So, next Thursday is a second midterm exam. Who could have thought? It feels like the first one was just yesterday. Yeah, it was much warmer, you know. So, next Thursday, November 5th, right here, for most students, as you know last time, we had an overflow and again, I apologize for this. I apologize to those students who didn't have comfortable seat. So, we'll avoid this situation, this kind of situation this time, we'll have some reading room. And unfortunately, that means that two sections, students from two sections, those which are led by Dario, will meet at a different undisclosed location. No, I just didn't want to write it. You can find it, Dario will tell you. If you are in Dario's section, Dario will tell you. And it's also on the class home page. It's in Barrow's hall. And if you're not in the section of Dario, then you don't need to worry about it. You should just come here, okay? And on the home page, you will find more information, but it's basically the same deal. Someone asked me for the cheat sheet for the page of formulas. Can we use the old one, the one from the first midterm? And the answer is no. You have to make a new one. It will also be on one side of it. It has to be on one side of it, on one side of a standard size sheet of paper, okay? But don't worry, we're not going to test you on the material of the first midterm anyway, so it doesn't really, it doesn't make a difference. The midterm will cover the material after the first midterm, okay? Everything that we've done up to now, since the first midterm up to now, up to this week, including this week. And there are, on this webpage, you will find problems for review, and you will also find a mock midterm. It's not posted yet. I will post it tonight, the mock midterm, okay? Any questions about this, about the exam? Next Tuesday, we'll have a review lecture as before in which I will discuss all this material and kind of try to put it in perspective, and you will have a chance to ask me questions about it also. Yes? Oh, so there will be no quiz next week. So it's the same deal as last time. Mock midterm, the idea is to give you problems which kind of look similar to what you should expect, of course, yes. Well, difficulty is in the eye of the beholder, of the beholder, right? So for some people, it will be more difficult, some people will be less difficult, but it should be in the same ballpark, okay? All right.  We talked about triple integrals, and we talked about special coordinate systems for triple integrals, the cylindrical and spherical coordinate systems, okay? So I wanna say a few words, general words about triple integrals. So when I say triple integral, it means that we have something like this, where E is a region in the three-dimensional space, something we denote by R3. F is a function, and dV is what we can think of as kind of a measure of integration, which we would usually write as dx dy dz, but in some order. But in fact, we know that we can use other coordinates as well, and today we'll talk more about the possibility of using different coordinate systems. So the basic setup for calculating this integral is, as I already explained last time, and as has been explained last week, but I want to repeat it one more time, is that you want to represent this triple integral as an iterated integral. In other words, you want to break it into steps where at each step, you are just evaluating a single integral, an integral in one variable, something which we learned before in one variable calculus. And the way you do it is, first of all, you find the projection of your three-dimensional region. So you have some three-dimensional region here, but the first thing you do is you find its projection on one of the planes, one of the coordinate planes. Let's say xy plane, but it could be xz or yz plane. So that's the projection d. Let's call it d. So this e is three-dimensional. This is three-dimensional, whereas d is two-dimensional. It's a two-dimensional region, which is in the xy plane, okay? So once you do that, you can rewrite your integral as an iterated integral where you have a double integral over d, and here you put dA, which is the measure of integration in the two-dimensional space, which is essentially dx dy. And then you have an integral of f with respect to the remaining variable where you have to put the limits. So there'll be some limits, alpha and beta, which in general will depend on xy. So here xy will be in d, and for each xy, the limits for the single integral will be from some alpha of xy to beta of xy. Now the way we write it is slightly confusing, and you saw that confusion last time at the lecture when we discussed it. It is important to remember that you read the integral from right to left. I mean, that's the way it's done. That's a convention. You read it from right to left, which means that you first do this one. You first, this is the first step. You first evaluate this single integral, okay? And then in the second step, you evaluate the resulting double integral. So once you evaluate this, you will get something which will depend on x and y. This will be an expression depending on x and y, which you then integrate over d in the xy plane. So we break it into two steps. The first step is a single integral with respect to the remaining variable along which you have projected. And the second step is integrating over the image of the projection, which is d. This is the image of the projection. So we can actually write it further, further as a combination of single integrals because this double integral in turn can also be written as an iterated integral, okay? So that means that writing integral over d, dA as an iterated integral with respect to x and y, we actually end up with three single integrals. So here we also have a choice. There is a choice. Which one goes first, x or y? Let's say, and what do we mean by which goes first? Which one will appear first or which one you will integrate first? As I just said, you read it from right to left. So this is actually, these roles are reversed, right? So let's write it like this. Let's say it's dx and then it will be dy and then finally it is dz and then you have your function f. Here I wrote f dz and here I wrote dz f. It doesn't really matter as we discussed last time. It doesn't matter in which order you write f and dz or f and dy and so on. I find this form a little bit more convenient to remember, more intuitive because the variable of integration is written right next to the integral. So you will not forget which one you're integrating. But the important thing to remember when you look at this formula is to read it from right to left. You read it from right to left which means that this is the first step. Integrate over z. I'm skipping the limits. Of course there will be limits. For example, here will be limits alpha of xy and beta of xy like on that blackboard. But let me just skip that. I'm just giving you kind of a schematics of this process. So the first step is integrating f over z. The result will be a function which only depends on x and y because we have integrated out z. Think of it as kind of removing the dependence on z by averaging out over z, over the effect of z. The result is a function of x and y. That function you integrate over y. This is step number two. This is the second step. And finally, you integrate the result which will now be only function of x over x so that the end result is actually a number independent of all of these three variables. So at each of the steps you get rid of one of the variables. You're kind of averaging out over that variable so that the result does not depend on that variable. If you are doing an integral and suddenly you notice that you have integrated over z but the result somehow depends on z. That you carry z somehow to the second or first or third step, you may have made a mistake somewhere. There should not be any dependence on z on the second step. There should not be any dependence on y and z at the third step, you see. So this is a very easy test to see if you are doing things in the right way, okay? Any questions about this? So it's kind of an rotational issue but you just have to remember that you kind of, this is, because it is actually, it is very, it makes sense because you're integrating this expression. Okay, but what is this expression? This expression is obtained by integrating this. I mean integrating respect to y, this expression.  It makes perfect sense that you go from right to left and not from left to right, okay? Now there is a very special case of a triple integral. There is a special case and that's the case when the function f which in general is just any function of x, y and z. Right, so f is really f of x, y, z and general could be anything, could be, you know, x squared times e to the y times cosine z or whatever, whatever you want. But there's a special case when this function is actually just equal to one. Okay, that's the simplest function you can imagine. Well, save zero perhaps but if the function is zero, the integral is zero so there is nothing to discuss. So let's talk about the case when the function is actually equal to one. And also let's assume in addition that our region in the three-dimensional space, e, is the region above this domain D, domain D in the two-dimensional space under the graph of some function of a function. Let's call this function f capital.  It's a function on e which is a region in 3D, on R3, on the three-dimensional space. But let's suppose that the region itself is, there is some function so let's draw a picture like this. So let's say D is a kind of a rectangular region. And here we have some function like this. So we have some, this is a graph of a function. So let's say that this is a graph of a function z equals f capital of xy, this. So this is what we discussed when we talked about double integrals. When we talked about double integrals, we discussed the fact that a double integral represents the volume under the graph. The volume under the graph. So the volume under the graph, I put it in quotations. What I mean is, first of all, that I'm assuming that this function is greater than or equal to zero on D. For otherwise it would not be above the xy plane everywhere on this region D. This is region D, which is now kind of a rectangular region. I have chosen to simplify things. Okay, and also when I say under the graph, I really mean under the graph but above the xy plane. And also on the sides, it's confined by the boundaries of D, which in this case are just vertical planes like this. Okay, so when we talked about double integrals, we discussed the fact that volume is equal to this volume under the graph is equal to the double integral over D of f capital xy da. So now I wanna show that you get exactly this answer if you also by calculating a triple integral but with the function one. You see, so this is one of the points which many people find confusing that the volume under the graph is actually given by double integral but it's also given by triple integral. But in the special case, in the special case is that the function which you integrate, the function of three variables is one. In other words, what I'm integrating now is dv. You see, this is the most general triple integral. It's an integral of f dv over e. But now f is one. So if you want, I can put one times dv but it doesn't matter, one times dv is dv. Okay, so what is it equal to? Let's use this formula. First of all, so I project onto d and in this case I surely can do that because my region has a very nice projection onto the xy plane specifically because it's under the graph of a function. And so I will have here dA and then I will have to put the limits here, alpha and beta and integrate my function. But what are the limits in this situation? In this situation, the lower limit is always zero. I'm always counting z from zero because the bottom lid of this region is just a rectangle on the xy plane. So that means z is equal to zero at the bottom. Here z is equal to zero across the bottom. So that's why here I actually put zero and then I'm integrating z from zero to something which lies on this graph. So that's sort of the top of this region. And so for the top point of that region, what is z equal to? z is equal to f capital of xy, right? So that means that my upper limit is f capital of xy. And then I have to put fdz, but f is one again. So it's one times dz, right? So this is very easy to find, right? This is just a very special case of this general setup. I have to take the anti-derivative to calculate this integral. I simply have to take the anti-derivative of one. An anti-derivative of one is z plus a constant, but anyway, we are going to take the difference of the limits. So it's going to be z with the limits f of xy and zero. So what is that? That's just f of xy. So this inner integral, which as we have agreed should represent the first step of the calculation, this inner integral is simply equal to f of xy, right? Simply because the function is one. This argument would not work necessarily for the general function. For general function, we'll have some f here, which depends on xy, right? And so xyz, in fact. So a general function depends on xyz. So you wouldn't be able to easily find the anti-derivative. But for the function one, the anti-derivative is z. I mean, this for sure we know, right? So that's the answer we get. And once we get this answer, let's substitute it here. So we get double integral of f capital of xy da. So that's the same answer as before, right? The answer before was the integral of f of xy da over d. And that's exactly what we got by integrating the function one over this entire three-dimensional region, okay? Does this make sense? Yeah? Any questions about this? So the conclusion is that this integral is actually equal to the triple integral over E, where E is this region inside. This is E. E is this three-dimensional region, which is drawn here under the graph. So one more time. The volume of this region under the graph can be represented in two ways. One is triple integral, but a very special function, namely function one, okay? And can also be represented as a double integral of the function f capital of which this is the graph over the region D. The reason why the two things are the same just is just the result of calculation of representing the triple integral as an iterated integral where at the first step we very easily evaluate what the result is for this special function one. That's what we get, okay? So now, of course, you can ask if the integral of the function one over this three-dimensional region is nothing but the volume of that region, what about the general case? What about an integral of a general function f of x, y, z? Is this also some sort of a volume? Well, there are two ways to interpret the triple integral for a general function. For a general function, function f of x, y, z. Not necessarily one, not one, but some general function. There are two ways to interpret two ways to interpret the integral. Well, the first way is, let's suppose you want to interpret it as some sort of a volume. And let's argue by analogy, okay? Suppose we have, let's go down one dimension lower and let's look at the integral of this function over the two-dimensional region, double integral. This is a double integral, but we interpreted it as a volume of something three-dimensional, you see. So a double integral of a general function actually corresponds to a volume of something three-dimensional, of dimension greater by one. So you start with a double integral, but you get a volume of something three-dimensional. That's not surprising, because likewise, if you have a single integral of a function one variable, what does this represent? It represents the area of something two-dimensional, namely, the area under the graph of this function F. All right, let's call it, let's choose a different variation for this function so that we don't confuse different things, right? So a single integral corresponds to the volume, but in this case, we call volume area, that's the terminology, proper terminology, area of something under the graph of this function, and this something is actually two-dimensional, even though you start with a single integral. If you start with a double integral, so two is the number of integrations, you get volume of something three-dimensional, right? So what will you get if you take a triple integral? You get a sort of a hyper volume of something four-dimensional, right? So I mean, that's, if you insist on interpreting the integral as a volume of sorts, you have to go to, you have to introduce one extra dimension, and so you will get kind of a hyper volume. I don't know, there is no good term for it.  Hyper volume of a four-dimensional region, region which is the region under the graph, under the graph, again, quotations of f of x, y, z. So you see there is a complete analogy. A single integral of a function g will be the volume or area under the graph of this function, and a double integral will be the volume, a double integral of a function in two variables will be volume of something in three dimensions under the graph, and the triple integral of a function f is a hyper volume of the four-dimensional region under the graph of this function, so you have to introduce an additional variable u and say that the graph is given by, so you have a four-dimensional space with coordinates x, y, z and u, and you write u is equal to f of x, y, z. Now, of course, we cannot really visualize it because we can only visualize three dimensions. There is time, so you can think of a region in space time if you want, and the extra variable u we can think of as time, but still it's kind of, so if you think about it, you can imagine this because you can just count, you can look at a certain region in three-dimensional space and you can count time from certain point zero to some other point which depends on x, y, z. So that will represent a kind of a region in space time, and what you're calculating is a hyper volume of this region. Now, this is, but still it's a little bit disconcerting because we don't really have good intuition for four-dimensional space, so it is natural to ask, is it possible to give it a three-dimensional interpretation? Well, we can go back to the double integral and ask the same question there. We can ask what if f capital, the general function, we know that this integral can be thought of as a volume of this 3D object, but can it also be interpreted in terms of just two-dimensional geometry? And the answer is yes. It is, if you think of this function as a density function, mass density function, then this will give you the total mass of the laminar or some kind of a plate which occupies this region. So this is something which we discussed last week, the total mass as a double integral. So in other words, you are thinking of, you can think of this calculation where the function is one. You can think of calculating the volume, but you can also think of it as calculating the mass. But in a special case, when the density function is one, if the density function is one, then calculating the mass is the same as calculating the volume. That's why we get the volume. But in general, the density function could be more complicated. Your object may not be homogeneous. It could have some heavier parts and lighter parts. And you want to kind of average out all of them. You want to find the total mass. And the total mass would then be the integral of that function or that density function. So from this perspective, you don't have to introduce an extra dimension. If your object, if you have a double integral, you have region D. You think of this function as a density function and you simply think that you're evaluating the mass of an object occupying this region where this function is a density function. And likewise, if you have a three-dimensional object which occupies some region E, you can think of the triple integral with the general function F as the mass of that object. You see? So that's the way to interpret it. So this is a total mass, total mass of a solid E with mass density function F. Any questions about this? So our last topic in this chapter about integration is about general coordinate systems and the general change of variables. We have already discussed some new coordinate systems in two and three-dimensional spaces. We have talked about the polar coordinate system and we talked about a spherical coordinate system. Also cylindrical, but cylindrical in some sense is not really a new coordinate system. It's really more of a cousin of the polar coordinates. So now we would like to talk more systematically about different coordinate systems in two and two-dimensional space. We can also do it in three-dimensional space, but we'll just focus on the two-dimensional space. And in 3D, the conceptually is the same, but calculations are much more complicated. So we basically will mostly confine ourselves to the two-dimensional case. And as always, when we talk about this conceptual matters, it's always good to look at a simplified version, which in this case would mean just single integrals, integrals in one variable. And see how things work for single integrals. So let's recall how this works. Change of variables. Let's change a variable in a single integral, f of x dx. So we know how it works. We simply write x as a function, let's say g of some new variable u. And then what we do is we substitute this g of u instead of x in the integral. But in addition, we realize that dx is not du, but rather dx is g prime of u. So what it means is that it will be wrong to just write the integral as f times du, but instead there is this additional factor, which is g prime. So we recognize that even in one-dimensional case, there is this additional factor. And this factor actually should be thought of as being conceptually the same kind of object, the same kind of phenomenon as the factors which we discussed in the case of polar coordinates, namely the factor r, and the factor which we discussed in case of spherical coordinates last time, which is rho squared sine phi. So these are all brothers in some sense. So they all appear for the same reason. I mean, although this is in one-dimensional case, this is two-dimensional case, this is three-dimensional case, but all of these factors are essentially, they are the kind of distortion factors, the area or volume distortion factors which happen due to the fact that we introduce a new coordinate system. So our task right now then is to understand what exactly this factor is in the case of a most general coordinate change. Not only in one-dimensional case, but in the case of two and three variables. And like I said, mostly two variables, you see. So we would like to develop a formalism of change of variables, but for double integrals. That's what we like to do. For which this r, this polar coordinate case will be just a special case. So let's try to figure this out to see how this works. So in the one-dimensional case, we have a given variable x and we have a new variable u. And we have a law which allows us to express one of them in terms of the other. And that's this formula x equals g of u. But now we have, now let's try to develop, let's try to think of something similar in R2. So in R2, we have two given variables or coordinates, coordinates x and y. And we would like to introduce two new coordinates. Suppose we introduce new coordinates. Let's call them u and v. So here's an example. Here's an example. Polar coordinates, something we have learned already. UV as this new coordinate system is our old friend polar coordinate system.  So when we do that, we should be, when we say that we haven't, we haven't introduced a new coordinate system, what do we mean by that? We mean that we can express each of the old coordinates x and y in terms of this new coordinates in a one-to-one way. In a way that establishes a one-to-one correspondence between points with x, y coordinates and points with u, v or our theta coordinates. In the case of polar coordinates, we know those formulas and we know that for every r and theta, where r is a positive number and theta is between zero and two pi, we have a particular value of x, y, and conversely. So it's a one-to-one correspondence. It's not strictly speaking correct, what I said, because of the special point zero, the origin, point O, the origin, for which if r is equal to zero, then theta is not well-defined. So except for this one point, this is this map which I would like to think, I would like to think of this formulas as giving me a map, a correspondence from between two coordinate systems. This is our true physical coordinate system, x, y. This is what we see. This is a blackboard and this is how we would represent points on this blackboard. And here is a kind of a fake auxiliary, imaginary coordinate system with coordinates r, r, and theta. All right, let me write like this. In fact, many of you will recognize that we have used this before when we tried to draw parametric curves and polar coordinates. We did use this kind of coordinate system. So it has nothing to do immediately with the plane because the way a point here will give us a point on the plane is through this formulas. But I would like to think of this formulas as giving us a map from this auxiliary coordinate system, this imaginary coordinate system, which we have in our head, which doesn't really exist, to the real coordinate system on the physical coordinate system on this physical plane. So for example, let's say I take the region where theta goes from zero to pi and r goes from zero to one. So that's a very simple region on this imaginary plane, right? It's a rectangle. But what does it correspond to in the physical world? What do you think? Circle. Okay, close, warm. So the circle will come if we fix r. If we just look at this interval, this interval will correspond to the part of the circle of radius one, which lies above the x-axis because I set from zero to pi. If I set from zero to two pi, it would be the entire circle. But because I set from zero to pi, it will be half a circle, right? Everyone agrees with that? Okay, but that's not the entire region, right? The entire region is composed not just by this interval, but also by all other intervals which lie beneath it, right? And so those intervals will correspond to circles of smaller radius, right? And so all together, here, they give me this rectangle. But in the physical world, they give me this half a disk, upper half of the disk of radius one. So you see something very interesting happened under this transformation. We started out with the rectangle and we ended up with upper half of the disk. And I would like to say that these formulas give me a transformation from this region to this region. What do I mean by this? I mean that each point here will correspond to a particular point in here. And in fact, this correspondence is one-to-one. In other words, no two points will go to the same point. Right, and like I said, this is almost true. Unfortunately, strictly speaking, it's not true for this interval. This entire interval where r is equal to zero will go to this one point. We'll go to this one point. So there is a subtlety. But the point is that we are studying here two-dimensional region. And something bad happens on this interval, which is one-dimensional. This is what mathematician would call measure zero. It's not measurable. Its measure is negligibly small. It's actually zero. So as far as computation of integrals is concerned, it's not going to change anything, okay? But if I wanted to be more precise, I would take, say, not this rectangle where r goes from zero to one, but I could instead take a rectangle where r goes from one-half to one. Then I would get, instead of upper half of the disk, I would get upper half of the annulus with the shorter radius one-half and the longer, larger radius one. So if I do that, actually, the mystery becomes, the transformation becomes less mysterious. You can certainly believe that I can transform this strip into this. What I do, I kind of twist it. That's what this transformation, given by this two-formal, does. It kind of twists and expands and stretches in a certain very controlled way, the fabric of this original domain, to create a domain in the physical world. That's what this information is, you see. And then if you think about the big one, it's sort of, if you get closer and closer to this red part, it becomes closer. The way I stretch it is I stretch it on this side in the same way, but I stretch it on the lower side, not stretch, but in fact, squeeze to make it shorter and shorter and shorter. And as I approach here, actually, the whole thing opens up into half of the disk. You see what I mean? You guys see what I mean? Yeah? Okay. So this is exactly the kind of formulas I'm talking about. This is a one-to-one transformation, one-to-one. So each point here corresponds to point here and conversely, each point here corresponds to one and only one point here. When I'm in this situation, I would like to be able to write a double integral of a given function f of x, y, d, a, as an integral with respect to u and v. And of course, so I'm kind of in the same situation as I was when I talked about the one-dimensional case. The first step would be to substitute the formula for x and y in terms of the new coordinates. So this would be, let's say, in general, here I have x is r cosine theta. So it is some function of r and theta. But in general, if I have a coordinate system uv, not necessarily polar, but more complicated, I'll give you an example in a minute of another change of variables like this. Let's say this will be some g of uv and this will be some h of uv. So the first step would be to just substitute g of uv and h of uv instead of x and y, right? And then just write du, dv. But this is not enough. The point is that there is always an additional factor. There is always a distortion factor here, which we are now going to calculate. An example of this factor is the factor r, which we have found in the case of polar coordinates.  What we want to do is to have a more conceptual way of deriving this factor, not just for polar coordinate system, but for a more general coordinate system uv as factor. And of course, it's easier to explain having a particular example in mind. Well, here's one example, the polar coordinates, but in polar coordinates, we have already calculated what happens. You have a question? Yes? You are asking, would this have the same area as this? The answer is no. I'm sorry? Very good point. So this is a very good question. So when I say one to one, doesn't it already assume that this will going to have the same area, right? And the answer is no. And it's very easy to see that. If you have, you know, if you buy a garment, which stretches, you can see immediately the answer, right? Unfortunately, none of what I'm wearing today can do the trick, but you can imagine that you have something which is stretchable. Maybe, no. Okay, let's just work with your imagination. You stretch it, right? A stretch is a perfect, a what? This, yeah. But my hands are, I don't want to spoil it. But that's the good one. Can I use yours? No. But you see what I mean, okay. So let's just use our imaginary clothes. So, okay, so when you stretch it, stretch is a perfect example of a change of variables because there is a one to one, each particle, we don't create any new particles, right? Clearly, each point goes to a particular point and it's one to one. It is like, okay, it's even easier to do in one dimensional case. You can stretch an interval and make it, you know, just say a map X goes to two X. You just, everything gets stretched by a factor of two, right? So it is one to one, but when you stretch, the area becomes bigger, right? So just the fact that it's one to one correspondence doesn't mean that the area's state is same. And that's exactly what we are fighting for. This factor is precisely the distortion factor for the area. But the point is, which is maybe not immediately obvious, is that it would not be a good idea to actually measure this area and then measure this area and take the ratio of these two areas, right? Because it would be okay if our function which we were integrating, function F, if we chose the function one, if we chose function one, then we would just be calculating the area itself. And so if the area was something before, the area, the two area would be this and the area which we would naively think is correct in the imaginary R theta coordinate system would be something else and there will be some distortion factor between them, right? But that's not a good way to go because you see the point, when I make this transformation, the distortion factor is not the same everywhere. Because I explained, for example, that on this guy, let's look at this guy, at the top interval. The top interval becomes the part of the circle, segment of the circle of radius one, right? Whereas this guy, which has the same length on the imaginary coordinate system, it becomes part of the circle of radius one half. So certainly this is shorter than this. So our distortion is not like just stretching, but it's kind of stretching in different way at different places. So what we need is a local distortion factor. You see a local distortion factor, which in other words, we want to see not how this entire area gets distorted, but we want to see how this very small area around a particular point gets distorted. And that's the local distortion factor and that's exactly the factor which we will put here. You see? Any other questions? To see how this works, maybe it's better to... Sorry? You have a question? No? All right. So I want to give you another example of coordinate change before we get to calculating this factor, just so that we have sort of a better variety of examples. So here is a second example, which is actually straight from the book. It's... And this I want to use as a kind of a motivation for everything we are doing right now because perhaps for you it's not so clear yet why we are discussing this issue. But, well, polar coordinates is already a good illustration because we know that some of the integrals become much simpler in polar coordinates. But here is another example where the coordinate system of choice is not polar coordinate system. So this is 15.9 number 20. And so let's say you need to calculate this integral over some region D, where D is described as follows.  So this is X plus Y equals zero. This is X minus Y equals zero. This is X plus Y equals two, or X minus Y equals two. This is negative two. And then there is another one, which goes like this. This is two, this is three. So it is actually rectangle because these two planes are parallel. Well, my picture is not so great, but they should be parallel and these two should be parallel. Also, right? So it is actually rectangle, but it's a rectangle which is rotated. So first of all, if you try to do this integral in the usual way by projecting onto the X line or projecting onto the Y line, it's going to be difficult because you see that first of all, you have to break it into pieces and then the bounds will be functions, will be some not very complicated linear functions. But then you look at the integrand and you see that it's going to really look awful if you try to do this, okay? So what this picture immediately suggests is that we should try to find another coordinate system in which this integral will simplify. And what is this coordinate system? Well, what if I told you that instead of calculating this integral, you would have to calculate the integral of U times E to the, let's say V, V times U times V, DU, DV, where over some region D prime, where D prime would be simply U going from, so the U going from zero to two, slightly different way, would also be rectangle, but this rectangle would be turned in the right direction. So this would be U and V and this would be two and three. This surely is much easier to calculate, right? Because this immediately breaks into iterated integrals and you find the answer right away. So what have I done? How can I get from this integral to this integral? I simply introduce new variables, U and V, where U is X minus Y and V is X plus Y. If I do that, that my bounds, which were X minus Y equals zero and become just bounds U equals zero and U equals two, which is what I did here, from zero to two and also V from zero to three, which is what I did here. So the region simplifies. It becomes, it kind of turns, we turn it so that it really stands straight like this.  But it is not true that this integral is equal to this integral because, as I said, there has to be some factor, which is introduced, some factor introduced here. And to understand why we should introduce this factor, it's very easy to just, so to look at the map, the transformation. So we have here, we have here this rectangle in UV coordinate systems. So see, this is an example also of this kind of transformation, but it's simpler because it's linear. And those of you who have taken or maybe are taking math 54 will recognize a linear transformation. In fact, it is a rotation by 45 degrees coupled with an expansion by a square root of two. So this is two and this is three. And there is, these formulas give me a map because actually here I have expressed U and V in terms of X and Y, but I can also go back and I can express X and Y in terms of U and V. And I will find that X is U plus V over two. How to find this? Well, this is U and this is V. Take the sum. When you take the sum, the Ys will get canceled and you'll get two X. So if you divide by two, you'll get X. And likewise, Y is equal to negative U plus V divided by two. Right? So that's this map. Each point here will correspond to a particular point here. But now, why should there be a factor, a magnifying factor or distortion factor? It looks like, well, this is a rectangle and this is a rectangle. Perhaps we can just get away without any factor. This is actually easy to find out because in this case, the stretching, the stretching that occurs is actually homogeneous. There's the same distortion factor or stretching factor everywhere across the entire region, unlike this case. So in this case, the stretching factor is actually R. So it does depend on the variables, on the polar coordinates, R and theta. In this case, it's going to be constant. And so to find out what it is, we could actually look at this globally. And we can just, in other words, assume that, suppose that instead of this function, I would just take function one. Let's suppose I take function one instead of this. If I take the function one, what I'm calculating is just the area of this guy. And this area is very easy to calculate, in fact. Because, well, you just have to use the Pythagoras theorem. So the point is that here you have three. And so then you can find what this is. This is the right rectangle. And this has 45 degrees. This is 45 degrees and this is 45 degrees, or pi over four. So this length is equal to this length divided by square root of two. So this is three divided by square root of two. Let me use yellow. This is three divided by square root of two. And this comes from a triangle, same kind of triangle, but now the long side is two. So this is going to be square root of two. Two divided by square root of two, maybe. So the area of this guy, the area here, and this is a true area in the physical world, is three over square root of two times two divided by square root of two, which is three. But the area of this guy, and that's the imaginary world, because we are using coordinates which are priori have nothing to do with the real world. We are using them for convenience in order to be able to evaluate the integral. The area here is two times three, so it's six. So you see what happened here is that the true area, if we were calculating the integral with the function f equal one, instead of this complicated function, we would get the answer three. This is a true area. This is a true area. I'm just simplifying the question. Question is to calculate this function, but if the function were actually one, we wouldn't need to make any integral whatsoever. We would just need to calculate the area which we can easily do by basic trigonometry, which I just did. You get three. But if you calculate it the way I'm suggesting now, where you put the U, DV over this imaginary region, D prime, you get six. Now, three is not equal to six. That's how you know that there is a factor. And now we know that this factor is actually one half. So it's actually the stretching factor, but it's really a shrinking factor instead of stretching factor. And we know that indeed it has shrunk. When we made this transformation, this guy has shrunk by a factor of two, if you count the area. I mean, its linear sizes have shrunk by square root of two so that the area, which is the product of the two sides, has shrunk by a factor of two. You see? So in this particular case, because the transformation is linear, the shrinking factor actually is a constant, is one half. In general, it's going to depend on the point where you are making the transformation. And we'll see now the general formula for the shrinking factor. But I want to use this example to illustrate that even in a very simple situation, supposedly, where you can actually draw this quite easily and recognize it as a rectangle and so on, it is much easier to find these new variables, U and V, which you can see, we kind of, they leap in your eyes. You see right away what the appropriate variables are because everything is expressed in terms of x minus y, x plus y. So why not use them as the new variables? The one, when you do that, the only thing you need to remember is a factor to put here. And the factor actually in this particular case, you can guess by measuring the areas. And the factor turns out to be one half. So the upshot of this discussion is that actually this integral is equal to this integral, which you can easily evaluate. And that's a very good illustration of the power of this method of change variables. But now we have to address the question of how to calculate this factor, this factor, question mark factor, in the most general case. So that we will now obtain some universal formula, which will service both this example, this transformation, as well as that example of polar coordinates. And it will give us in this case the distortion factor of one half, and it will give us in this case the distortion factor of r, which we have already calculated before. So how to find the distortion factor? Well, for this, we simply have to compare the areas just the way I have compared here, but locally. Instead of globally, you just look at the areas on a very small scale in a small neighborhood of a point. In other words, instead of this whole thing, this whole image, comparing the area of the whole thing and this whole thing and dividing, you want to take the area here where the sides are some delta u and delta v, so that this area would be delta u times delta v. And you want to compare that to the area which you have in the real world.  So let's draw this general picture.  And here from v zero to v zero plus delta v. And then we'll take delta u and delta v to zero, so as we did before. And let's look at the image of this guy under this transformation. And it can become bigger or it can become smaller depending on what the transformation is. So I will draw it as if it becomes bigger just so that it's easier to draw. But in principle, it doesn't have to become bigger, it can become smaller. So in general, this rectangle is not going to be a rectangle, right? Because we know that in the polar coordinate system, a rectangle goes to the annulus or part of the annulus. So there is no reason to expect that even a small rectangle will go to a rectangle. It gets distorted in a certain way. And so it will sort of be close to a rectangle, but it will not be quite the same as a rectangle. So here is what it's going to look like in general. Where, what do I mean by this picture? I mean that say this side goes to this side, right? This side goes to this side, kind of to distinguish between them. And then let's use a different color for the other sides, for the vertical sides. So say this one will go here and this one will go to, will go here. So that's what happens to this rectangle under this transformation. And what we need to do is to calculate the area of this guy. We need to calculate this delta A, the area of this guy. The area here we know, it's delta U times delta V. We'll have to calculate this area as, as delta U times delta V times the factor. And that's exactly what we need. Then this will be the guy which will put, the factor which will put in the integral. So how to do this? Well, here again, we use one of the methods which we developed before, namely cross product. We know that areas can be calculated by using cross product. So what we're going to do is we're actually going to approximate this. I mean there's no way for us to calculate this area exactly. What we'll do is we will shift it slightly, we'll deform it slightly to make it into a parallel ground. By taking the tangent, by taking the tangent vectors here. So the old idea again is at work, which is that you're on a very small scale and we are at a very small scale, which is the point. On a very small scale, you can actually approximate smooth curves by their tangent lines. And so the areas will be very well approximated by the areas enclosed by those tangent lines. So in this, in the case at hand, it simply means that instead of this area of this curvy region, I will take the area of this rectangle. And so what is the area of this rectangle? So delta A will approximate by the area of the rectangle. And I don't want to get into too much detail here. So I will just give you the answer. If you want to trace all the steps, you can look in the book. It's explained in great detail. But basically, basically you need, once you talk about tangents, once you talk about tangents, you talk about derivatives. So what you'll need to do is you need to take derivatives of these functions. You have x is g of u v and y is h of u v. And so what you'll do is you will have r of u v, which would be g of u v, which would be like x times i plus y times j, which is g of u v i plus h of u v j. And this area of the parallelogram will be approximately equal to the following.  So why it's exactly this? I will skip because I will not have time to finish. But I think you can see clearly the basic idea, something we've learned before. How do you find tangents? You find tangent by taking derivatives. And so that's, since I have replaced my rectangle, sorry, my kind of curved rectangle by the straight, sorry, my curved parallelogram by straight parallelogram, I use tangent lines. Because I use tangent lines, these vectors can be expressed in terms of partial derivatives of r with respect to u and v. So this is the first vector, this is the second vector, and I take the cross product. So if you want to know more how to get this formula, just look in the book. It's a very simple calculation. But once you get this, you can calculate very quickly the cross product by using the methods which we've developed. So you have this i, j, k. And then you're going to have d, dg, du, dh, du. And then you'll have dg, dv, dg, dv, dg, dv. Sorry, dg, dv, dh, dv. And then you have zero, zero. And then you have to take the absolute value of this. So this, and so you see that the only contribution will come from k, and it will be this determinant. So it's dg, du, this guy times this guy, dh, dv, minus dh, du, dg, dv. So you see, you have g and h, right? You have this formulas. So you kind of know from the beginning that there's a distortion factor will be given by some sort of derivative, or some combination of derivatives. So it's sort of natural to think in the following way. You've got two functions into variables. So you have four possible derivatives you can take. You can take two parts of derivative of this function, two parts of derivatives of this function. And then you have to assemble them in some way into a meaningful quantity. And that's how we assemble them. We assemble them into this two by two matrix and we take its determinant. So it's a really combination of those four partial derivatives. It was a very special combination. And what I claim is that that's exactly the factor. So the factor is sort of the distortion factor, which I denoted by this question mark is equal to this, dg d, well, let me write it. It's a little bit nicer to actually write it like this. dg du, but g is like x, right? So I can just write dx du and then h is like y, right? So I have x is g of uv and y is h of uv. So it's dx du, dy du, dx dv, dy dv. So you kind of assemble them in this way, where this is like x goes along the first column, y along the second column, u is in the first row and v in the second row. You can actually transpose these two factors. If you only can switch rows and columns, it's the same. So in the book, actually, I think it's written the other way, but it doesn't matter. And we will actually have a notation for this. It's like taking d of xy over d of uv. And this is called a Jacobian of this coordinate change. And that's the factor which you need to introduce. That's the distortion factor, because that's the area. You see that the area of this parallelogram, which approximates the physical area of the image of this guy, of this rectangle in the imaginary space, is equal to this factor times du, delta u, delta v. Whereas here, you have delta u, delta v. So what's the difference? What's the ratio between these two? The ratio is this factor. And we find this factor by using cross product, because that's how you find the areas of parallelograms. So you can appreciate again the kind of a nice, this is a nice technical tool. And then we end up with this formula. So let's see if indeed we are going to get what is expected, what we had expected on different grounds, slightly different grounds, in the two cases which we have discussed, the polar coordinates, and this problem which we talked about, the integral we talked about. So in the case of, I made a slight mistake here, which I forgot to write delta u, delta v. Because you had the factor of delta u and the factor of delta v. So I just pulled them out of the cross product. So they will appear here. So let's check. Suppose that you have r and theta. And you have g is r cosine theta, and h is r sine theta. So we have to take this derivative with respect to r. So we've got cosine theta. And then we take the respect to theta. So it's minus r sine theta. And then we've got here the roots of this function with respect to r, that's sine theta. And then the roots of this function with respect to theta. So that's r cosine theta. So we take this, this, minus this, this. So that's r times cosine squared theta plus sine squared theta, which is r, as expected. That's the distortion factor we had found earlier for the polar coordinate system. Right, and finally, let's calculate, and finally let's calculate the other one. So here we have x and y are given by this formulas. So what we need to do is, we need to take the derivative of this, of the first function with respect to u. So it's one half. With respect to v, it's one half. And the second function with respect to u, it's minus one half, with respect to v, one half. You calculate one quarter plus one quarter is one half, as promised, right? So there is one important point which I want to make and we will stop, which is that sometimes you'll get a negative number. So you always take the absolute value. The point is that we are not keeping track of orientation here, which we should have. In other words, you could switch u and v, for example, and this will make the two rows switched.  So in fact, if you're careless, which we are kind of careless, you might get a positive or negative answer. So the cure for this is you always put absolute value. That's the formula. The correct formula is always put absolute value at the end, so that the result is positive. All right, so I'll see you on Tuesday.


So, full disclosure, I am not a rising expert. I don't really spend a lot of time, you know, tweaking my desktop for screenshots. And I don't post a lot on r slash Unix porn or anything like that. So I'm by far no rising expert. But for those of you who are not familiar with the term rise, that is basically used to describe a Linux person's desktop. Basically is when somebody talks about my rice, they're talking about their customized Linux desktop. And when they talk about rising their desktop, they're talking about customizing it and, you know, getting all these cool terminal applications on the screen and, you know, in a certain layout so they can take a nice pretty screenshot and put it on r slash Unix porn. And I found this really cool tool. If rising your desktop is something that's important to you. I found this great auto generation wallpaper utility called packwall. If you're on an arch based system, what this does is it creates a wallpaper based on the number of installed packages on your system. It creates a dependency graph of your installed packages and makes a wallpaper out of it. So if I switch over to the browser and go to GitHub, you will find the source code for packwall over on GitHub. And all you need to do is just get clone this repository and you can use it. But again, it's kind of designed to be used with arch based systems. And there is a a you are package for packwall. What you need to do is if you have yay installed, you need yay space dash capital S space packwall dash yet. And that will install packwall for you. And there's a screenshot here. It shows you exactly what it does. It creates this nice little graph based on the number of packages you have installed. And you can customize the colors by default. I think it uses like a solarized color scheme. There's some customization options. If I scroll down a little bit, there are some flags and the flags, including changing the background color and the edge color and various node colors. And let me show you this in action. So let me switch over to my desktop here. So I am on Qtile today. This is the Qtile tiling window manager. And to see packwall in action, let me zoom in here so you can see exactly what is going on. Let me clear the screen. And if I simply run packwall because I installed it from the AUR package, those of you that download it and just do a get clone, what you would do is you would run the packwall script from the directory that you cloned. And it would look something like this. You would do a period slash packwall.sh from the cloned repository. For me, I've got a binary on the system, so I just need to run packwall. And let me run packwall and let me switch to an empty screen so you guys can see what will happen to the wallpaper. Now running packwall does take several seconds for it to finally generate the wallpaper because it's got a lot of information. I guess that it actually has to run through before it actually creates the wallpaper. And it has created the wallpaper. Now I cut out some of the dead air in this video, but it took maybe 30 seconds on my machine to generate the wallpaper. And the default wallpaper scheme, the color scheme is the Solarized Dark scheme. But we can customize that a little bit. Let me pull up a terminal and zoom back in here. And what I'm going to do is I'm going to go back to the GitHub page. And if I take a look at some of the available flags here, one of them is dash B for background color. So if I go back to the terminal here and I do a packwall space dash B and then in quotes, I need to do the RGB color. And you could also do RGB colors as well. If you wanted to add some transparency for me, what I want to do is I want to do a solid color here. And I'm going to do it the same color as my bar color. So I want my wallpaper background to be the same color as the bar color. Same color as my terminal color as well. We will give this a few seconds and the wallpaper should change. And the new wallpaper has been generated with the background color that I wanted already. That's starting to look nice. That's a pretty sexy riced desktop. I could almost take a screenshot of this and put this over at r slash Unix porn.  I think this looks better than all the weird Japanese anime that so many people post over at Unix porn. Let me see if I can customize this a little bit better. So let me get my terminal back up. And I'm going to keep going with some of the flags and options as far as the colors. So it looks like I have about seven or eight different options for colors. And I think what I want to do is I want to open up a new terminal. And I think I want to open my X resources fall because I've got my terminal color scheme loaded in my X resources. And what I'm going to do is I'm going to plug in some of these colors into the options that are available here in the pack wall documentation. All right. And what I did is I said what six or seven different colors here. So I set the background and then dash E, which is the explicit node color, dash D, which is just a node color, P, which is the orphaned packages, F, which is foreign nodes, U, which is outdated nodes. And then the dash S, I think is the color of the lines. They're calling it the edge color and the documentation. So let me just hit enter and generate this graph and we will see what it looks like. And if it looks good, I will start using that wallpaper and my new graph was generated. The only thing is I have so many installed packages on my system. I have 2000 packages installed on my system. That is very bloated for an arch based system. And I have so many lines that it's basically nothing but a dark circle in the center of this graph. You can't even see like, I think there's some inner rings of circles as far as the installed packages, but you can't see it because there's so many lines being drawn. But I did notice in the documentation that other than just RGB colors, I could also do RGB A colors so we could add some transparency. And I think what I want to do is add some transparency to this last one, the dash S flag, which is the line colors. And it needs to be a lot of transparency because there's so much going on. I need to, you know, put plenty of transparency in this thing. So let me add a couple of digits to the end of that particular color and wait a second. And hopefully once we make this quite a bit more transparent, we can actually see more detail in the center of this graph. And yeah, I like that a lot better, adding a pretty good bit of transparency. Now at least let some of the background and then some of the circles within the inner circle here actually bleed through a little bit. I'm pretty happy with what I've done here, just spending a few minutes on this. Now, beyond what I have done, there are some some other things you could do with Pac-Wall. So getting back to the documentation, some other stuff you could do with Pac-Wall. Now in the repository here on GitHub, they have this file here. 90-pacwall.hook. If you add this to your arch install, so what you need to do is you need to place the the hook, place that particular file 90-pacwall.hook, place that in slash user slash share slash lib alpm slash hooks.  Now, because the Pac-Wall generation takes like 30 seconds on my machine, it may take up to a minute on on lower powered machines. I don't know. I probably would not use this. I probably wouldn't because if you install and remove packages often, and you're going to trigger this auto generation every time you install or remove a package, that's a lot of time that it takes to generate this wallpaper. So I probably would just generate the wallpaper every couple of weeks or whatever myself. I probably wouldn't bother doing this each and every time I make modifications to the system. It should also be noted that if you install Pac-Wall from the AUR, it actually does not place 90-pacwall.hook on your system. So it makes you go out and get this file yourself and place it in that particular directory because they understand the people that created Pac-Wall understand most people probably don't want to add that hook. So all in all, I'm pretty happy with this. Why don't we take a screenshot and post this over at r slash Unix porn? Now, the folks over at r slash Unix porn require you to have at least two windows open in your screenshot. So I'm not exactly sure what I want to open here. I think one of the really neat things I could add to the screenshot is my file manager VIFM. So if I open VIFM, I opened on the wrong screen here. And then if I do a capital G to go to the bottom of this particular directory somewhere in this directory and my home directory, Pac-Wall actually generates Pac-Wall dot ping. That is actually what you are seeing as far as the background here, the background wallpaper. So if I resize VIFM just to get it in the display here for my screenshot, I think that should be one of the programs I open. And I might as well open up Doom Emacs, my Qtile config in Doom Emacs. Since I'm in Qtile and I might as well add a Neo Fitch to the screenshot. Let me go ahead and take a quick scryte and I'm going to go ahead and upload this to r slash Unix porn. Now, before I go, I need to thank a few special people. I need to thank Michael Gabe, Heplo, Nate Corbinian, Mitchell, Entropy UK, ArchVTBI 30, Chris Chuck, DJ Donnie Dillon, George Lewis, Omri Paul, Sean Tobias and Willie. These guys, they are my highest tiered patrons over on Patreon without these guys. This episode about rising your desktop with Pac-Wall, it wouldn't have been possible. Now, the show is also brought to you by each and every one of these ladies and gentlemen. These are all my supporters over on Patreon because this channel is sponsored by you guys, the community. There are no corporate sponsors here at DistroTube. If you would like to support my work, check out DistroTube over on Patreon. All right, guys, peace.


hey guys so we are currently at the grocery store and me and Madison just need to get like more frozen fruit because we've been having like smoothies or i don't know Madison hasn't really been having them that often but i've been having a smoothie every morning it's something that i'm doing to you know get my health back in check because at least i know that i have a smoothie in the morning that makes me feel better you know it makes me just feel like i got some good food in me at least throughout throughout like the day you know this is one thing that i do to stay healthy and just make sure i'm getting my fruits in and like the vegetable powder that i put in so we're currently at the grocery store and we are just going to get restock on frozen fruit and Madison wants to start making smoothie bowls as well so we want to just get like toppings for that and do some cans of coconut milk because that's like really good to thicken the smoothies up and stuff so we're literally just going to go in and out just get some frozen fruit and a little bit of toppings for the smoothie bowls that we want to make because we just had a matcha smoothie bowl and it was so good so that inspired us let's go in we're on my way to this little health food store near me it basically has like a lot of like organic and just like it's like a health food store it's all like a big grocery store it's like a little little health food store and i love going to those types of places because it's like they have like healthier options and instead it's more pricey but like you can go if you only go there for certain things it's okay so if you guys like want to start eating healthy i suggest that you find like one of those little like healthy little stores near you and then you can just go in there and then i don't know it just makes it kind of easier so i'm going there because i want to pick up some ingredients for dinner i really want to make veggie wraps for dinner and i also want to pick up some matcha powder because we ran out of matcha i bought honestly the last bag i bought was like months ago and it lasted us a while so we need a new bag and honestly i can only find matcha powder at this little grocery store like it's literally not sold anywhere else oh my god i can't even listen to music because madison took my aux cord it's still gonna be a good day see when little things like that happen like little things that'll make you mad like i don't have an aux cord right now so i have to listen to the radio it's like you can't like get mad at that you have to look at the positive sides and be like you know what it's fine like i i can listen to a radio station today here's some new music hopefully so gotta keep a positive mindset let's go okay guys i just got out and i'm going to give you guys a little bit of a passenger seat grocery haul so i got these um vegan cheese pasta i absolutely love i got two bags i got some rice for the wraps i'm gonna make tonight some peppers for the wraps and i finally got some more matcha powder and i got some limes because i'm gonna need it for the sauce i'm gonna make over here i just wanted to try this new oat milk brand that i've never seen in my life before so i was really intrigued and we're also out of oat milk so and then i got green onions and cucumber and i got vitamin e vitamin e and iodine is supposed to help with endometriosis so i got that and then the iodine that i got and this is supposed to help as well with endometriosis i don't really know the scientific explanation but if i'm completely honest but apparently this can at least help so i'm gonna try it and see how i feel because i do not want to suffer another period and yeah i got some sesame oil as well because we had none and it's really good when you make like stir fries and stuff and yeah that's all that i got okay so now's the time of day where i hit like a slump like i need a coffee i need some sort of caffeine to keep me going throughout the day because honestly in the morning when i wake up i'm fine like i wake up and i don't need immediate caffeine but then once i hit this time of the day like midday one two three o'clock that's when i feel like i need a burst of energy like i need i need something to help me to keep going throughout the day especially if i'm like planning on having a workout or if i'm gonna practice or something that leads me to today's sponsor which is EHP lab so thank you so much for sponsoring today's video and they have really freaking good energy drinks you guys like literally my brother tried them when i first got the package and he's like holy shit this is the best energy drink i've ever had and i'm like i know they're so good i was literally just saying the other day and i was like joel we have no like good energy drink companies like i didn't know really any good ones now let me tell you so they sent me five can i not talk they sent me five flavors um also the cans are literally so like textured i don't know how to describe it like you would need to get it yourself to like really know what i'm saying but they're they have like a really cool like you can kind of hear texture into them and they feel so cool like i've never really experienced a can like this before that was my first impressions i was like holy shit these cans are so cool but anyways so the five flavors are peach candy rings kina colada guava paradise passion fruit and then strawberry kiwi so i'm gonna crack them all open and i'm gonna kind of give you guys a little bit of a taste test they are giving away a freaking tesla yes a tesla so if you guys buy a pack of energy drinks you can enter to win by posting a photo on your story putting a picture of yourself with the energy drink on your story and tagging oxy shred i will have like the details down below okay got them all open so first up is the pina colada that is just so good like this is dangerous because they're honestly too good that like i want to drink like 10 a day that is so good it's like the perfect amount of bubbliness found even makes sense carbonation there you go it's a perfect amount and i've been drinking these throughout the week usually i burp like when i have like um like a pop or something and it's like really strong like carbonation i like burp it's a great amount of carbonation okay this is the peach candy rings that is so good literally tastes like fuzzy peach next up we got guava paradise so but i literally love freaking guava like that might be my favorite one yet i don't know why but there's just something about guavas that i love okay next one passion fruit this one tastes like a fruity a fruity punch we love that and this one is the strawberry kiwi that is so freaking good my top is the i think it's a guava and pina colada i do like myself some tropical drinks they're all so good though but i think those are my two favorite flavors so these actually contain l clarinetine i think it's called and smart energy so it's not going to give you those like jitters you know when you have like way too much caffeine or like the wrong type of caffeine and you're like literally like you can't you get anxious i haven't experienced anything with that with these drinks it's been like literally great and literally the fact that it has on me when i drink these it's like i'm more focused i can feel like i can just accomplish my tasks it just helps me like you know give me that pump throughout the day to keep going and just it makes me feel like i just woke up and i feel like refreshing you know what i mean also these are going to give you zero crash they're full of vitamin c and vitamin b give you that nice natural energy literally after i drink one of these i just feel good like i just feel like i can manifest anything and accomplish anything because the effect that it has on me like i just i just feel like i can crash the rest of my day but if you guys want to shop them i will have a link down below and i also have a coupon code for you guys which is bento 10 i believe for 10 off and it'll be like everything will be in the description and don't forget you can literally win a tesla if you post a picture and i selfie for yourself on your instagram story and tag them yeah i'm honestly so happy that they reached out to work with me because i'm not even kidding you when i say i've been looking for like some good energy drinks i just never really found anything that i really liked but this game changer yeah so thank you to hp loves for sponsoring this video again and you don't forget to click the link in the description i think i'm going to sign up for a gym membership today like a new gym because there's one that opened up like closer to my house than like the old gym i used to go do so i think that's what i'm going to be doing tonight right now i think i'm going to go downstairs and make dinner like start the veggie wraps i'll show you guys how i make them it's so easy and so healthy but yeah so i think i'm just gonna go downstairs start dinner and then go sign up for the gym membership and who knows i might even end up working out tonight wow that was so exciting i really hope i can but if we can't go to the gym tonight then we're going to do an at-home workout so here's the vegetables that are going to be going in carrots um peppers cucumber green onion and then i'm also going to add rice so that's hot rice into them and then that's all i'm gonna do and these are the rice um wrappers that i'm using i'm pretty sure you can probably get these anywhere like they have them at literally any grocery store yeah literally all you have to do is put these in warm water like one at a time and then they soften up and then you can like wrap them like a wrap okay so here are how the rice wraps turned out um they actually make quite a bit i don't even mean to make this much but you made a bit and if you guys are stacking them like if you make this and you want to stack them on a plate make sure to put parchment paper in between each layer or else they're all going to like stick together because like the rice wraps are kind of sticky so yeah i made a lot and i just made like a peanut sauce and i honestly i bought the recipe for this because i made it a few times and basically all i did was put peanut butter um hot water lime and maple syrup and then soy sauce so that's what i did for the sauce actually coconut aminos but that's like a soy sauce substitute um so that's what i did for the sauce i can link a similar recipe down below because i just like didn't follow a recipe but any recipe is basically gonna have like the same components to it but yeah so good and so healthy and it's a good way to get your vegetables on so as for my health goes in terms of like dieting and being healthy i don't diet at all like i really don't diet the only restrictions that i put on myself is like i try and limit my dairy as much as possible and i'm thankful that like i don't have any like medical conditions um like celiac where i have to stay away from something so it's more of like a preference like dairy i just don't like it because it's um makes my zima more inflamed and it's not like i have to stay away from it because technically i could eat it but it there is consequences to it but yeah so for monday to friday i tell myself like no dairy like if you're gonna eat any dairy make it like one percent of the week and then on the weekend i'm more like lenient like last weekend i had a pizza and it was fine like as long as i limit my dairy then it makes like me feel a lot better it makes me happy and so i just like have little things that i like to stick to and during the week i always make sure like that my snacks are healthy like if i'm watching like a show at the end of the day i'll make sure to grab a healthy snack rather than grabbing like a butter popcorn bag for example and then like friday night saturday nights i just kind of let myself eat what i want so i have the popcorn and i'll have something more unhealthy um if i feel like it and also everything in moderation like i've learned that it's so important too much of anything is all is honestly not good for you either so you have to see that and yeah i just like try and listen to my body that's also very important to do um and i just honestly do that and eat basically what i feel like is good for me so that's gonna differ every person like you can't look at what i eat and copy that exact same thing you can't look at what the celebrity eats and copy the exact same thing because you don't have the same body as them like everybody has different function functioning bodies so it's like you really can't do that okay you guys are gonna think i'm crazy it's literally 11 o'clock at night right now and i'm going to the gym i just signed up to the new gym and well we are going to the new gym because i kind of want to scope out the equipment first of all i want to scope it out when no one is there because i feel like you just look stupid you know what i mean not because of that but like you know i just want to get a feel of the gym when like there's barely gonna be anyone there so we're literally gonna go to the gym i don't think i've ever been to the gym this late but it's gonna be an experiment i'm really excited and it's gonna be a fun time we're gonna get a nice ass workout and but we're also gonna be really easy on ourselves because we haven't worked out a long time so i just finished my first workout in so long and let me tell you it really feels so freaking good to be back in the gym because now i can like be on a consistent schedule like i'm gonna want to go like at least three times a week that's like my minimum goal just because i feel like with going three times a week i can still accomplish more goals but it's not like pushing myself too much so i'm gonna try to go monday wednesday friday that like aligns with my schedule but obviously sometimes my schedule is like different so it'll just depend on the week but i feel like monday wednesday friday is a solid schedule because then you get the weekend off and it's not like too much because for me personally i don't like going to the gym like i don't know i would say more than five days a week because i just like having at least like a couple days of rest i think that is so important at least like the way my body works like i just need my rest and i know that i'm gonna push myself too hard if i go too often so i think like the perfect medium for me is like three days a week maybe four maybe five i don't know honestly it'll just depend on the part i'm feeling that week and what i want to you know accomplish but that's like going to be my minimum goal three times a week but i'm really so happy that i'm back at it i'm literally so happy this room by the way is so nice it's literally just like a little studio and i was like blaring in the windows when i saw it and i was just dancing because it feels so but just have like a dance party if you ever feel stressed out or anything just like literally blast your favorite song and start dancing i promise you don't make you feel literally so much better i'm just like release this energy because sometimes like when we feel stressed or anxious it's like pent up energy and it's like you need to let that energy go and you need some ways to exert from your body and dancing is such a good method like even if you're suck at dancing like i literally look so dumb dancing and just sort of like can't receive like it's so important to exert your energy especially when it's pent up like that and that's why we get anxious that's why we get like stress is because we have so much energy pent up in our bodies that need we need to you know let loose and just let it go and dancing makes me so happy like if i'm in a bad mood i'll just put my favorite song and just like dance to it like honestly dancing isn't great just like moving your body to a song is just so fun and it basically always feels so freaking happy so tip of the day is sergeant do more dancing i promise you it'll make you like so feel so good it's currently 12 o'clock it is literally 12 o'clock am and i'm at the gym that is so weird but you know what we love it you love it um so i think i'm just gonna pack myself up and go home now because it's 12 am and i still need a shower and dare say do a little bit of editing oh my god no i can't i need to go to bed i need to go to bed i'm gonna clean up my mat here and then get going


Welcome! My name is Bonnie Petrie. I'm one of the reference librarians and I'm going to show you how to get articles this library doesn't have using Interlibrary Loan's ILLIAD service. Whenever you run into a situation where you find an article that you need for your research and there's no full text, Interlibrary Loan can usually order it for you. Ordered articles will be delivered electronically and they usually come in about two to three working days, Monday through Friday. They can even come on the same day depending on how far they have to go to get the item that you need. And this is a free service for students. So do make use of it. And I'm going to show you how. I'm on the front page of the library's website and I'm going to scroll down just a little bit. And I'm going to show you the link that gets you access to your free ILLIAD account. It's right over here under our Quick Links, Request Interlibrary Loan. And you might be wondering what's the difference between the ILLIAD account for Interlibrary Loan and my library account. Let me show you a chart that I made. So the ILLIAD or Interlibrary Loan account allows you to order articles that we don't have. You can order in normal circumstances which we're not in right now. In normal circumstances you can order books and media and other kinds of things that are not found in the CSU system. And you can also keep track of your orders, look at what you've ordered in the past, what's going on with the present. And it also allows you to go in and pick up your articles electronically. My library account does different things. It is going to keep track of what you've got checked out from the FAO library. And again during normal times which we're not in now, you can order books from other CSU libraries and my library account tracks those. And you can also pay fees or fines online. So those are the differences between those two accounts. I'm going to go back to the page and I'm going to get into a database. I'm going to go to choose a database button and we have over 150 databases here. We've divided them up into big subject areas. I'm going to scroll down and get into psychology and do a very simple search so we have something to look at here. I'm going to enter the psych info database and here we are in the psych info database. I'm going to scroll down and I'm going to click in the box for scholar peer reviewed. I am not going to click in the box that says full text. If you see a box that says full text and it's sitting on the search screen of a database, do not click in it. You may have heard this hint before but it bears repeating. If you click in that box for the full text and when it's on the search screen, it turns off a special system that we paid extra for that gets you the maximum amount of full text possible across all of our 150 databases. If you turn that off, you're just going to end up with the full text in that one database instead of getting everything that you possibly could get. So leave that box alone. We tried very hard but we could not make them go away. So just ignore them if you see them on the search screen. But if you do see something for scholarly or peer reviewed, that's a good box to click in. I'm going to go back up to the top and I'm putting in just a very simple search so we have an example to look at. We've got 161 items in our list here. And what we're looking at are the brief records for the articles. The title of the article comes first and then the authors. And you can see the title of the journal. This one here is Accident Analysis and Prevention. And then the other parts of your citation, the volume, the issue, and your date and your pages. We're also getting some keywords here or some subjects. I'm going to point out the blue button here, search for full text. That's the system that we paid extra for that gets you the maximum amount of full text possible. You're going to see that popping up everywhere throughout our databases. I'm going to scroll down here. I'm going to stop on number three and let's say that we need this for our research. Exploring reasons for varying support for the status quo among Ontario's moose hunters. So I'm going to click on search for full text because I don't have HTML or PDF, anything else going on there. And it's automatically going to launch a search that will find it no matter where we have it, if it's possible. Ah, okay. So it's telling us that the FAL library does not have a copy. It does give us a reminder and a link so we can check for a free version in Google Scholar. Maybe we'll get lucky, let's see if we do. And although we're seeing our article come up here, it's just the brief information. If there was full text that was free, it would be over here on the right hand side. If you click through this title link here, you're going to find that they will gleefully charge you rather a lot of money for the full text of that article. So we can't get the article that way. I'm going to go back and I'm going to get this article through the Interlibrary Loans Iliad service. So you can click the link right there. And what it's going to do is take you into your Iliad account. You can set up an Iliad account ahead of time before you even plan on using anything. And you can do that from the front page of our website. Or if you haven't set it up already, when you place your first order it will remind you to complete that information. And it's just contact information so we can get in touch with you. I'm already logged on and so since I clicked on that link to request it through our Iliad service and because I already have an Iliad account, it's gone and pasted all of the details of the citation into the online form. So now all I have to do to order that is scroll down and click Submit Request. Now I don't really need this article so I'm not going to click on that button. I'm going to scroll up and just show you a little bit of what your Iliad account will look like here. So you've got the online forms for different things. You're going to be using the one for article, which is the one we're on right now. And then here's the other sections that you can look at. If you send away for an article and it comes in, you're going to get an email that tells you it's ready to pick up. You go to our website, log in to your Iliad account. And once you're in you're going to go to the section for electronically received articles. I'll show you this section in my account. And here I've got two articles that I've ordered. They will stay here for 30 days and you can download them and keep a copy for yourself and use it for your research. You also have in the menus here on the left hand side. You can check what you ordered in the past and do some other things there. But we're just focusing on articles right now. And once you send for an article, remember it takes about three working days at the most. Interlibrary loan will try to get it to you even more quickly if they can. And when you get that email telling you it's ready to pick up, you want to go to the front page of the library's website and go right here, request interlibrary loan. Even though you've already requested one and you've got one, that's how you get into your account. And if I click there, it will take me back to my account since I'm logged in already. There's one other scenario that you may encounter and that's when you are not online in a database with something to click on to see if we have the full text or not. And you want to be sure that you don't miss anything that we already have here before you send away for an interlibrary loan.  This is the title. You have to look it up. Or maybe a friend did. Or maybe you found something that looked good in a printout of an article where they have a list of references and you found something that looks good and you want to find out if we have it here. This is what you do. Start off on our homepage and you're going to use one search. And here I've entered the title of the article that I've heard about that I want to find. Tardigrades will outlast us all. And once you start typing in there, you're going to get this dropdown menu and you want to choose articles. Now one search will search all of our databases at once, which can be overwhelming if you're doing a subject search, but it's great when you want to find out do we have the full text of something. Okay, so we do get an exact match. Tardigrades will outlast us all. There's our citation information there. And notice, I'm going to just scroll up a little bit here. Notice that our article says delivery options here, whereas the next one down says full text available. This is actually where it says in the green letters, full text available. This is one searches version of our blue button search for full text. This has to look different there. This one is letting us know we're going to have to order something through Enter Library Loan. So let's click on that. And again, we get that link that reminds us to check for a free version in Google Scholar. Let's see if we get lucky. And we're not lucky today. We don't even get a match for our article except in a list of references here, not the main thing. Okay, so back we go. And again, we'll click on get this article through Enter Library Loan, the Iliad service. And it pops us right into the article request form. It fills it out for us. You want to take a look and be sure that it does a good job there. I see that it's a little bit off on this. It's got a little stray mark there. You might want to fix that before you actually click submit. But that's how you would go about finding full text of your articles through the Enter Library Loans Iliad service. And that's the end of this workshop. Good luck on your research.


Hello and welcome to the session. I am Deepika here. Let's discuss a question which says If probability of A is equal to 6 upon 11 probability of B is equal to 5 upon 11 and probability of A union B is equal to 7 upon 11 finds probability of A intersection B probability of A upon B and probability of B upon A Now we know that the conditional probability of the event E given that F has occurred that is probability of E upon F is equal to probability of E intersection F upon probability of F provided probability of F is not equal to 0 So this is a key idea behind that question We will take the help of this key idea to solve the above question So let's start the solution Now in part one we have to find the probability of A intersection B. We are given probability of A is equal to 6 upon 11 probability of B is equal to 5 upon 11 and probability of A union B Is equal to 7 upon 11 Now we know that probability of A union B Is equal to probability of A plus probability of B minus probability of A intersection B So this implies probability of A intersection B is equal to probability of A plus probability of B minus probability of A union B and this is equal to now probability of A is equal to 6 upon 11 plus probability of B which is 5 upon 11 minus probability of A union B which is 7 upon 11 and this is equal to 11 minus 7 upon 11 and this is equal to 4 upon 11. So probability of A intersection B is equal to 4 upon 11. Now in part 2 we have to find the probability of A upon B. Now we are given probability of B is equal to 5 upon 11 and probability of A intersection B this is equal to 4 upon 11. So according to our key idea probability of A upon B is equal to probability of A intersection B upon probability of B and this is equal to 4 upon 11 into 11 upon 5 and this is equal to 4 upon 5. So probability of A upon B is equal to 4 upon 5. Now in part 3 we have to find the probability of B upon A. Now we are given probability of A is equal to 6 upon 11 and probability of A intersection B which is equal to probability of B intersection A and this is equal to 4 upon 11. Now according to our key idea we have probability of B upon A is equal to probability of B intersection A upon probability of A. Now probability of B intersection A is 4 over 11 and probability of A is equal to 6 upon 11. So probability of B upon A is equal to 4 upon 11 into 11 upon 6 and this is equal to 2 upon 3. So probability of B upon A is equal to 2 upon 3. Hence the answer for part 1 is 4 over 11 for part 2, 4 over 5 and the answer for part 3 is 2 upon 3. So this completes our session. I hope the solution is clear to you. Okay bye and take care.


Our meeting today with King Hassan and the delegation is an important milestone along the road toward a common objective, a just and lasting peace in the Middle East. His Majesty King Hassan and his colleagues were forthright and helpful to us in clarifying the positions of the Arab League adopted last month in Fez, Morocco. I've clarified for His Majesty and the delegation a number of points concerning the peace initiative that I announced September 1st. We listened to one another's views in spirit of goodwill, understanding, and mutual respect. Peace in the Middle East means achieving security for all the states in that troubled region, security for the Arab states and security for Israel, and a sense of identity for the Palestinian people. We must pursue these goals rigorously, thoughtfully, and in close consultation with all here. And like our guests today, we all share the hope of peace. Our mutual goal of peace and the road to it lies through a negotiating process which I hope can be resumed in the very near future. I hope they realize all the delegation and His Majesty how welcome they are here in the United States and how much we appreciate their coming here. President, I and my colleagues, I would like to assure you our would like to express our gratitude for this wonderful reception we received today and the hospitality which we have received here today. We also would like to thank you for the depth and the goodwill which we have witnessed here today as we reviewed with you the hopes of the Arabs and their objectives which is peace, peace with justice, and dignity. I personally believe that we will find in the framework of Security Council Number, Security Council Resolution Number 242 and in 338 and your program, your plan for peace and the decisions of the Fez Conference, we will find in all these which will save us a lot of time in order to achieve our noble aim and objective which is peace and coexistence and construction for the welfare of the region and all mankind. Thank you very much again, Mr. President.


After the White House welcome, Pope Francis got down to business with his bishops at St. Matthew's Cathedral.  I know how much the wounds of these last few years have weighed on your spirit, and I have joined my efforts to yours in the efforts of succoring those victims. And when we bring succor, then the victims are healed and we have to hope that such crimes will never repeat themselves. The Pope praised bishops for their handling of the issue, but survivors advocates wished he would focus more on the victims. But that wasn't the only topic on his agenda. My family's immigrants from Mexico as well, so that one touched home and just being there for the immigrants. The Pope reminded Americans of our own immigrant past. I follow closely the immense efforts you have made to welcome and integrate immigrants who continue to look to America like so many others before them. It was absolutely on point about what's going on, you know, opening your heart to immigrants and not being judgmental and taking care of people who have been hurt. Before leaving the United States, the Pope will also be visiting New York and Philadelphia. In Washington, D.C., I'm Christina Ernie.