AI Assistant
Transcript
00:13:680Paolo Guiotto: Probably.
00:16:700Paolo Guiotto: He thinks that this is still night, I don't know. So, since it is black, I'm supposed to write with white, I guess.
00:26:30Paolo Guiotto: Let's see. Okay, maybe this is too strong. Let's see this.
00:32:30Paolo Guiotto: So today, we enter in a new topic. On the notes, the next would be differential equation, but we will skip this for the moment.
00:43:360Paolo Guiotto: And we jump to Chapter 5 about multiple supposed to be.
00:50:330Paolo Guiotto: Multiple… into grass.
01:01:380Paolo Guiotto: Now, you already know what is an integral.
01:05:280Paolo Guiotto: So you learned that, for a function, F of real variable, F of X, huh?
01:14:220Paolo Guiotto: with the X in some interval.
01:17:710Paolo Guiotto: Baby.
01:20:800Paolo Guiotto: Real value, though.
01:24:210Paolo Guiotto: that must verify two technical conditions for the definition, which are F… bounded.
01:36:840Paolo Guiotto: Well, one technical condition is already inside this definition, so we work on a closed and unbounded interval.
01:48:30Paolo Guiotto: It is.
01:50:570Paolo Guiotto: defined… D… Integral.
02:03:350Paolo Guiotto: of F, on the interval AB,
02:11:760Paolo Guiotto: which is rated with this symbol, integral from A to B fax.
02:17:20Paolo Guiotto: EX.
02:20:340Paolo Guiotto: Now, this quantity, this is a number, How's it.
02:25:680Paolo Guiotto: object is just a numerical value, is a number that represents a geometrical quantity. If the function f is positive.
02:36:930Paolo Guiotto: the integral on AB.
02:39:810Paolo Guiotto: of… F… Is a way to define the area
02:48:910Paolo Guiotto: Of a particular plane figure, which is called the trapezoid, of our firm.
02:56:600Paolo Guiotto: Which is basically the following. So imagine this is the Cartesian plane, the x-axis, the Y axis.
03:05:410Paolo Guiotto: we have the interval AB, somewhere on the x-axis. The function is positive, so something like this.
03:16:370Paolo Guiotto: and the…
03:17:930Paolo Guiotto: trapezoid is the plane figure delimited above by the graph of F, below by the interval AB, and left and right by these two lines. So, the area of this figure. Now, what should I use?
03:35:370Paolo Guiotto: No.
03:37:400Paolo Guiotto: It doesn't work in this color? Okay.
03:40:590Paolo Guiotto: So let's use the colors that that.
03:43:110Paolo Guiotto: Let's see what's… So this, in pink, is the trapezoid.
03:52:960Paolo Guiotto: defined by the function f, we can give an analytical Shape to this thing.
04:00:840Paolo Guiotto: It's a set of points, so these are points.
04:04:810Paolo Guiotto: XY, it coordinates XY, well, you can see that the X is… the abscessa must be between A and B, so these are points XY of the Cartesian plane, R2, such that the X is between A and B,
04:23:510Paolo Guiotto: About DY, this is DY.
04:27:310Paolo Guiotto: You are between 0, which is on the x-axis, and the biggest possible value for that abshesa is this one, and this value is F of X. So, for Y, the range is from 0 to F of X.
04:43:230Paolo Guiotto: So this is the, say, the analytical definition of that pink set, which is called the trapezoid of F.
04:54:60Paolo Guiotto: Now, the integral provides a method to compute the area of that figure.
05:01:370Paolo Guiotto: So, the interesting fact is that with the area is a concept known since the answering,
05:12:130Paolo Guiotto: since the Greeks, and even, actually, before, but they, for the first time, introduced a way to define areas of plain figures, but at that time, it was limited to very few
05:27:500Paolo Guiotto: And special figures, like, of course, rectangles, triangles, and circles.
05:35:910Paolo Guiotto: And it was a real, say.
05:39:790Paolo Guiotto: strong achievement, the Archimedian, calculation of, the area, delimited, by an arc of parabola. So imagine that this is the parabola Y equal X squared.
05:56:130Paolo Guiotto: Archimede was able to compute this area here, that today, we would say is the integral from 0 to B,
06:04:920Paolo Guiotto: of X squared that, you learn to compute this in the high school, basically. So, but for that period, it was a very…
06:15:310Paolo Guiotto: a great achievement. This is big cube divided 3, by the way. And, so this formula, obtained by Artemedes, was really a great achievement, because it was the first,
06:29:80Paolo Guiotto: Plain figure, for which he was able to compute… anyone was able to compute the area outside of those figures of the elemental geometry.
06:41:400Paolo Guiotto: Of course, our committee had not the mathematics to introduce the operation of integral, but, basically, its ideas, were those at the base of, the concept of integral.
06:58:140Paolo Guiotto: Now, we will review this, because we have to extend the main goal of today is the following.
07:08:30Paolo Guiotto: to extend…
07:13:850Paolo Guiotto: extend this…
07:19:380Paolo Guiotto: Operation, the operation of Integral.
07:24:20Paolo Guiotto: to the case where we have a function, not of one single variable, but of several variables. So, to the case
07:38:600Paolo Guiotto: when… the function… F…
07:42:450Paolo Guiotto: is a function of several variables. It's a function, for example, of two variables, or three variables, which will be basically most of the examples we will do, but in general, a function of any number of variables. So, say.
08:01:910Paolo Guiotto: and variables.
08:04:900Paolo Guiotto: Why not? Because in many circumstances, it happens to be, These kind of functions.
08:12:220Paolo Guiotto: So, we want to, produce,
08:15:650Paolo Guiotto: A definition of integral for a function f, in general of a vector variable, so let's say X arrow.
08:25:250Paolo Guiotto: So, borrowing this symbol, then we will return on this to understand what is the philosophy behind this writing. So, you write the integral here from A to B. A to B is a way to represent the range for X.
08:41:620Paolo Guiotto: So now, imagine that in general, if the function is a function of an array, that array is not in an interval, because it's in a subset of Rn. So, let's say, on a domain D, we will just write the integration domain down here.
08:59:780Paolo Guiotto: And we will still use this notation DX arrow, we will see what is it.
09:05:460Paolo Guiotto: So we want to produce this.
09:09:280Paolo Guiotto: Now, why is so important this concept?
09:13:490Paolo Guiotto: Well, first of all, so, let's say, why…
09:20:710Paolo Guiotto: We do this. Because, for example, a first point is that we can extend
09:28:820Paolo Guiotto: This idea, which is the idea of providing a general method to compute the areas of plane figures. Why I say general? Because,
09:40:830Paolo Guiotto: by using a function f.
09:44:80Paolo Guiotto: As you desire, you can model a profile there, and you can get the shape of the figure you want, more or less.
09:52:720Paolo Guiotto: Now, imagine that I have a function of two variables. What… what is that figure? So, if I am in two variables, I mean that my domain is now in a plane XY, so I have a function f of variable XY,
10:10:390Paolo Guiotto: Let's… let's try to understand this figure.
10:14:150Paolo Guiotto: So, we have a domain, D, which is the domain for the function f, which is a subset of the plane XY.
10:23:130Paolo Guiotto: Okay? So here, there are points with coordinate X. Y.
10:29:640Paolo Guiotto: The function f is defined on that domain, so if we want to represent that function, we need a third axis. Let's call it Z axis.
10:37:780Paolo Guiotto: where we put values of function f, so at this point, XY, let's say that the value of the function is this one, this is F of XY,
10:48:950Paolo Guiotto: So, with these three numbers, X, Y, plus the third coordinate, which is the value of the function at point XY, we have a point in the space R3. Now, this is in R3.
11:04:120Paolo Guiotto: exactly as it happens here. This point here, this is the point with abscisha X coordinate f of x.
11:14:290Paolo Guiotto: So when you move X in the interval AB, what you obtain is a Y, no? When now you move XY into domain D, what you will obtain is
11:24:430Paolo Guiotto: hopefully a surface, because at each point of this domain, we have a point somewhere in space. So we may imagine that Varang point XY in the domain D, what we get…
11:38:470Paolo Guiotto: The set of all these points, huh?
11:43:730Paolo Guiotto: will be something like a curved surface in space. So this blue thing is the set made of all points of type, X, Y,
11:54:50Paolo Guiotto: F of XY… when XY varies in D, Well, similarly to this one.
12:05:140Paolo Guiotto: this… well, let's color in blue also this one. The set of these points, this is the set of points X, F of X, when X is in this domain, for this case is in the interval AB.
12:19:210Paolo Guiotto: And this is nothing but the graph.
12:24:710Paolo Guiotto: off… the function F, okay? So it's say the…
12:30:330Paolo Guiotto: upper boundary of that figure. We may say that here, it's like the roof of that building.
12:37:500Paolo Guiotto: So, the trapezoid now is what kind of set, is whatever is between that roof, the graph of the function, and the basement, which is the domain D. So, it's a solid object.
12:54:120Paolo Guiotto: Okay, so in this case, the trapezoid is the region that we should color inside here. Now, you understand that it's a bit difficult because it's a solid, but we can easily write what is it, the trapezoid. The trapezoid of F
13:12:700Paolo Guiotto: is the set of points, we are in space, exactly as here. We add points XY in F2, where X is between A and B, which is the domain of the function.
13:23:370Paolo Guiotto: and the Y, the second coordinate, is between 0 and the value of the function. Now, here we will have points X, Y,
13:33:870Paolo Guiotto: Z, in R3, such that the first two coordinates, XY, belong to the domain D.
13:43:520Paolo Guiotto: And the third card here, the Z, is between the quarter zero
13:49:720Paolo Guiotto: and the value of F at point XY.
13:54:130Paolo Guiotto: So this trapezoid is now a solid object, a solid domain of R3.
14:02:20Paolo Guiotto: And therefore, by extension, if the integral from A to B of the function f of x
14:09:940Paolo Guiotto: yields the value of the area of the trapezoid, I would expect that, similarly, if I have a definition of integral for the function F,
14:22:480Paolo Guiotto: of XY, let's say, borrowing the notation, the XDY, then we will see that it has a meaning to write this. This should be the… not the area, because this trapezoid is not a plain figure, is not a surface, but it's a solid. So it will be, rather, the volume
14:42:840Paolo Guiotto: of the trapezal.
14:46:60Paolo Guiotto: So, in other words, a first important reason is that an integral
14:57:70Paolo Guiotto: Of, a function of two variables.
15:01:490Paolo Guiotto: of F equal F of XY.
15:05:660Paolo Guiotto: provides… or should provide. The goal is to show this, provides a method
15:19:380Paolo Guiotto: to compute…
15:24:690Paolo Guiotto: volumes…
15:28:950Paolo Guiotto: of solid… figures.
15:37:690Paolo Guiotto: So, we… basically, we have a method to compute the volume of certain solids that are described in this way.
15:46:80Paolo Guiotto: Actually, so this is, let's say, the first, the first, motivation. Why should we extend the integral to the case of functions of, a generic number of variables? Here we have a motivation for the integral in two variables.
16:04:230Paolo Guiotto: Actually, a second motivation is the following, that the integral still, this one, for functions of two variables, provides an even more general method to compute areas of plane figures, so it extends directly this one. Let's see why.
16:23:130Paolo Guiotto: So… Integral… of functions.
16:30:400Paolo Guiotto: of X, Y, extends… Calculus.
16:40:830Paolo Guiotto: off.
16:42:280Paolo Guiotto: areas… False.
16:45:640Paolo Guiotto: Lane.
16:47:950Paolo Guiotto: Dillos.
16:50:650Paolo Guiotto: You may say, but we already have a method, it's the integral in one variable, right?
16:56:190Paolo Guiotto: Yeah, but that method has a strong limitation from the geometrical point of view, because it allows to compute the area of a figure of this particular type.
17:06:970Paolo Guiotto: whatever is between the graph of a positive function and the axis. So now, imagine that I want to compute this area.
17:19:60Paolo Guiotto: I think a plain figure may like that.
17:23:290Paolo Guiotto: So, how do I compute this area? By using the integral you have seen last year.
17:30:70Paolo Guiotto: So, the point is that this white line is not the graph of any function of X, because for this X, I should need 1, 2, 3, 4 values, and that's not a function.
17:44:790Paolo Guiotto: And even rotating, you know, there is no, no preference in the variables, so X and Y are the same from the genetic point of view. Even if I look as a function of Y, you see that, for example, for this Y, I have that there are two X, so this is not…
18:03:270Paolo Guiotto: This, this, this…
18:08:130Paolo Guiotto: Well, and let's say that the area delimited by that white line It's not a trapezoid.
18:17:960Paolo Guiotto: this… is not.
18:21:140Paolo Guiotto: a trapezoid.
18:23:820Paolo Guiotto: of any function F. So, how should I compute the area?
18:28:840Paolo Guiotto: So, the area… of this, let's say, let's call it, I don't know, D.
18:36:530Paolo Guiotto: It makes sense to call… to call it the…
18:39:530Paolo Guiotto: So, area of D is what?
18:42:690Paolo Guiotto: Do we have a method to compute such areas, or not?
18:47:50Paolo Guiotto: Now, here, there is a simple idea why…
18:53:610Paolo Guiotto: the integral of a function of two variables would be… could be the answer to this problem. Because imagine that now we transfer this figure in space, so we take… of course, it won't be the same, you understand, but let's say that this is our domain, okay?
19:13:400Paolo Guiotto: So the domain D is,
19:16:350Paolo Guiotto: is, this one, no? In plain XY.
19:21:680Paolo Guiotto: Okay, now, define a function, F, which is constantly equal to 1 on that domain. So, what is the graph of this function? It's just a repetition at quote 1, so all points here will have quote 1 of that shape. So, if this function
19:41:320Paolo Guiotto: F of XY is trivially, constantly equal to 1 for every XY in D.
19:49:550Paolo Guiotto: They… they…
19:58:250Paolo Guiotto: No? Now, the trapezoid, so what is between the blue surface and the basement, it's a solid, it's a sort of,
20:09:850Paolo Guiotto: Geometical object, well-known geometical object, like…
20:13:90Paolo Guiotto: you know, if you take a disk, and you just translate upward of one unit, you get a cylinder, no? So, this thing is, we may think it's like a cylinder, no? Basically, we have a basement, and we just translate up of
20:29:220Paolo Guiotto: One unit. So we have this solid, and we may expect that the volume Of the trapezoid.
20:38:240Paolo Guiotto: For this, function.
20:41:380Paolo Guiotto: Of course, I have not yet defined anything, so these are just guess, no? I'm doing, it's just a guess I'm doing on this figure, so… but what should be the volume of this trapezoid?
20:53:750Paolo Guiotto: Yeah, because it should be the area of D multiplied by H. How do you compute the area of the cylinder? You do the area of the base times height, no? You multiply just by height. So since the height is 1 here, this should be equal to the area of D.
21:10:630Paolo Guiotto: So, we would get this formula. So, area.
21:13:970Paolo Guiotto: of D would be the volume of the trapezoid, that is the integral on D of the function 1 dx dy.
21:22:860Paolo Guiotto: So you see an integral, a special integral, the integral of just this function…
21:28:390Paolo Guiotto: very simple function, constant equal to 1 on the domain should provide a method to get the area of D for more general figures with respect to those we can compute with ordinary one-variable integrals.
21:42:60Paolo Guiotto: So this is a second reason to, introduce this operation.
21:47:420Paolo Guiotto: Now… And there are other reasons, for example, a third reason,
21:57:520Paolo Guiotto: Well, there are reasons that comes from physics, for example. I don't want to enter into those things, but I want to talk about a completely different reason that also explains why we need to consider a generic
22:13:460Paolo Guiotto: integration for a function of n variables.
22:17:520Paolo Guiotto: Now, another motivation comes from probability.
22:27:450Paolo Guiotto: Now, the question is…
22:29:540Paolo Guiotto: I don't know what you… what you studied about probability, I don't know even if you have done any probability so far. No.
22:37:460Paolo Guiotto: But… You know what, genetically, what this word means, okay?
22:43:420Paolo Guiotto: Now, let's say that,
22:49:100Paolo Guiotto: how to set. Well, let's say that,
22:56:10Paolo Guiotto: Give me one second just to think about how to…
23:02:910Paolo Guiotto: Well, let's say that probability is a branch of mathematics that specializes in
23:09:220Paolo Guiotto: in defining, you know, to give a ground to the basic ideas that we intuitively may have about probability. So we want to say… we want to compute things like the probability that some…
23:27:170Paolo Guiotto: random experiment, which is represented by the outcome of this variable X. This variable X is actually a function defined on some set omega. The set omega is called the sample space.
23:45:240Paolo Guiotto: which contains all the possibilities, all the outcomes of a certain experiment. For example, you toss a coin, there are two possibilities, head and tail, omega is made by two states. That could be the two letters, H for head, T for tail, okay? You… you…
24:05:800Paolo Guiotto: you roll a die, you have 6 possible outcomes, for example, so the state space is made by 6, states. So omega is made by 6 elements, 1, 2, 3, 4, 5, 6.
24:18:190Paolo Guiotto: And omega can be much more complex, depending on what kind of system you are modeling. But let's say that this X is just a function.
24:26:350Paolo Guiotto: defined on that sample space with values in R. And you want to give a meaning to things like, what is the probability that X is in a certain interval AB? Now, the outcome of this experiment is in the interval AB.
24:41:330Paolo Guiotto: Now, a way to describe these quantities is through integrals. So, saying that this is an integral from A to B of a certain function, which is called the probability distribution of the variable X, where this function fx
24:59:00Paolo Guiotto: is a positive function. It's called probability.
25:04:680Paolo Guiotto: density.
25:08:170Paolo Guiotto: Let me explain in a second, what does it mean? Such that the integral from… on the interior line from minus infinity plus infinity of f of x is equal to 1.
25:22:580Paolo Guiotto: In such a way that, you know, probabilities, well.
25:26:80Paolo Guiotto: In the popular interpretation probability are things like 50%, 70%, things like that. In mathematics, we consider numbers between 0-1.
25:35:750Paolo Guiotto: zero is nothing. One stands for 100%, no? So 1 half means 50%. It's much better, instead of thinking into percentages.
25:46:250Paolo Guiotto: And, exactly as we consider better radiance for measure of angles with respect to, degrees, no? Which is…
25:56:680Paolo Guiotto: The measure you learn first, maybe.
25:59:850Paolo Guiotto: Now what… why this, function of X is called the probability density? Because, imagine that, this interval AV is very small, let's say that approximately this quote, this, is the sum of these products, so let's say that,
26:17:10Paolo Guiotto: the probability that the capital X is equal to little X is, let's say, proportional to the length of the interval. Sorry, I'm saying that.
26:27:890Paolo Guiotto: If I take a little interval, this is saying that probability that X belongs between X and X plus H,
26:35:740Paolo Guiotto: Where in our intuition, h is very smaller, so these two points are very close, you know? This is the integral from X to X plus H of, well, let's change the letter here, FXYDY.
26:50:910Paolo Guiotto: And we may say that,
26:53:790Paolo Guiotto: If the interval is very small, and the function fs is good, so, for example, continuous, that function won't vary too much between X and X plus H, so that
27:07:230Paolo Guiotto: we may assume that that function is constant and equal to the value of, for example, at point H, and then what remains is the integral of 1, which is equal to H. So basically, it says that the probability that
27:21:480Paolo Guiotto: the outcome belong… is between X and X plus H, so it is X with an error of
27:28:370Paolo Guiotto: at most H, is exactly proportional to H, and this is the coefficient of proportionality.
27:36:10Paolo Guiotto: When this happens, you say that this is a density, because it is not a probability, but probability per unit of length. So when you multiply by a length, you get a probability. That's what you say.
27:47:320Paolo Guiotto: So this is why it's called the density, no?
27:50:860Paolo Guiotto: So, this is… was, this was just to say that, integrals.
27:57:40Paolo Guiotto: of these particular functions, so positive function with the total integral equal 1, why there is this second condition? Because when you put the probability that X is any real number, you should get 1, because
28:10:210Paolo Guiotto: Except. It's like to say, probability that tax is any outcome, well, of course, the probability will be 1, no? Probability that tax is whatever.
28:19:440Paolo Guiotto: It's a sure event, so it's a 100% event, and this imposes a constraint to this function fx.
28:28:300Paolo Guiotto: So, the point is that, however, we use integrals to compute probabilities. This is for a variable X, which is a numerical Scala value. Now, if you have an array, so if
28:42:970Paolo Guiotto: the outcome X is made, actually, of several outcomes.
28:49:40Paolo Guiotto: X1, X2, XN, because your experiment is based… your output is based by n different experiments, X1, X2, XN. So, similarly, you would imagine that the probability that this array, X1,
29:04:610Paolo Guiotto: Xen.
29:06:500Paolo Guiotto: belongs to D, Could be computed through an integral.
29:11:950Paolo Guiotto: Now, the integral will be an integral of the probability density, but this probability density now should reflect the fact that the variable has n component, so also the probability density will have n components, so X1, XN.
29:30:490Paolo Guiotto: So this becomes an integral in n variables.
29:38:280Paolo Guiotto: on the domain
29:40:210Paolo Guiotto: So, it is natural to have defined this operation to give a meaning to probabilities of arrays of variables. Now, since this n is not a geometrical dimension, but it's rather the number of experiments you want to consider. So, you want to consider 1 million of experiments.
29:57:860Paolo Guiotto: That's an array of 1 million of components.
30:00:580Paolo Guiotto: So that's not a plain integral, it's an integral in one million of variables. But in any case, having this defined, I have a basement, to… to…
30:11:220Paolo Guiotto: where I can ground the calculus of probabilities. So, probability theory also demands the development of this kind of integrals.
30:22:80Paolo Guiotto: I'm pretty sure that you are going to study probability sooner or later, because you cannot do informational engineering without any probability. Nowadays, probability is the basic language of complexity, because
30:37:580Paolo Guiotto: Most of phenomena are too… so complex that we cannot pretend to know with exact precision. For example, weather forecast.
30:48:690Paolo Guiotto: In principle, we have law of physics that tests how the atmosphere evolves in time, so I could do a prediction.
30:58:690Paolo Guiotto: knowing where clouds are moving, I know… I could know values of pressure, temperature of the atmosphere, and so on, so at the end, understand if it's going to rain or not, no?
31:11:970Paolo Guiotto: Now, the equations that drive this are differential equations. We have seen a bit of this in the first year, but even much… a kind of equation which is much more complicated of what you have seen. So this makes this equation basically impossible to be solved exactly.
31:30:770Paolo Guiotto: That can be only solved in an approximate way, at least to nowadays, no pages, but it's about two, three hundred… two, three centuries that mathematicians try to study this without, no real success, because the equations are too much complex.
31:48:580Paolo Guiotto: And moreover, even if you would be able to solve those equations, you would need to know the exact configuration at a certain moment of the atmosphere. So, I should need exactly, point by point, what is the state
32:03:730Paolo Guiotto: Of the flow, the airflow in the atmosphere, so the velocity of the airflow at each single point of the atmosphere, which is practically impossible.
32:12:470Paolo Guiotto: Because we do not have any way, any means to measure this thing with such a precision.
32:18:660Paolo Guiotto: We may know about the pressure at some few points on the surface of the planet, and maybe somewhere in the middle in the air, but we do not have the pressure of air at each single point.
32:32:200Paolo Guiotto: So this means that to do a prediction with such an approximate data means that even if you know how you can do this prediction, you lack of information, so you are going to introduce necessarily errors.
32:46:270Paolo Guiotto: So that's why predictions are made on a probabilistic idea, no? It's like when you toss a coin, you know?
32:55:600Paolo Guiotto: In principle, the movement of the coin, when you flip the coin, is driven by the law of physics, the classical physics, Newton's law of gravitation.
33:07:600Paolo Guiotto: Okay, maybe there is a little bit of friction because of air, but you know, this movement of rotations is, in principle, it is possible to study. We have the equations. These are the Newton equations.
33:20:870Paolo Guiotto: The problem is that the equations are so complicated that nobody would use the Newton equation to say that if the coin falls head or tails.
33:30:950Paolo Guiotto: But you say a much rough thing, saying that we have 50% of possibilities that it will be had, and 50% of possibilities that will be paid. So it means less information, but much quicker and easier.
33:46:710Paolo Guiotto: And so, it changed a lot, because in the first case, I have a beautiful theory, but I cannot say anything about the prediction. In the second case, I can do a prediction, but of course, it won't be exact, no? So that's why probability is so important, and…
34:03:530Paolo Guiotto: Modern probability is based on this kind of thing, so…
34:07:770Paolo Guiotto: you necessarily need to have tools to deal with these quantities, and these tools are all integrals. So that's a big reason to introduce multiple integrals. So, now.
34:20:860Paolo Guiotto: I hope I gave you
34:23:650Paolo Guiotto: A certain number of motivations why we need to develop an integration for functions of several variables.
34:35:360Paolo Guiotto: Okay, once we, understood this, so the problem becomes, how can we define the interior?
34:44:00Paolo Guiotto: Now, you already know that for one variable integral.
34:50:650Paolo Guiotto: I don't know how you have seen this, but the point is that there is a certain process that, at the end, yields the definition of integral. That's a complex process.
35:04:180Paolo Guiotto: which is based on some intuitive geometrical idea that I will refresh now, but the key point is that at the end, I… with that definition, I don't know how to compute an integral in practice. So…
35:17:810Paolo Guiotto: We need the definition, because otherwise we don't know of what we are talking about, but normally, as in many cases in mathematics, it happens that the definition is almost never the good recipe to compute a quantity.
35:31:880Paolo Guiotto: So there are basically two steps. Step one, I need to define the object in an appropriate way, okay?
35:39:210Paolo Guiotto: And step two, I have to develop tools to compute that object. These two things
35:45:850Paolo Guiotto: may… may have not relations between them, okay? So let's start with the construction. Since the construction is more or less basically the same of one variable integral, but more complex, because we have several variables.
36:02:410Paolo Guiotto: And it is not so important in the calculation. I will do the construction
36:10:580Paolo Guiotto: in a sort of light way, without not so many details, but I will give you an idea of how the definition of integral is made. So, let's start with step one.
36:25:180Paolo Guiotto: definition… off.
36:28:860Paolo Guiotto: into life.
36:30:840Paolo Guiotto: Well, this is exactly as in the case of one variable. This is made by different steps, sub-steps, let's say. So let's say 1.1 case…
36:44:510Paolo Guiotto: off…
36:45:630Paolo Guiotto: functions, positive functions, which is the important case for the concept of area of trapezoid, and here would be volume of trapezoid, or more general things. Then we will do the extension to the case
37:00:880Paolo Guiotto: of… F, which is a real value, so positive, negative, doesn't matter.
37:09:460Paolo Guiotto: actually… Well, we won't do here, but there would be a third case, huh?
37:16:300Paolo Guiotto: So, I put between commas because we are not going to do, or between parentheses, case of…
37:23:810Paolo Guiotto: F with, which are C-valued.
37:28:140Paolo Guiotto: Now, Why is she valued? Well.
37:32:700Paolo Guiotto: These are another motivation that comes from certain tools you will see in the far future, okay, not now, if you survive until the far future, okay?
37:45:410Paolo Guiotto: Now, the far future is that, especially in information engineering, in certain fields, it is convenient to calculate things by thinking in terms of complex numbers, because certain operations are easier then.
38:02:90Paolo Guiotto: It's than having real numbers.
38:04:910Paolo Guiotto: Just to mention you an example. So, if you think to the complex representation of, to the trigonometric representation of a complex number, you may represent a complex number, Z, as something like rho cosine theta plus i sine theta, right?
38:24:750Paolo Guiotto: And this is also the algebraic representation of this.
38:29:240Paolo Guiotto: Now, you can also use another representation. Maybe this one will become a little bit more clear at the end of our course, which is the exponential representation. So, let's say that this E to Y theta
38:43:800Paolo Guiotto: If you don't know what is it, it's just a way to call this guy here.
38:49:370Paolo Guiotto: Now, the point is that, imagine that you have to do this simple thing. You have to multiply two numbers written in this form. So, Z is raw cos theta plus I sine theta, and W is R, say, cos phi plus I sine theta.
39:07:780Paolo Guiotto: Now, you can do these calculations, not particularly long, I want to do this.
39:12:600Paolo Guiotto: So you multiply by, according to the algebraic rules, so the algebraic rules means that i times i is minus 1, and so on. And you will get that, after a lot of calculations, that this product is equal to raw R,
39:29:580Paolo Guiotto: times cosine of theta plus phi plus I sine.
39:36:240Paolo Guiotto: Theta plus phi.
39:39:780Paolo Guiotto: Okay, you get this formula.
39:42:600Paolo Guiotto: But if you do this product here, with this notation of the exponentials, this result is immediate, because if you do the product, you would get what?
39:53:240Paolo Guiotto: Z times W is R times rho, you see immediately here. Then you have the two exponentials. You know that the product of the exponential is the exponential of the sum, and that's the key algebraic thing that makes calculation easier. So immediately, I see this formula.
40:12:20Paolo Guiotto: Which is exactly this one.
40:14:130Paolo Guiotto: So, in a second, I got the same result, yet if I do algebraically, I have to remind of addition formula for sine, cosine. I do a lot of algebraic multiplications to get this one. So, if I use the numbers in complex form, instead of thinking as
40:32:00Paolo Guiotto: if they have… they have an algebraic form like A plus AB, I do calculations much easier.
40:37:890Paolo Guiotto: So there is a reason to do calculations also for functions which are C-valued. Maybe you don't… you don't see in the real life a C-valued function, no, because we deal… we work in a real world, apparently.
40:54:70Paolo Guiotto: But, sometimes it is more convenient, and especially with signals and things like that, complex valid functions are important.
41:03:160Paolo Guiotto: So, in principle, we would need also this kind of extension of integral, which is actually a simple trivial algebraic extension. Maybe I will tell one more about this.
41:14:740Paolo Guiotto: Okay, now, this is the first, and the second part of the program would be, how… Do we…
41:24:240Paolo Guiotto: Computer.
41:27:550Paolo Guiotto: an integral of a function f.
41:30:840Paolo Guiotto: An integral on several variables, so let's write in the array form.
41:36:770Paolo Guiotto: Because with the definition that we will see in the process, in the step one.
41:41:870Paolo Guiotto: It won't be clear exactly how to do this calculation.
41:45:840Paolo Guiotto: But that's not a novelty. It's the same that happens with one variable integrals.
41:51:770Paolo Guiotto: Now, here you will see that there are two fundamental tools.
41:56:600Paolo Guiotto: The number one is called the reduction formula.
42:08:260Paolo Guiotto: Our, as the name says, this is a reduction to what?
42:12:520Paolo Guiotto: Well, what basically this formula says is something very intuitive and simple. If you have to compute, for example.
42:20:820Paolo Guiotto: An integral of a function of two variables, like, this one.
42:29:890Paolo Guiotto: You can reduce this calculation to two integrals in one valuable each.
42:36:870Paolo Guiotto: So there is a way to rewrite this integration as two integrations, one in X, and followed by one in Y.
42:46:940Paolo Guiotto: So, two one variable integrations, this means that two operations that we already know how to do.
42:53:760Paolo Guiotto: We suppose that you all know what is integral in one variable.
42:58:710Paolo Guiotto: So, if you don't remind very well, please take a couple of hours to review this, because we are going to use without… without refreshing things. And if, for example, if I have an integral for a function in three variables, or something like a three-point integral.
43:18:230Paolo Guiotto: we have that this can be reduced to three integrations, each in one of the variables. So this will be a practical way, because since we know how to handle the one variable integrations, by iterating one variable integration, we will get
43:35:330Paolo Guiotto: The, multiple value… value integrations.
43:39:670Paolo Guiotto: So, let's say that, very, very roughly, this means that, for example, integration in two variables, in the X, the Y, can be split into two integrations, something like, I decide first to integrate, for example, in X, I do this integration.
43:59:220Paolo Guiotto: Once I've done this integration, there is no more X, because X has been integrated, so there is only Y, and they do a second integration in Y. Of course, the problem is what should we write down here, what are the integration domains, but this is the idea.
44:15:60Paolo Guiotto: And the second, fundamental
44:18:150Paolo Guiotto: Technique is the… a technique that you already know, which is the change of variable.
44:29:40Paolo Guiotto: So this is not a novelty, because you know that sometimes to compute an integral for a function of one variable.
44:39:40Paolo Guiotto: It could be reasonable to introduce a new variable, something like Y is some function of X, huh?
44:48:250Paolo Guiotto: And transform this integration into an integration in the new variable Y.
44:54:130Paolo Guiotto: So now you know that something happens here, so you do not integrate anymore from A to B, something changed here, so I will refresh all this at the moment, but…
45:05:670Paolo Guiotto: Since this will come in…
45:08:540Paolo Guiotto: probably at the end of this week, take, or please take these days to review how it works integration in one variable, because we are going to use both here and here, okay?
45:20:520Paolo Guiotto: As you will see.
45:23:250Paolo Guiotto: Okay, so this will be the two techniques, and so what we will do is to see how they work and do practice, okay?
45:33:680Paolo Guiotto: So this is, let's say, the master plan of this, of this partner.
45:38:650Paolo Guiotto: Since we have done 45 minutes, if you want, we can take a 5-minute break, and then we… Continue.
45:57:570Paolo Guiotto: Okay, so let's launch into this, construction.
46:03:750Paolo Guiotto: But there is something, it's still, so, we say the definition…
46:15:890Paolo Guiotto: So, one definition… off.
46:20:590Paolo Guiotto: Integral.
46:24:110Paolo Guiotto: So, we say that we start with case F positive.
46:29:140Paolo Guiotto: So, 1.1Ks… F… function of Voliable XM.
46:39:260Paolo Guiotto: The final domain, D of Rn.
46:43:90Paolo Guiotto: Positive, so with values in 0 plus infinite.
46:48:440Paolo Guiotto: Now, it is better if we quickly refresh how it works, the same construction for the one variable integral, because we are going to do basically the same thing, but we have to adjust ideas, as you will see. So, let's…
47:06:10Paolo Guiotto: reminder.
47:10:00Paolo Guiotto: I'll… it works.
47:17:310Paolo Guiotto: 4… the one variable integral from A to B of F of X.
47:23:990Paolo Guiotto: Yes. So we have, a function
47:28:170Paolo Guiotto: F of one single variable, X, defined on an interval AB, And it is positive valued.
47:39:730Paolo Guiotto: We must assume, as we will see in a moment, that F is bounded.
47:49:380Paolo Guiotto: So, in this construction, there are two key assumptions. One is F must be bounded, and the other one is the interval must be closer than bounded. So, it's very particular. It doesn't work if the interval is open, for example, or if the function is not bounded.
48:04:810Paolo Guiotto: Now, what is the idea?
48:07:100Paolo Guiotto: We want to define the area of the trapezoid, which is the plane region delimited above by the graph of F and below by the interval AB. So how do we do this?
48:19:900Paolo Guiotto: Well, since, of course, the idea is the idea of computing an area, so, we use elementary areas. This, this is a very handsome idea, it's not a humili's idea, that is an idea to fit the area…
48:34:670Paolo Guiotto: the area.
48:36:320Paolo Guiotto: with certain rectangles, and we do the rectangles in this way. We take the interval AB, and we divide into subintervals, so we take
48:46:830Paolo Guiotto: the interval from A to B.
48:49:80Paolo Guiotto: We introduce division points, okay, that, for convenience, we call T0.
48:55:630Paolo Guiotto: Or better, sorry, X0.
48:59:310Paolo Guiotto: What is this?
49:01:50Paolo Guiotto: X0 equal A, then X1, X2, in general, XK, XK plus 1,
49:09:590Paolo Guiotto: Until a certain, I see, and I'll use the N for that.
49:15:750Paolo Guiotto: Well, let's call it the X, subarities, sub… How'd be good.
49:23:10Paolo Guiotto: So this is, this introduced the concept of subdivision, which is a set that's usually denoted with the letter pi, of points X0 equals A, strictly less than X1, strictly less than X2, etc.
49:40:600Paolo Guiotto: XK less than XK plus 1.
49:45:720Paolo Guiotto: etc, less than X capital N, which is the final point, E, is a subdivision division… off.
49:58:470Paolo Guiotto: Aiden.
50:00:250Paolo Guiotto: Now, these intervals of the subdivision, so the generic interval from XK to XK plus 1, will be the basis of this
50:11:330Paolo Guiotto: Of this rectangle, which is, we start defining rectangles which are inscribed into the
50:20:70Paolo Guiotto: Trapezoid. So, the biggest possible is this one, with, hate…
50:27:310Paolo Guiotto: Here, the hates, if this is the interval from XK to XK plus 1,
50:32:890Paolo Guiotto: the, height, let's say HKE, is what?
50:37:680Paolo Guiotto: Well, you see that to be inside, the head must be smaller than the minimum value of F on that interval. So, formally, I should write the minimum of values of F when X is in the interval from XK to XK plus 1.
50:55:930Paolo Guiotto: Now, since I'm not assuming anything about the regularity of F, in particular, I'm not saying F continuous, if F is continuous, this quantity is well-defined. Otherwise, there could be this because of Weissler's theorem, no? Maximum, minimum.
51:12:90Paolo Guiotto: For a continuous function on a closing barrel, the interval always exists.
51:17:40Paolo Guiotto: But, avoiding this restriction, I can always define, let's say, the surrogate of minimum, which is the infamom.
51:26:310Paolo Guiotto: which coincide with minimum when minimum exists, otherwise this always exists, whatever is at. So this is the quantity that it is really used to do this construction.
51:41:390Paolo Guiotto: Since the function is bounded, this quantity is finite.
51:46:880Paolo Guiotto: Now, the area of this rectangle here is now, defined by… so, well, let's write here.
51:58:780Paolo Guiotto: The area of this rectangle is the product base times height. The base is that interval from XK to XK plus 1, so the length would be XK plus 1 minus XK, so this is the length of the
52:12:530Paolo Guiotto: base times the height, which is the value HK.
52:17:510Paolo Guiotto: I repeat this for each of the intervals. I will have different rectangles with different aids and different bases.
52:28:330Paolo Guiotto: And basically, at the end, I have, a figure like this one.
52:37:540Paolo Guiotto: So the sum of the areas of all these rectangles will be…
52:42:700Paolo Guiotto: an approximation by the fact, because you see that the area should be bigger, and so I define this quantity, which is now called the inferior sum.
52:52:850Paolo Guiotto: which is denoted by these symbols, associated to the subdivision pi, because it depends on the subdivision, which is, by definition, the sum of the areas we computed here. So just a copy, sum of areas, XK plus 1.
53:08:910Paolo Guiotto: minus XK,
53:11:130Paolo Guiotto: times HK, and when I sum for K, that goes from, technically speaking, the first interval is X0, X1, so the first K, sorry, K plus 1, the first K is 0.
53:24:00Paolo Guiotto: And the last interval is X sub n minus 1XN, so the last k is n minus 1. Well, that's just for the…
53:33:770Paolo Guiotto: technical procedure. Now, this quantity is well defined as a number, positive number, But it's not infinite.
53:43:490Paolo Guiotto: Okay? So for every subdivision, this is called the inferior
53:52:240Paolo Guiotto: some… Off, on.
53:59:510Paolo Guiotto: subdivision, Bye.
54:03:130Paolo Guiotto: Pi is not 3.14, it's… it's just the name we give to this set of points.
54:10:300Paolo Guiotto: So, each subdivision, yields an inferior sum.
54:17:920Paolo Guiotto: Now, none of this will be the area in general, of the trapezoidal, and all of them, there will be approximation by defect from below of the area.
54:30:760Paolo Guiotto: So, in principle, now, to define the area of the trapezoid, I should take the best of this approximation, which is the best.
54:42:280Paolo Guiotto: The best is the biggest, because they are approximation by default. So, I should say that, let's call area of trapezoid…
54:51:950Paolo Guiotto: of F, the…
54:54:160Paolo Guiotto: In principle, I should say, I put between commas because it's not exact, this definition, I should take the maximum of these sums over all possible subdivisions pi.
55:08:160Paolo Guiotto: But again, this is a maximum of a set of numbers, and in general, there is no maximum for a set of numbers. So we replace this, which is maybe not defined, with the supremum of the inferior sum
55:24:720Paolo Guiotto: Over all possible subdivisions.
55:27:870Paolo Guiotto: Now, the point here is that we could repeat the construction
55:33:720Paolo Guiotto: with a completely symmetric argument in this way. So again, take the interval at B, let's take a subdivision of AB, our function.
55:44:730Paolo Guiotto: But now, let's construct an approximation by excess of the area. I do the same. This is the interval from XK to XK plus 1, and now I build a rectangle that contains the trapezoid in that part. So I should take a rectangle that contains
56:03:490Paolo Guiotto: this part of the graph, no? So in this case, I have a different age, HK.
56:12:200Paolo Guiotto: What is this one? Now, as you can see, this coincides with the biggest possible value of the function. So, as above, I should take… above, I…
56:22:10Paolo Guiotto: The idea was to take the minimum of F in the interval. We cannot do that in general, so we took the infamum. Here, we should take the maximum, so the idea would be, let's take the maximum of the values of F,
56:36:50Paolo Guiotto: on the interval XK.
56:38:370Paolo Guiotto: XK plus 1. Again, if the function is continuous, this maximum exists. Otherwise, I prefer to say, let's take the supremum of values f of x, which extend the maximum, it coincides with the maximum when the maximum exists. Otherwise, this second always exists while the first
56:57:810Paolo Guiotto: Might not exist, so… And this one is finite, again, because F is bounded.
57:05:190Paolo Guiotto: So you cannot have supremum equal plus infinity, because F has a bound above.
57:10:630Paolo Guiotto: Okay, now, the area of this new rectangle
57:15:990Paolo Guiotto: This area is equal, again, by the product between the length of the base
57:23:80Paolo Guiotto: So XK plus 1 minus XK times DH, which is the number HK, capital HK.
57:29:580Paolo Guiotto: And when you do… you repeat this construction for every of the intervals of the subdivision.
57:37:50Paolo Guiotto: You get a figure made of many rectangles that contains
57:44:810Paolo Guiotto: the trapezoid. So the sum of a K, again, for K going from 0 to capital N minus 1, this time now defines what is called the
57:57:850Paolo Guiotto: Superior sum of the subdivision pi.
58:02:920Paolo Guiotto: So this is an approximation by excess.
58:11:240Paolo Guiotto: Of the area of the trapezoid, because as you can see.
58:14:960Paolo Guiotto: The trapezoid is inside this, this,
58:18:940Paolo Guiotto: This, let's say, this skyline of… rectangles.
58:23:750Paolo Guiotto: And, therefore, the area of the trapezoid will be smaller than the sum of the areas of the rectangle, so that's an approximation by XS.
58:33:470Paolo Guiotto: For each subdivision, I have also an approximation by excess.
58:39:40Paolo Guiotto: As an approximation by default here.
58:42:390Paolo Guiotto: So, now…
58:44:190Paolo Guiotto: I should say, well, let's take the best of the approximation by excess. What is the best approximation by XS is the smallest, so the minimum. I can do that in general, so I will take the informal of these numbers.
59:00:810Paolo Guiotto: Over all possible subdivisions pi.
59:05:400Paolo Guiotto: This quantity is now the best Approximation.
59:12:700Paolo Guiotto: Bye.
59:14:170Paolo Guiotto: excess… off… area.
59:19:820Paolo Guiotto: of trapezoid.
59:22:150Paolo Guiotto: which is similar to the previous one we defined here. This is the best approximation, by the fact
59:37:890Paolo Guiotto: of the area.
59:40:800Paolo Guiotto: of trapezoidal.
59:44:810Paolo Guiotto: So, since we have two numbers that potentially might be different, so we call this, with this, underline, so we call this, inferior.
59:58:540Paolo Guiotto: area.
00:01:140Paolo Guiotto: off… the trapezoid either.
00:06:380Paolo Guiotto: And, we call that one the superior area.
00:12:60Paolo Guiotto: Of the trapezoids.
00:16:840Paolo Guiotto: Superior.
00:21:10Paolo Guiotto: area.
00:23:700Paolo Guiotto: Now, by definition, of course, we do not prove
00:27:940Paolo Guiotto: Also, this is not part of our cause, but it is clear that it is… clear.
00:37:230Paolo Guiotto: that, intuitively, at least, the best of the approximation by excess will be bigger than the best of the approximation by defect. So the inferior area will be less or equal than the superior area.
00:53:410Paolo Guiotto: So now we have, like, two candidates to be…
00:57:180Paolo Guiotto: The, area of the figure.
01:00:340Paolo Guiotto: If they are the same, okay, we can… we can choose one of them. It's the same. But if they are different, it's not clear what should be the area. Now, one may wonder, is it possible that they are different? Now, the problem is that it is possible.
01:18:960Paolo Guiotto: There is a remark.
01:23:460Paolo Guiotto: it can… Happened.
01:29:300Paolo Guiotto: That.
01:31:310Paolo Guiotto: The inferior area is strictly less than the superior area.
01:37:200Paolo Guiotto: There is a pathological function that makes this possible. I probably… you have seen the function. The example is this. Take AB, the interval 0, 1,
01:52:20Paolo Guiotto: and take the function F, made in this way. It's a function which is 0 when x is rational.
02:01:70Paolo Guiotto: And in the interval 0, 1,
02:04:110Paolo Guiotto: And it is equal to 1 when x is irrational.
02:08:340Paolo Guiotto: In the interval 0, 1.
02:10:590Paolo Guiotto: This function is called the Dirichlet function.
02:26:510Paolo Guiotto: And you cannot really see how this function is made. We cannot do a plot, because this function would be like this. If you have a point in 01, in X, which is a rational number, we plot 0. So, if this is rational, I have 0. If this is irrational, plot 1.
02:46:320Paolo Guiotto: Now, you know that rationals and durationals are everywhere.
02:51:360Paolo Guiotto: Because of the property, which is called the density of ratios and the ratios in reals, so this means that whatever is the interval you take, there are always, in the interval.
03:02:110Paolo Guiotto: at least 2, and in fact, infinityals and erasional. So you would see something like this, a function that continuously jumps up and down between 0 and 1,
03:14:470Paolo Guiotto: Okay? So, it's a completely discontinuous function. In this case, it can be proved that the inferior area is 0, and the superior area is 1.
03:27:160Paolo Guiotto: So you have that, it can be… they can be strictly different. So, it can happen that, they are not equal.
03:35:850Paolo Guiotto: Now, this happens, of course, for a quite strange function. You may think that that's a function of a sorcery of mathematicians, something like this, but that's not actually a strange function. For example, in probability.
03:54:590Paolo Guiotto: This is the even too much regular function, because
03:59:160Paolo Guiotto: In probability, in random phenomena, you have, normally, irregularity. If you look at how a noise affects a signal, you see that noise is continuously changing, impacting on the signal, because it's the name, it's noise, it's not a regular sound that affects it. So you look at signal of a mobile, for example.
04:21:660Paolo Guiotto: The mobile filters so you don't see noises, you don't hear noises, but the real signal is very noisy.
04:28:230Paolo Guiotto: So it's very irregular, and you may imagine as it is the natural signal of your voice, which is affected by lots of magnetic fields and so on, and so this signal, the real signal, is very irregular.
04:44:510Paolo Guiotto: So you can see as the composition of two quantities. Some between the natural signal plus noise and noise is something like this.
04:52:140Paolo Guiotto: Okay? That acts more or less randomly on the true signal. So these kind of functions are interesting for a probability, not for analysis, where we use tools like limits, continuity, derivative, that they demand.
05:08:750Paolo Guiotto: very good functions. So, for analysis, this is a bad function, but for probability, it's an interesting function.
05:16:340Paolo Guiotto: So don't think that this is an exotic example that is used just to show you that there are functions for which it happens that the two areas are different.
05:26:980Paolo Guiotto: However.
05:28:660Paolo Guiotto: we have a definition that says if the two areas are the same, so if the best of the approximation by default coincides with the best of the approximation by X, then we say that
05:43:100Paolo Guiotto: we say… That's it.
05:48:230Paolo Guiotto: F.
05:49:240Paolo Guiotto: is integral.
05:57:720Paolo Guiotto: on the interval AB, And we define the integral.
06:04:940Paolo Guiotto: And… the integral from A to B of the function F, Jeez.
06:11:530Paolo Guiotto: The common value, so either the inferior area or the superior area, it's the same because they coincide.
06:20:370Paolo Guiotto: So, this is the story for F-positive. Of course, as you can see from this process, you have a definition, a good definition.
06:29:360Paolo Guiotto: But in practice, it's impossible to proceed in this way to compute these areas. This can be done in very specific cases, but it cannot be done in general.
06:41:430Paolo Guiotto: At a certain point, you discover that there is a connection with differential calculus through the fundamental theorem of differential calculus, that… of integral calculus, that provides you a method to compute integrals. But this is another story, okay?
06:58:750Paolo Guiotto: Now, what we want to do is to take this construction and try to repeat the construction for the case of a function of several variables. Now, for
07:10:900Paolo Guiotto: simplicity, I will limit to the case of a function of two variables, but as you may understand, the idea can be extended. So now.
07:22:00Paolo Guiotto: Let's… try… to adapt, this… idea.
07:35:80Paolo Guiotto: to define… the integral on a certain domain, D, of a function f of xy
07:45:310Paolo Guiotto: you will see that the notation DXDY is natural, exactly as the notation f of x dx is natural. Because if you think how this quantity arises, we have a question here, at the end, you start from this sense.
08:02:690Paolo Guiotto: And these sums are basically products K times basis. The basis is an increment of X, so that's why we write the X.
08:12:810Paolo Guiotto: And that H is basically the value of the function.
08:16:729Paolo Guiotto: So, you have sum of value times the increments of the value.
08:22:729Paolo Guiotto: And we do the sound when these points range from A to B. Now, this symbol here, as you know.
08:31:460Paolo Guiotto: This, is, is an S, in fact. The origin of this is, stylized the S that stems from South.
08:39:100Paolo Guiotto: from some, from A to B.
08:40:810Paolo Guiotto: Of course, it's not.
08:44:210Paolo Guiotto: Visa.
08:45:649Paolo Guiotto: I think it's not complicated, but the intuitive idea is that this writing reminds us of this, of this process, that we build the area by doing signs or failures.
09:00:60Paolo Guiotto: base times 8, so this is the area. Now, we will do basically the same here. The unique difference is that now this will be a volume.
09:08:420Paolo Guiotto: But let's think about, if this stands for sum, we are doing some values of F, that would be still 8, times DXDY is now a product of 2 lengths, so it's an area.
09:22:460Paolo Guiotto: So it would be area of something times 8, that's the volume of the parallel repeated. And in fact, you see this in the construction.
09:32:500Paolo Guiotto: However, there is a first problem.
09:35:510Paolo Guiotto: Forms.
09:37:520Paolo Guiotto: F function of XY, which is defined on domain D, which is a subset of R2.
09:46:149Paolo Guiotto: And we take a positive valued.
09:49:500Paolo Guiotto: still F-banded.
09:53:620Paolo Guiotto: Because, as you will see, we have to say that a certain int soup must be 5.
10:00:460Paolo Guiotto: Now, the first problem is that, is the domain.
10:04:630Paolo Guiotto: In the case of one variable integrals, we never reflect on this, because on the real line, intervals are the natural domains for most of functions.
10:15:740Paolo Guiotto: In the case of a function of two variables, There is not a natural… The shape for the domain.
10:23:540Paolo Guiotto: So, first of all, there is not an interval, because we are in two variables, so what we have, it's a plane, XY,
10:32:180Paolo Guiotto: So, let's say the idea closest to the idea of interval could be a set made of X varying into an interval and Y varying into another interval. That set is a rectangle.
10:47:10Paolo Guiotto: So the analogous of an interval in Cartesian plane would be a rectangle, no? Because interval, I interval.
10:56:640Paolo Guiotto: contained in R means that you have a set of points, X in R, where X is between certain value A and certain other value B. That's an interval.
11:09:60Paolo Guiotto: what is an interval? I…
11:13:140Paolo Guiotto: interval, let's put between commas, because it's not going to be an interval, in R2.
11:19:240Paolo Guiotto: Well, if I have to think that something which is closer to this thing, I would say it's a set of points, XY, because now we are in a 2, so we have points.
11:30:290Paolo Guiotto: Where X is between some A and some B.
11:34:630Paolo Guiotto: And Y is between some C and some D.
11:38:490Paolo Guiotto: So, like this figure, AD and CD. And that set is a rectangle.
11:46:120Paolo Guiotto: Okay? So, similarly, an interval in R3 will be a set of points, XYZ, with X between A and B, Y between C and D, and Z between E and F. And this is a parallel epiphid, no?
12:02:950Paolo Guiotto: But as you understand, in the case of playing, an interval is a very special set.
12:08:560Paolo Guiotto: And normally, for example, very natural domains are not of this type. Think about a disk, a very simple figure. It's not an interval. So, disk…
12:21:430Paolo Guiotto: is… not… and Intubac.
12:27:250Paolo Guiotto: So…
12:28:180Paolo Guiotto: And the normal domains, which are, for example, we have seen, we learned that when we study the functions of two, three variables, domains are defined to indirect quantities, things like that.
12:41:730Paolo Guiotto: They are never just rectangles. You can get any kind of shape, so the first problem is that there is not a natural domain, as there is a natural domain for functions of one variable.
12:54:490Paolo Guiotto: So, we have to decide how to start this first problem. So, we attack this by saying, okay, let's do this definition when the domain is a rectangle, then let's see what can be done for the general case. So, let's…
13:13:310Paolo Guiotto: D, for this case, contended into R2, B, It… rectangle.
13:21:640Paolo Guiotto: So we can, for the moment, as you will see.
13:25:780Paolo Guiotto: borrow the same idea of the one-dimensional construction of integral. So, let's say that we are in plane XY, our domain is a rectangle.
13:37:710Paolo Guiotto: So, made of an interval AB for DX, and an interval CD for Y.
13:45:500Paolo Guiotto: Okay?
13:47:150Paolo Guiotto: So…
13:50:160Paolo Guiotto: let's do two figures. This is R2, and this is the domain. Then we do a second figure, which is the space figure.
14:00:150Paolo Guiotto: So again, we have a domain, which is a rectangle in a plane XY.
14:06:160Paolo Guiotto: So this is the domain.
14:08:360Paolo Guiotto: So again, this is, AB.
14:11:700Paolo Guiotto: This is C, D.
14:14:360Paolo Guiotto: And we have a function defined on this, a positive function defined on this domain. So we have to imagine that to each point of the domain, we have a value. So, at the end, the graph of the function will be a surface.
14:31:780Paolo Guiotto: By the inch.
14:33:260Paolo Guiotto: It is not.
14:34:470Paolo Guiotto: Flat here, could be over there.
14:39:810Paolo Guiotto: You see this, huh?
14:43:860Paolo Guiotto: It's a short face.
14:45:420Paolo Guiotto: It's the roof of this, of this house.
14:51:520Paolo Guiotto: Not straight, okay, but you understand, that's not… that's the problem of doing videos in this case.
14:58:490Paolo Guiotto: Okay.
14:59:730Paolo Guiotto: Now, what we want to do is the volume between the blue roof and the basement there.
15:06:780Paolo Guiotto: So exactly as we did here, we wanted the area between the graph of F and the interval AB. What we did is divide the database mate, AB, so the interval AB into subintervals.
15:21:620Paolo Guiotto: How can we divide now this domain, D, which is a rectangle? We could divide, first, the interval AB,
15:29:100Paolo Guiotto: introducing a subdivision of points from A to B, so let's call these points XK, but we also divide CD. Let's introduce this, we call, not with the same letter to, because we have two independent, no, indexes. So, in such a way that,
15:48:270Paolo Guiotto: this division of AB and the division of CD will determine a subdivision of the rectangle in small rectangles.
16:01:290Paolo Guiotto: Okay? So you can see here. So if this is XK, XK plus 1, and this is YJ, YJ plus 1,
16:13:450Paolo Guiotto: If I put… I take all points of the rectangle D, for which X is between X and XK and XK plus 1, and Y is between YJ and YJ plus 1, what I get is a little rectangle down here, you see?
16:32:500Paolo Guiotto: So instead of doing a subdivision of the segment with intervals, I do a subdivision of the rectangle with sub-rectangles, no? Which is the rectangle you see down here.
16:46:130Paolo Guiotto: Now, this will be the basement of what a parallel pipede?
16:51:640Paolo Guiotto: Which will be…
16:53:900Paolo Guiotto: in two cases. Case 1, it will be inscribed into the trapezoid. Case 2, it will contain the trapezoid. So, now the construction is very similar, because I defined this eight, let's call.
17:09:470Paolo Guiotto: H. Now, it depends on the two indexes, because I have an XK, XK plus 1 on the x-axis, and YJ, YJ plus 1 on the y-axis, so let's say that this depends on the two indexes, K and J,
17:23:90Paolo Guiotto: And for the little h, this will be the infamum of the values of the function f.
17:29:520Paolo Guiotto: at point XY, that belongs to that rectangle, so let's use some shortening, let's call it RKJ…
17:40:130Paolo Guiotto: Well, this is the rectangle RKJ.
17:45:710Paolo Guiotto: Okay? You take points… No.
17:49:700Paolo Guiotto: You take points.
17:52:120Paolo Guiotto: Here.
17:53:940Paolo Guiotto: And you compute the smallest value of F on that segment. Now, the volume of this parallel pipede that you have here, which is inscribed, will be the volume of this
18:08:990Paolo Guiotto: will be a volume KJ, let's say, will be the product between the area of the basement, which is the product of length of XK plus 1 minus XK, this is the side on the x-axis, YJ plus 1 minus YJ, this is on the y-axis.
18:28:880Paolo Guiotto: times the hate HKJ.
18:34:10Paolo Guiotto: Okay?
18:35:650Paolo Guiotto: Well, let's introduce some shortening for this notation. Let's call this guy DXK.
18:42:300Paolo Guiotto: And this will be DYJ.
18:46:250Paolo Guiotto: Okay? So whenever I write DXK, it will be XK plus 1 minus XK, DYJ will be YJ plus 1 minus YJ.
18:54:580Paolo Guiotto: So now, this is a single volume. I repeat this for all the rectangles of the subdivision for the domain D.
19:03:970Paolo Guiotto: And I sum up all the volumes, so I get what is the inferior sum associated to this subdivision pi, where pi is now a double subdivision, so a subdivision for points X, so it will be a subdivision like XKYJ,
19:22:630Paolo Guiotto: for index K and J varying from 1 to, let's say, N, or if you want, we can also admit that we have n points on the axis X and M points for the
19:38:20Paolo Guiotto: Y axis, it doesn't matter. So this is the sum of all over all KJ,
19:45:360Paolo Guiotto: We should be precise with the notation, so it should be 0 to capital n minus 1 of these volumes, volume
19:54:390Paolo Guiotto: KJ… So, this is the sum of a KJ of, say, HKJ,
20:04:370Paolo Guiotto: D, X, K, D, Y, J.
20:10:800Paolo Guiotto: Okay? This is the inferior sum. Now, this is… this… Ease… and approximation.
20:22:290Paolo Guiotto: by defect, off.
20:26:320Paolo Guiotto: Not the area of the trapezoid, because now it will be a volume. Of volume, of the trapezoid.
20:35:200Paolo Guiotto: of F.
20:37:210Paolo Guiotto: So we give a name, to the best of these approximations. So we define the… well, sorry.
20:44:520Paolo Guiotto: the inferior volume, of the trapezoid of F.
20:54:830Paolo Guiotto: So, this is the best of the approximation by the fact.
21:00:250Paolo Guiotto: since these are values that are supposed to be smaller than the true volume, to take the best one, I will have to take the biggest possible, so the maximum, or better, the supremum of these inferior sums over all the subdivisions.
21:18:670Paolo Guiotto: In a similar way, we will build the best of the approximation by XS. So, in a similar…
21:31:950Paolo Guiotto: Wait.
21:33:800Paolo Guiotto: we define.
21:36:840Paolo Guiotto: the superior sum associated to a subdivision pi, which is, again, a sum of KJ. Now we will have a capital HKJ, this 8, times DXK.
21:51:430Paolo Guiotto: DYJ.
21:53:420Paolo Guiotto: Well, this number…
21:57:350Paolo Guiotto: is now an 8 in such a way that the parallel epiphet contains, so exactly as above, I have to take the maximum of the function on the little rectangle, so the maximum is replaced by the supremum of the function, FXY, 4.xy.
22:15:900Paolo Guiotto: in the rectangle RKJ.
22:22:490Paolo Guiotto: And, the best of these approximations And the…
22:28:550Paolo Guiotto: That we call the superior volume, of the trapezoid.
22:35:720Paolo Guiotto: of the function f, which is now the best of the approximation by XS. These are approximation by XS, so they are supposed to be bigger than the
22:46:90Paolo Guiotto: through volume, so I have to take the smallest possible, which is the interim of this superior Samsa, over all possible subdivisions pi.
22:58:830Paolo Guiotto: Exactly as for the one-dimensional case, in general.
23:06:320Paolo Guiotto: The superior volume is greater than the inferior volume.
23:12:320Paolo Guiotto: And they might be different.
23:15:430Paolo Guiotto: There is a similar example, as the Dirichlet function, and it… Might.
23:27:590Paolo Guiotto: B… That the superior volume is actually greater than the inferior volume.
23:34:370Paolo Guiotto: But if they are equal, we define this as integral functions. So, definition… we say, That.
23:47:270Paolo Guiotto: F.
23:50:480Paolo Guiotto: is integral.
23:57:230Paolo Guiotto: on.
23:58:670Paolo Guiotto: D.
24:00:330Paolo Guiotto: rectangle.
24:03:00Paolo Guiotto: If… The superior volume coincides with the inferior volume.
24:09:480Paolo Guiotto: In that case, the common value is what we call the integral.
24:13:870Paolo Guiotto: In this case.
24:18:650Paolo Guiotto: the integral on domain D of the function f F of X, Y, DXDY.
24:26:50Paolo Guiotto: Now, you see why there is this notation, the X to the Y. This reminds of the fact that when you do these volumes.
24:33:710Paolo Guiotto: You see that there is a predator.
24:35:960Paolo Guiotto: value, capital H or little h, which is basically
24:39:850Paolo Guiotto: the maximum or the minimum of the function on that rectangle. You have to think that if the function is continuous.
24:47:110Paolo Guiotto: the two values will be very close, because the rectangle is very small. When the rectangle is very small, there is no difference between minimum and maximum, and they coincide with the value of the function. This is the idea, no? So you have product of F times DX times DY. That is reflected from this notation. So, we define this as the common value of these two.
25:10:390Paolo Guiotto: quantities.
25:13:610Paolo Guiotto: Well, a remarkable fact is that this proposition
25:18:820Paolo Guiotto: Of course, to check directly with the definition that the function, here we are still with positive functions, I remind you, right?
25:28:400Paolo Guiotto: That this is true is difficult, because it's infeasible in general, to compute the,
25:35:560Paolo Guiotto: the upper volume, the lower volume, the superior volume, the inferior volume. It's impossible in practical cases. So, it is good to have nice conditions that ensures the existence of this quantity.
25:51:230Paolo Guiotto: And exactly as for integrals in one variable, you may remind that there is an important factor that continuous functions are integral.
26:01:300Paolo Guiotto: Okay? This is also here. If the function f is continuous on D, rectangle.
26:12:80Paolo Guiotto: F positive.
26:14:140Paolo Guiotto: then… there exists the integral on the… of the function F. Well, very often for this, let's say.
26:22:590Paolo Guiotto: general results, we do not write integral f , it is understood that this is the double integral.
26:32:910Paolo Guiotto: Okay, so we know that continuous functions are integral
26:37:990Paolo Guiotto: Before we move to the case,
26:41:280Paolo Guiotto: No. Okay, this is… this is the definition of integral if the domain is a rectangle. Now, what if the domain is not a rectangle?
26:57:990Paolo Guiotto: what…
27:00:340Paolo Guiotto: if, as it is in general, because in general, the domain won't be a rectangle. If domain D is not
27:08:810Paolo Guiotto: a rectangle.
27:12:170Paolo Guiotto: Now, we divided the discussion into two sub-cases.
27:17:20Paolo Guiotto: sub K is 1, we assume that the domain D is bounded first.
27:22:770Paolo Guiotto: So… If, domain D is bounded.
27:32:470Paolo Guiotto: Now, if it is bounded, Whatever is the domain D,
27:39:360Paolo Guiotto: Okay, like this, it is bounded.
27:42:220Paolo Guiotto: It will be contained into a rectangle. There will be a big rectangle, a rectangle big enough that contains that domain, right?
27:52:180Paolo Guiotto: So let's call that rectangle. So… don't exist.
27:58:630Paolo Guiotto: There exists, R, rectangle, such that domain D is contained into rectangle R.
28:09:210Paolo Guiotto: Okay, my function is defined on D.
28:12:620Paolo Guiotto: I want to extend, in a trivial way, to the full rectangle R,
28:18:400Paolo Guiotto: by just putting 0, the function, here. So I say, let's say, let… Let's extend…
28:31:640Paolo Guiotto: F, function of XY, To…
28:37:420Paolo Guiotto: just defining that, F… well, this is a new function, in fact, FXY is the old FXY when the point XY is in the
28:52:650Paolo Guiotto: and 0 when XY is not in D.
28:59:350Paolo Guiotto: Now, there is a standard way to write this instead of using a different letter. We identify this in this way as F of XY,
29:09:760Paolo Guiotto: Times this quantity here, one… One… what works if it doesn't work rightly?
29:19:380Paolo Guiotto: 1D, X, Y, Well, this function is also called the indicator.
29:29:620Paolo Guiotto: off… The…
29:32:890Paolo Guiotto: So what is the 1D? It's a function which is 1 when point XY is in D. 1D is a function which is equal just to 1 if point XY is in D,
29:45:40Paolo Guiotto: and 0 if point XY is not in the
29:51:120Paolo Guiotto: I know that if XY is not in the domain of F, this quantity has not any meaning, okay? It's no F, but we…
30:00:690Paolo Guiotto: That's fine, it's not in F, whatever is this value, we would multiply by 0, so we would see the number here, and we would get 0. So we normally just write this as F times the indicator of D.
30:16:730Paolo Guiotto: So, in this case, we give this definition.
30:22:660Paolo Guiotto: So, F function on domain D contained in R2, positive valued.
30:31:110Paolo Guiotto: bounded.
30:35:260Paolo Guiotto: F bounded, on, D bounded.
30:44:780Paolo Guiotto: we define
30:49:180Paolo Guiotto: the integral on D of the function F as the integral of… on R,
30:56:870Paolo Guiotto: of F times the indicator of D, Where?
31:03:540Paolo Guiotto: R is a rectangle, that contains the domain D,
31:11:440Paolo Guiotto: And of course, provide this right integral makes sense.
31:16:810Paolo Guiotto: Provided.
31:21:120Paolo Guiotto: these… mates.
31:26:940Paolo Guiotto: sensor.
31:32:90Paolo Guiotto: So, now we have extended the definition of integral.
31:36:640Paolo Guiotto: For a positive bounded function to a bounded domain.
31:42:30Paolo Guiotto: The final step to have the definition of integral for the generic domain DE Is do the following.
31:49:790Paolo Guiotto: So now, imagine that we have a domain D, which is not necessarily bounded, so imagine that can be unbounded.
31:57:930Paolo Guiotto: You cut off this on a finite rectangle.
32:03:270Paolo Guiotto: And you compute the limit when this rectangle gets bigger. So, let's say that this is rectangle from minus n to n in the horizontal, and in vertical from minus n to L. So, it's a square.
32:17:00Paolo Guiotto: So let's call RN the rectangle, minus NN times itself. So it's a square.
32:30:710Paolo Guiotto: Okay, now this is bounded, and if I intersect RN
32:35:840Paolo Guiotto: this rectangle. With domain D, I have a bounded domain.
32:41:470Paolo Guiotto: Because it is exactly what you see, colored in red here.
32:47:960Paolo Guiotto: So this is, in red, the D intersection RN.
32:53:110Paolo Guiotto: So what I do is, I compute the integral on the intersection Rn, of my function F.
33:01:640Paolo Guiotto: And then I send n to infinity if this limit exists.
33:08:500Paolo Guiotto: And I call this, by definition, the integral on D of the function F.
33:15:690Paolo Guiotto: provided that
33:21:190Paolo Guiotto: the… limit.
33:25:210Paolo Guiotto: exists.
33:28:610Paolo Guiotto: Fine.
33:33:490Paolo Guiotto: And now we have the definition of integral.
33:36:90Paolo Guiotto: For positive function on any domain, okay?
33:40:200Paolo Guiotto: Let's stop here, and tomorrow we will continue this construction, now taking functions which can be also negative, positive, negative, both signs.