We have been discussing mapping from R to RN, curves in n-space. In this lesson, 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 - it is a mapping from RN to R. When you were doing all of the math and the functions that you have been dealing with for all of these years, were functions of one variable, x. That was a single variable calculus. In multivariable calculus, you have more than one independent variable, and then you do something to those variables and you end up spitting out a number.
We can plug in vales for x and y to investigate how the graph looks.
Note that if x = − y or y = − x our z - coordinate is 0, for example 0 = 2 + ( − 2) or 0 = ( − 3) + 3. Our z - coordinate also increases (or decreases) rapidly if the x and y - coordinates have equal sign.
Our graph is therefore a plane slanting upwards:
Graph f(x,y) = [x/y].
We can plug in vales for x and y to investigate how the graph looks.
Note that no values exist when y = 0, since it is undefined on our function. Also whenever x = y our function takes the value 1, this is offset whenever y > x or y < x.
Our graph is therefore a curved surface increasing whenever x and y have the same sign, decreasing otherwise:
Graph the level curve of f(x,y) = xy for k = − 1.
We let k = f(x,y) and graph the function in the xy - plane.
So − 1 = xy is our level curve, we can solve for y and graph y = − [1/x].
Graph the level surface of f(x,y,z) = x2y − z for k = 0.
We let k = f(x,y,z) and graph the function in the xyz - space.
So 0 = x2y − z is our level surface, we can solve for z and graph z = x2y.
Sketch the level curves of f(x,y) = 10 − 2x + 5y for k = − 2, − 1,0,1,2.
We equate f(x,y) to each k and graph the functions in the xy - lane in order to sketch the level curves.
For k = − 2 we obtain − 2 = 10 − 2x + 5y or − 12 = − 2x + 5y.
*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.
Functions of Several Variable
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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
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
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
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 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