1 independent variable, and you have one dependent variable.0771

Take a look at what we have done.0780

When you have a function of one independent variable, for example, let us take f(x) = x².0785

We can form the graph, right? We can form the graph by plotting x and f(x).0800

What we can do is we can take the x, we can do something to it, spit out a number, and the number that we spit out we make it the y coordinate.0812

Now, we are taking this and we are graphing it and what we end up getting is a graph in 2-space.0820

Now we can do the same thing with a function of 2 variables.0826

Let us let f be a mapping from R2 to R, so now we are dealing with 2 variables and now a point in R2 is the domain.0830

So, a point in R2 is just a vector.0851

When we operate on this, when we operate... and you will use that term used -- I used it a little loosely because there is something called an operator which we will discuss later if we actually talk, well, we will talk about functions from vectors to vectors, from like R2 to R2, or R3 to R3.0858

Those are technically called operators, but we use the word operator just to mean that you are doing something on some object here, some vector.0878

When we operate on this point, we get a number right? We get a number.0884

Now, what I can do is I can take the point, (x,y), and f(x,y), the number that I actually end up getting, and now I end up getting a point in 3 space.0897

When I graph these points, the x, that f(x,y), what I end up with is now I get a surface in 3-space.0907

With a function of one variable I get a graph in 2 space, with a function of 2-variables I get a surface in 3-space.0917

This is about the limit, because again we can only sort of work geometrically with 3-space. That is the only thing we can draw and represent.0927

So, let us just draw this graph and see what it might look like.0934

Here is our standard 3-space, this is x, this is y... excuse me, this is z... so I have something like... so this is some surface.0937

Now (x,y) is some point. That is our independent variable, a vector in (x,y).0955

Well, when I do something to (x,y), I spit out a value. That value is the z component, so we use that as our -- the number that we spit out is our dependent variable, z -- that also, because we have 3 numbers, x,y,f(x,y,), we can actually turn that into a graph.0962

As it turns out, this actually gives you a surface in 3-space. This is very, very convenient, very beautiful.0983

Let us go ahead and just do some examples, that is sort of the best way to make sense of this. Let us do example 1 here.0993

Example 1 for function of several variables.1003

We will let f(x,y) = x² + y². The function that we had mentioned earlier.1007

This is a function of 2 variables, you are mapping R2 to R, here is what it looks like. The best way to do something like this is if you hold one of these constant and just think of it as x,y, = x², you are going to get a parabola in the x,z plane.1015

Remember f(x,y) = z, so another way of thinking about this z is equal to x² + y², right? Because our f(x,y) is going to be our z component, our third number.1032

What this looks like is... this is actually a paraboloid, in other words it is a parabola that starts at the origin and goes up along the z axis, all the way around it.1048

If this is the z-axis, the parabola goes up like that, so it is actually a surface, and it is really quite beautiful.1068

That is what you get. That is all that is happening. You are taking this function of several variables, of 2 variables here, it is algebraic, but we can because we have 3-space, we can still represent it and we have this surface.1076

We have this extra geometric intuition that we can use, that we can exploit, and that is very nice, it comes in very handy.1088

Each point on this paraboloid is exactly the point (x,y,x²+y²), that is it.1097

A random point is just that. That is all that is going on.1109

Now, let us define -- give a definition -- a level curve of f when f is a mapping from R2 to R, so this is specifically from R2 to R, is the set of points in R2 such that f(x,y) is equal to c, a constant.1114

Let us talk about what this means. Now, let us go ahead and use this example that we did.1157

We had f(x,y) = x² + y².1164

Well, a level curve of this is the set of points (x,y) such that x,y equals some constant c, 4, 5, 9, whatever.1174

What these are, these are level curves, so let us think about what this actually means.1185

This is the z-axis, and we said that the paraboloid actually goes up along the z-axis.1191

Again, all of the points are (x,y), x² + y², this surface. The points can be anywhere along that surface.1196

When (9x,y) which is x² + y² equals some constant c, what you get is this c here is that the z value is fixed.1205

In other words, z is the same for all of them. So what you get is basically something like this, you get a curve, you get that curve.1220

When I look above, down the z-axis, what I am going to get is something that looks like this.1230

Now, this is the y-axis, this is the x-axis, and the z-axis is coming straight up.1238

Well, the parabola is coming straight up at me, so that when I am looking at it from above, x² + y² equals some constant c, it is actually a curve in the x,y.1244

Or maybe this curve if the constant is smaller, or this curve if the constant is bigger.1258

Notice that if the x² + y² = c is the equation of a circle.1262

When you have a function of several variables, when you have a function of 2 variables in this case, since we are talking about 2 variables.1268

That function of two variables happens to equal a constant what you end up getting is a curve in the x,y plane.1275

In fact, a whole bunch of curves, a family of curves depending on what C is.1284

In this particular case, you get this curve, or maybe you get this curve, or this curve, or this curve.1288

Because f(x,y) is the z-component. Well, if z is fixed, that means z is fixed, x, y can be anything, what you get in this case is this curve, or this curve, or this curve, but z is fixed.1296

We call those level curves and they are very important in math and physics and in engineering. We will talk a little bit more about that in just a moment.1312

I am going to define the next level of something called a level curve called a level surface.1325

Let us see, true space, ok, yes. So let us do another example here. Example 2.1331

Now, we will let f(x,y,z), so this time it is a function of three variables, equal 2x² + y² + 3z².1344

Now, f is a function from R3 to R. Our point that we get is (x,y,z,f(x,y,z,)).1364

This is a graph in 4-space.1380

We cannot draw a graph in 4-space, but we can work with it algebraically, that is the whole idea.1383

Now, when we take the set of points (x,y,z), such that f(x,y,z) is equal to a constant, we are going to get something like this.1391

2x² + y² + 3z² = some constant C. 5, 10, 15, sqrt(6), -9, whatever it is.1400

This is an implicit relation among 3 variables. What this is is an actual surface in 3-space.1413

We had a function of 2 variables a mapping from R2 to R.1425

When we set that function equal to a constant, what we get is a curve in 2-space.1432

Here we have a function of 3 variables, a mapping from R3 to R.1437

When we set this function up equal to a constant, what we get is a surface in 3-space, a graph in 3-space.1444

That is the whole idea. So, imagine if you will, let us just sort of see if we can draw what something like this sort of looks like.1453

In this particular case, let us go ahead and take a slightly different function, make it a little more uniform.1465

x² + y² + z² = C. Now I have just changed all of the coefficients to make them equal to 1 and 1 and 1.1470

What you get... this is... you should recognize it, it is the equation of a sphere centered at the origin.1479

So what you get are these spheres, these surfaces, theses spheres, these shells of different radii.1487

Those are called level surfaces. These level surfaces are actually very important.1495

For a function of 2 variables, we call them level curves because it defines an implicit relationship between the variables x and y, when they are equal to some constant.1501

In 3-space we call them level surfaces because they define some implicit relationship between the variables x, y, and z.1511

They are places where the function is constant. That is the whole idea. Where the value you spit out stays the same. They end up being very important.1516

Now, I am going to finish off by writing one thing.1529

In physics, if a function from R3 to R gives the potential energy -- not the potential energy of -- if a particular function from R3 to R, a function of 3 variables, gives the potential energy at a given point in space, then f(x,y,z) equal to c... when you set that function equal to a constant... these things are called equipotential surfaces.1532

You also hear them called isopotential sequences... surfaces, not sequences, surface.1602

So what they are basically saying is that if I have a function which gives me the potential energy at various points in space, this is going to end up being some graph in 4-space, if I set it up being equal to a constant, it is the points in 3-space where the potential is the same.1613

So, I have this surface that tells me where the potential energy is the same. This is profoundly important.1630

The idea of a potential surface... and in a minute we will talk about it for thermodynamics... these things are profoundly important.1635

The notion of a level curve and a level surface come up all the time in science and engineering.1646

Now, if f(x,y,z) -- we are just throwing out some examples so that you know that these are not random, weird mathematical things. These are profoundly important.1652

If f(x,y,z) gives the temperature at a given point... umm, yea that is fine, the temperature at a given point in space, various points in space...1663

Then, the function equal to C, in other words what we call the level surfaces of this, are called isothermal surfaces.1687

Again, the idea is... what is important here is to have the notion of the idea of a function of several variables, I think that is reasonably clear.1705

Instead of having 1 variable, you are just having several, 2, 3, 4, independent variables, you are operating on that vector, on that point in space, and you are spitting out a number.1716

That is what it means. We use the word function every time the thing that you spit out is a number, is a point in R.1723

The idea of a level curve, and a level surface, it is where these functions of several variables take on a constant value.1731

Again, I might have some function which is some bizarre surface, but there are points on that surface.. in that space, where the function itself is a constant value.1739

It stays the same, and those curves in 2-space, or those surfaces in 3-space become profoundly, profoundly important, and they show up everywhere in physics.1753

So, we will go ahead and stop it here for this particular lesson.1764

Thank you for joining us here at educator.com, we will see you next time. Bye-bye.1768