WEBVTT mathematics/multivariable-calculus/hovasapian
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Hello and welcome back to educator.com and welcome back to multivariable calculus.
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We have been discussing mapping from R to RN, curves in n-space.
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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.
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It is a mapping from RN to R, so let us just jump right on in and start with a definition.
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We will let c(t) be a curve in RN, in n-space.
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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.
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You remember the velocity is just the norm of the derivative, of that curve.
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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.
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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.
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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.
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That is it. Let us go ahead and write it in an extended form so that you see it a little bit more clearly.
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In extended form, we can write... so c(t) is going to be some function x(t) and some function y(t), right?
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Two functions, it is a vector function, it is an example of a mapping from R to R2, so there are 2 functions.
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Now if I take c'(t), that is going to equal x' and y'.
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I am going to go ahead and leave off the t's just to save some notation.
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So, the norm is going to equal x'² + y'² all under the radical sign.
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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.
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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.
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This is it, nice and straight forward. Let us go ahead and do some examples.
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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.
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Example 1. We will let x(t) = (cos(t),sin(t)), where t > or = 0, and < or = to 2.
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We want to find the length of this curve from 0 to 2.
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Well, let us go ahead and find what x' is, and then we will find the norm.
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x'(t) = the derivative of cos is -sin, the derivative of sin is cos(t).
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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.
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Of course sin² + cos², that is a fundamental identity, it is called the Pythagorean identity.
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It is equal to 1, so the sqrt(1) = 1, therefore our length = the integral from 0 to 2 of dt.
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Just 1 dt, because that is the definition. You find the norm, and that is 1 dt.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Let us see what we get. So example number 2.
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Now, this time we will let c(t) be equal to cos(t),e(2t),t.
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Now we have a curve in 3 space, and we want to find the length of the curve as t goes from 1 to 3π/2, some value.
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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.
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That is the derivative of that curve, that is the tangent vector.
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Remember the derivative, that is what it was, we thought about it as a tangent vector at that particular point along the curve.
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That is what we are integrating, is we are just integrating all of these tangent vectors along the curve and adding them up.
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So, the norm, so c'(t), the norm of that is equal to sin²(t) + 4 × e(4t) + 1.
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Now, our length is equal to the integral from 1 to 3π/2 of... oh sorry, this is under the square root sign.
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sin²(t) + 4e(4t) + 1, all under the radical sign, dt.
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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.
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Your best bet, we have plenty of wonderful, powerful, symbolic algebra programs. Things like Maple, Mathematica, MathCad, all kinds of things.
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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.
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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.
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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.
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So let us go ahead and begin to talk about what functions of several variables really are, and we will take it from there.
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We will draw a line here, and we will do functions of several variables...
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So, it is exactly what you think it is.
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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.
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x is the independent variable, y is the dependent variable, so that is single variable calculus.
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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.
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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.
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So, a mapping from R to RN is a curve in n-space.
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Now, what we are going to do is just the opposite.
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Now we do functions that take a vector, a point in n-space. 3-space, 4-space, 2 numbers, 3 numbers, 4 numbers.
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We are going to map them to a number.
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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.
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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.
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R, that is it. 1-space. We sort of reserve that for functions.
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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.
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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.
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Let us say f is a mapping from R2 to R, defined by f(x,y) = x²+y².
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Now instead of f(x), we have f(x,y). We are taking a vector, a 2 vector, a point in 2-space.
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We are doing something to it, the components, and that is it. For example f(1,3) is going to be 1² + 3².
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It is going to give me a value of 10. It is going to spit out 10, a specific number.
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Let us say f is a function defined from R3 to R, and defined by the following.
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f(x,y,z), generally for 2-space we use x,y, for 3-space we use x,y,z.
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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.
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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,.
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So equals say 1/x + sin(x,z)... excuse me y,z... that is it.
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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.
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That is all the function of several variables is, more than 1 independent variable, and you have one dependent variable.
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Take a look at what we have done.
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When you have a function of one independent variable, for example, let us take f(x) = x².
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We can form the graph, right? We can form the graph by plotting x and f(x).
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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.
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Now, we are taking this and we are graphing it and what we end up getting is a graph in 2-space.
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Now we can do the same thing with a function of 2 variables.
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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.
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So, a point in R2 is just a vector.
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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.
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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.
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When we operate on this point, we get a number right? We get a number.
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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.
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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.
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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.
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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.
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So, let us just draw this graph and see what it might look like.
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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.
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Now (x,y) is some point. That is our independent variable, a vector in (x,y).
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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.
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As it turns out, this actually gives you a surface in 3-space. This is very, very convenient, very beautiful.
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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.
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Example 1 for function of several variables.
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We will let f(x,y) = x² + y². The function that we had mentioned earlier.
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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.
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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.
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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.
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If this is the z-axis, the parabola goes up like that, so it is actually a surface, and it is really quite beautiful.
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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.
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We have this extra geometric intuition that we can use, that we can exploit, and that is very nice, it comes in very handy.
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Each point on this paraboloid is exactly the point (x,y,x²+y²), that is it.
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A random point is just that. That is all that is going on.
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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.
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Let us talk about what this means. Now, let us go ahead and use this example that we did.
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We had f(x,y) = x² + y².
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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.
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What these are, these are level curves, so let us think about what this actually means.
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This is the z-axis, and we said that the paraboloid actually goes up along the z-axis.
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Again, all of the points are (x,y), x² + y², this surface. The points can be anywhere along that surface.
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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.
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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.
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When I look above, down the z-axis, what I am going to get is something that looks like this.
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Now, this is the y-axis, this is the x-axis, and the z-axis is coming straight up.
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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.
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Or maybe this curve if the constant is smaller, or this curve if the constant is bigger.
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Notice that if the x² + y² = c is the equation of a circle.
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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.
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That function of two variables happens to equal a constant what you end up getting is a curve in the x,y plane.
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In fact, a whole bunch of curves, a family of curves depending on what C is.
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In this particular case, you get this curve, or maybe you get this curve, or this curve, or this curve.
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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.
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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.
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I am going to define the next level of something called a level curve called a level surface.
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Let us see, true space, ok, yes. So let us do another example here. Example 2.
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Now, we will let f(x,y,z), so this time it is a function of three variables, equal 2x² + y² + 3z².
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Now, f is a function from R3 to R. Our point that we get is (x,y,z,f(x,y,z,)).
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This is a graph in 4-space.
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We cannot draw a graph in 4-space, but we can work with it algebraically, that is the whole idea.
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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.
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2x² + y² + 3z² = some constant C. 5, 10, 15, sqrt(6), -9, whatever it is.
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This is an implicit relation among 3 variables. What this is is an actual surface in 3-space.
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We had a function of 2 variables a mapping from R2 to R.
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When we set that function equal to a constant, what we get is a curve in 2-space.
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Here we have a function of 3 variables, a mapping from R3 to R.
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When we set this function up equal to a constant, what we get is a surface in 3-space, a graph in 3-space.
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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.
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In this particular case, let us go ahead and take a slightly different function, make it a little more uniform.
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x² + y² + z² = C. Now I have just changed all of the coefficients to make them equal to 1 and 1 and 1.
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What you get... this is... you should recognize it, it is the equation of a sphere centered at the origin.
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So what you get are these spheres, these surfaces, theses spheres, these shells of different radii.
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Those are called level surfaces. These level surfaces are actually very important.
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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.
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In 3-space we call them level surfaces because they define some implicit relationship between the variables x, y, and z.
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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.
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Now, I am going to finish off by writing one thing.
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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.
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You also hear them called isopotential sequences... surfaces, not sequences, surface.
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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.
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So, I have this surface that tells me where the potential energy is the same. This is profoundly important.
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The idea of a potential surface... and in a minute we will talk about it for thermodynamics... these things are profoundly important.
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The notion of a level curve and a level surface come up all the time in science and engineering.
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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.
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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...
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Then, the function equal to C, in other words what we call the level surfaces of this, are called isothermal surfaces.
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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.
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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.
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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.
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The idea of a level curve, and a level surface, it is where these functions of several variables take on a constant value.
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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.
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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.
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So, we will go ahead and stop it here for this particular lesson.
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Thank you for joining us here at educator.com, we will see you next time. Bye-bye.