Hello, I'm Simon Benjamin, and this is lecture one of my course on Fourier series, Fourier transforms, a little bit of Fourier transforms, and partial differential equations. This is a, I really enjoy teaching this because it's a very visual and intuitive kind of mathematics, which gives us a set of tools, the Fourier series stuff, which then allow us to tackle a bunch of very interesting and otherwise very hard problems, which become super easy because of the tools we've learned. So it's, I like it, it's eight lectures in all, and this is number one. In this first lecture we're going to remind ourselves what even odd and periodic functions are, what are those terms mean, and then we'll start exploring Fourier series. Now, along with this course goes some notes that explain things that I'm saying in a slightly different way and contain a lot of the equations and examples that we'll look at, and so for those notes, they are available online in a bunch of different places, but if you have trouble finding them, go to SimonB.info, and that is where you'll always be able to find the notes that go with this course. I'll make sure they're there, and you might like to have a copy and work through as we go through the lecture. Alright, let's begin. So, what is an even function? What does that phrase mean? Let me just draw a couple of examples, and I think that will nail it for us. So I'm going to throw out my axes, just x and y axes. Come on, let's make it nice and straight, using iPad's cheap mode. Now I'm going to draw on a function, and this function is just going to be a sort of, well, let's start with a sensible function, the best of my ability to sketch it, which is not very great. So that's the side, and now I'll try and suggest. Yeah, that'll do. It's not amazing. So what I'm trying to draw, now it looks, what I'm trying to draw here is just x squared. So my axes are x and f of x. x squared is an even function. An even function is one where you get the same result from feeding in x and minus x. x squared is a great example, because of course when we square something we lose the sign. Minus two squared is four, same as plus two squared. So in general an even function f of x, or we can say f of minus x is equal to just f of x. That is an even function. What's another example of an even function that we could quickly think of? Well, we could go over to periodic functions if we want to, and then the obvious even function to think about would be cos. Move things around a bit. Cheating. I'm not stop cheating. Just being neat. Okay, here I go with my cos sketch. That's pretty poor. That'll do. So cos of course goes from plus one to minus one, and oscillates and completes one period in when x goes from zero to two pi. But for our present purpose is the interesting thing about cos. Not very happy with the sketch. The interesting thing about cos is that it gives us the same thing, whether we feed in an angle or the minus of that angle. So in other words, yes, cos of minus x is equal to cos of x. That is a second then even function. Even functions. There we are. All right, let's talk about the odd function or the family of functions that are odd rather than even. And so I've given myself a new pair of new set of axes down here. Let's just draw on some examples. The easiest example I can think of of an odd function is just f of x is equal to x. So just a diagonal line on our diagram. f of x equals x. Now the reason for that is being an odd function is that the magnitude is the same for if we change the sign of x, the magnitude of our function stays the same, but it changes its sign. So in other words, f of minus x is equal to minus f of x. And of course is the case if it is x itself. But another example, well, let's go on to our trigometric functions. We've already had cos. Of course, the partner to cos is sine and sine is an odd function. It could be difficult to draw. Considering this course needs a lot of sine and cos stuff in it, we'll have to hope that I get a bit better at sketching them as we go along. That's not too terrible. It's a little bit better than my cos actually, I think. Let me grab it. I haven't quite got it on the origin there. Which is the crucial point. So our sine function, of course, has the property that sine of x or sine of minus x is just minus sine of x and therefore it is an odd function. We could carry on and add further lines on here. The next thing I'd be inclined to try and draw on would be x cubed, which of course recovers. So x, cares about its sine, x squared loses the sine, x cubed again cares about the sine. But I think that's probably made the point clearly enough. An interesting point about a odd function is certainly if it's continuous, it needs to go through the origin. Because if you think about it, if it's a smoothly changing function and it has to get from a value for a positive x to the minus of that value for negative x, then as we approach the origin, we're just going to have to go through zero. Okay, so now, how would we formally, we've already put down sine and cos as a couple of our examples. That's led us into periodic functions, which is what we'll be thinking about a lot. How do we write down the fact that a function is periodic in a sort of neat, neat way as an assertion? What we would write is that a function is periodic with period p if f of x plus p is just equal to f of x. And we understand this to mean for all values of x in all the expressions I've written up to now defining even, odd, and now periodic functions. Of course, we don't mean this to for just some particular value of x. We mean that for any value of x, if we add on p, we just get f of x back again. So that's, in fact, copy my reasonably good sine, sine of course being a example of a periodic function. Let's just duplicate that sine function from above. Let me shrink it a little bit. Get fit. And so what we're saying is that any point that we choose on here, if we look ahead, a distance of p will find, we must find the same point. So we can show this by just doing this little trick. As we add on p, we must always go ahead to obtain the same value. Or we would if my drawing was perfect, which of course it isn't. That's not too bad. So that was just a brief recap on even functions, odd functions, and periodic functions. Now, it's time to play around with the ingredients of the Fourier series. The Fourier series is a sum of sines and causes and maybe a constant. That's it. Sines and causes are functions that we're pretty familiar with. We've been sketching them. What does it mean to add them together? Well, let's think about what happens if we add together a sine function and another sine function. And I think for this, the best thing to do is switch over to a math package that allows me to just type stuff in and we'll plot it and now the diagrams will actually look smart and accurate. Okay, so I'm gonna just stick my iPad on with its keyboard. So I've come over here to a maths package. It's actually the package called Mathematica, which is a very powerful and general maths package, and it doesn't actually run on an iPad. But what I'm doing is I'm looking at the desktop of another machine on my iPad so that I can do everything on the same device for you. Let's create a function and we'll just make it a good old, I don't know, cos of x and we'll then plot it. And I guess that the notation here is fairly obvious. We're gonna plot from minus 10 to 10 and we will need to just shrink that down a little bit so it doesn't fill the whole screen. There we are. Well, there we are. That's a cos function and a nicer one than I was able to draw for you. When I was sketching. But let's now put together that cos function with another one. So I'll add on and I'm just gonna randomly essentially mash the keyboard here. I'll wait the second cos function by a different amount and I'll also change the frequency by boosting it up by this other number that I'm just mashing in here. What will happen now? So we're putting in a second cos function. What do we expect to see? Let's just find out. Okay. Yeah, it doesn't seem to have made very much difference. Let's look, I mean it has made a difference, of course, in that now it's not a simple cos anymore. But let's zoom out a little bit and look from say minus 20 to 20. See what we get. Okay. We seem to have here a function that actually looks quite, so it's still an even function. And that's not surprising because we're putting together to even functions. That is inevitable. It has to be an even function. But is it a periodic function? Does it repeat itself? It doesn't seem to, right? So it looks like we certainly don't see a repeating pattern here. We see the mirror effect from it being an even function. But there's no sign of it repeating. We could try going out a bit further before it gets just too fiddly to plot for us. And now we are getting a kind of sort of a sense of a repetition. And that's because we've got a basic cos function. And on top of it, we've written a higher frequency cos function. But if we looked at this carefully, we would find that it isn't actually perfectly repeating. The points that seem like they might be corresponding will be slightly off. There won't be a repeat on this length scale. And in fact, where I typed in these keyboard mashing numbers, if I'd have put in, for example, square root two, in fact, I can just do it. Two to the power of 0.5. And so there, again, we're just seeing, yes, it's an even function. We may sort of, something may catch our eye and it looks like it's repeating itself. It actually isn't ever perfectly repeating itself. And now it will never repeat itself. Because root two is an irrational, that is to say, the digits required to specify it go on forever. And it will never come back into sync with the basic cos function. But that's not what I want. I don't actually want anything as sort of challenging as that. I want to put together cos functions and sine functions, by the way, and have them create for me a new periodic function. And so in that case, I better stick to ratios between the frequencies of the different cos functions that are just integers, essentially. If I do that, let's see what happens. In fact, if I take away the root two here, and I just make this cos plus a weighted cos 2x, now that's a periodic function. It doesn't look very useful. We could home in a bit on the, just to stop us having quite so many repetitions. This is a perfectly periodic function. As I say, it doesn't look like I'm going to want to use it for much, but it's periodic. Now, is there a way for me to add together signs and causes that it will give me something actually more interesting than this? Well, let me just type in an example. So here is cos, I'm going to actually not have cos 2 of x. I'm going to have cos 3 of x, and then I'm going to have cos 5x divided by 25. Let's see what this one does. I'm going to do anything stunningly different, but it may give us a hint that adding together causes, or indeed signs, can sometimes be useful. So now we've got a little bit of an interesting one. This function is, it's cos like, in a sense, that seems to have the usual period. It has to have period 2pi, for example. And excuse me, it has to have period 2pi because we've added together just integer multiples of basic x. So in the same 2pi range that was the period of our basic cos of x, our cos of 3x must complete three full cycles and again be back to the beginning and similarly 5, and if the even terms are in there, they also would have to sync up. So it must have the same period as the slowest or the longest period version, which is 2pi here. But it seems to be doing something interesting in that we seem to have sharper corners. In fact, we don't normally get corners in a sine or a cos function, right? We seem to be developing corners and something approximating straight line elements. And that's indeed the case. You can see perhaps, or you've already noticed that there's a pattern here, that we put 3x divided by 9, which is 3 squared, and 5x, the cos 5x term gets divided by 25, which is 5 squared. And if we add a couple more. All right, so I actually crashed there, so I had to rebuild the point we were at. But what you can see is I've put in another function g, which is the same thing, except I've added the next logical term in the sequence. Of course, we don't know why at this stage in the course these numbers are the ones to go for, but just repeating or continuing the sequence, just pattern recognition, we see that the next logical thing would be cos 7x over 49. We can compare whether that sort of does anything interesting. Let's home in, by the way, on the point near the origin. And we can see that, yeah, okay, so we're now, this is a zoom in on what looks like a somewhat sharper angle than we would normally see when we do signs and causes. And let's compare, that we can actually get Mathematica to plot both functions, the one that only goes up to cos 5 and the new one that goes up to cos 7x and put them on top of each other and see what is the effect of adding, what is only a little term, because it's divided by 49. And we see that the new line, the brown line, even has a slightly sharper turn than the previous one. So we're getting the hint here that the more terms we add, as long as we stick to this pattern, the closer we get to something that's not so cos-like, which is to say straight lines meeting at a sharp angle. So let's just see only function G and go back to our range from minus 10 to 10, just to look at that again. Yeah, that's looking pretty sharp and that's actually a triangular wave, that shape is called, and that does come up in various areas of science that we want to analyze a triangular wave. Let me give you one other instance of why this could be interesting. If I home in to just minus pi to just pi, and now if I shift this up a bit, maybe it will just suggest exactly what I'm after. So if I shift it up by one. So now this is still only a crude approximation because we've only got four terms, but if you imagine this becoming perfect, so that it's two straight lines coming to a sharp angle, I want to suggest to you that that looks like something that you will have encountered many times in your life. Imagine a guitar. So I'm not a guitar player, I imagine sort of holding a guitar like this. The guitar strings are fixed at each individual string. It's fixed at two points, which I want to suggest are like the points at either end of this figure. And then if I pluck my guitar string, I might lift up the string with my nailed or my plectrum, and maybe I pluck it right in the middle, in which case the string will momentarily be forming exactly a shape like that before I release it. And then we might ask what happens next. Now it's going to take us a while before we can answer that question, but we will be able to answer. We'll actually be able to predict the full physics of what will happen next when a guitar string, for example, is released. That will require us to understand Fourier series a bit. That will take us a few more lectures of that, and then we'll need to understand the wave equation. But answering real world questions like that becomes possible because problems like this starting function, which is, for example, a stretched string of our guitar just before we release it, they don't look like signs and causes, and they would be apparently quite difficult to feed into the differential equations that we'll meet in the second part of the course. But if we can learn how to break them up into signs and causes, and this example suggests that we could do that, then we'll find that those otherwise difficult equations just kind of melt apart and are extremely easy. It's beautiful to see, but we'll have to wait a little bit before we actually do that. So now if I come back over to my sketching area, let me pose the following question. Suppose that any periodic function could be written as a sum of signs and causes. In the little example we saw there, the triangular wave looks like it could be written, at least to a very excellent approximation, as a bunch of causes. We've got it as a pretty decent approximation just by choosing four terms. But how would we find out the correct amount of cause and sign? So let me, in fact, make that proposal a little bit more specific. Let's suppose that I have some function, I'll need a pen for that, maybe a blue pen. I have some function f of x, and it's a periodic function. So I can use this x plus p. Now, in fact, just to keep things simple, I'm going to say it has period of 2pi. We'll see how to generalize that later, but it will keep things neat. So now I'm thinking of some period 2pi function. But it's periodic, but it could have any old shape. So what kinds of shape could it have? It certainly could have the triangular wave that we were just looking at. That's a horror idea. Let me try a little bit harder to make the thing.  So sure, it could be this triangular wave that we've already met, but it could be any number of other things. It could be, for example, another very famous periodic function that doesn't look much like a sign or a cause, would be the square wave. So that would be a function that is either 1 or 0, and it just switches from one to the other abruptly. So that's, yes, it's periodic, but it's just made of horizontal components and perfectly vertical components. So that's not very sign or cause-like either. And anything else that I could write on here is also a legitimate example for what I now want to ask. I want to say, suppose that with enough signs and causes, we can create these periodic functions. And I'm just thinking of one such function. Maybe it's the triangular wave. How would I figure out how much sign and cause to have? So what I would say is we're going to assume that it's possible to write f of x, as well as defining it as a triangular wave or a square wave. It is possible to write it as a sum that is made of, well, bear with me, I'm just going to write out the proposition and then we'll talk about it. So a0, and for reasons that will become apparent, I'm going to divide it by two. Don't worry about that. Plus a sum.  And then, so don't worry, I will talk through this expression. I need to move it a little bit, make a bit of space. Draw a little scribbly diagram up a bit. I'd like to get this one on one line. So this is now a fairly complicated looking expression. The first thing to say is that these symbols a, n and b, n are just some constants.  And we want to find out what those constants should be. We can see that these sums run from n equals 1 to infinity. So this sum, this first sum, just for any who are a little bit unfamiliar with this notation, what this means is that we're going to have a, the first value is a1 cos of x plus a2, another constant, a different one, cos of 2x plus and so on. And it goes on forever. But of course, these constants could be 0. And so then the sum could actually just practically stop at a10 in that a11 and all higher ones might be 0. That would be fine. It fits within our description. Our description is super general, so it just allows any amount of any cos of any integer multiple of x. There's any amount of cos 3x, there's any amount of cos 37x, whatever we like. And similarly, the symbols bn are used for the constants that say how much of the sign terms we would like to have. Now, my claim is, or the claim we're investigating is that I could create any periodic function I want by this very general prescription of adding together suitable signs and causes. The only constraint is that the function I am going to try and create will have to have period 2pi. And that's just because I didn't want to make the expression for the sum of terms look even more complicated by putting in divide by p, divide by p as the period of the function. So I was happy to accept, sure, it's a 2pi period function. I can generalize that later. But in terms of the shape of the function, I'm claiming now, or I'm investigating the claim, that we can make anything we want. So if I just put on something crazy here, that could be, that does look pretty crazy. If I just copy that, if I just duplicate it and sort of try and join it up with itself, okay, it's got a bit big, let's scale it down a bit. Oh, doesn't want to scale there with me. There we go. So this is a periodic function. It does repeat itself. And so it goes on forever. And the claim would be that I can make this using signs and causes if only I can find the right amounts of sign and cause. If that claim is true, what are the amounts of sign and cause? Let's target first this term here. It's the constant term. So if we can get anything, maybe we can deduce what that has to be. Given some kind of function we've been told to look at like this brown one that I just randomly made up. Now, what does changing the value of that a0 do? So if we had found out all the terms and we'd correctly built this function and now we just, for fun, tried altering a0, what would the effect be? It would be to move our function up and down, because it's just an additive constant. So if I sort of put it back roughly where I found it, how could we work out what the correct value of a0 is? Yes, it moves the function up and down. Well, if we're interested in how much of the function is above the x-axis and how much is below over one complete cycle, because of course it just repeats. So the first observation would be if we can understand our function, if we can successfully reproduce it over the range, say, 0 to 2 pi, then we will have 1, because the sum of terms that we've written down here is necessarily going to repeat itself. The whole thing will repeat itself every 2 pi. So we can just focus on that range. So let's just kind of mark that off in our diagram. So we're going to focus going from here to here. This range, 2 pi, is all we need to analyze. We get that. We necessarily have got the function working everywhere. So yes, what should be the value of this constant a0 that shifts things up and down? If we think about this function and we're asking, all right, how much will it need? Is it mainly above the axis? Is it way above the axis? Is it below the axis? The simplest thing we might start to write down mathematically to explore that would be to think about doing an integral over one period, because that will add up all the area here. And I guess we could change color because it will subtract off the area there. And therefore, we would have some kind of mathematical measure of how much our function likes to be above or below the x-axis. And then we could apply the same condition onto our sum here where we're trying to discover the correct values of these constants. But we would notice something really neat, which is that when we do an integral, let's even just do it now. So suppose that we, given the periodic function that we've been told to recreate as a sum of signs and causes, sure, we could do that if we wanted to just say, all right, fine. It must integrate to something even if it's zero. So let's say that it integrates to some constant C when we take our function and go from 0 to 2 pi of f of x dx. In my diagram, it's just the green area minus the red area. But now, what happens when we substitute in for f of x this proposed new way of expressing it? So I'll copy that so that I don't have to keep writing it. And we'll just stick an integral symbol on it. Let's give ourselves a new screen. There we are. So now we want to do the integral from 0 to 2 pi. Make my pen a little bit sharper. Make a bit of space of our function. And that is going to be equal to, of course, the integral over this whole thing. And we can, of course, break up an integral that's over a sum of terms into each term that we're interested in. So I'm going to break it up just according to these sums. So 0 to 2 pi, oops. 0 to 2 pi here. And 0 to 2 pi here. And 0 to 2 pi here. Okay. So we're saying that, what did we call it? Some value C is what we must get when we perform this integral. But here's an interesting thing. The sum, the integral, well, let's take one of these terms, this one, and investigate what we're talking about here. It's the integral I missed off my dx out to dx. So it's the integral of a sum of terms. So I can just move the integral sign through, and there's a constant, which also can come out in front of my integral. So in fact, I can just write the integral in here and make it a sum of integrals. So to say it in words more clearly, the integral of a sum of terms is the sum of the individual integrals of each term. So what? How is that helping? It's helping a lot, because n here, of course, has to be an integer. And the integral from over one complete period of a cos function is just zero, always. So let's draw what we mean here. We're taking cos, we're going over one, that's a horrible cos. I mean, that's slightly, that's horrible. So when we integrate cos from naught to two pi, we'll get zero. And if we integrate cos of two pi, we're still going to get zero, because it doesn't matter what the, as long as it's an integer. So as long as it does have period two pi, the integral over one complete period of cos is always zero. And the same for sine, actually. So all these terms which have a n and b n, so a1, b, a1, a2, a3, a4, a4, a4, and all the b ones, all these unknowns just vanish. It doesn't matter what their value is for the moment. They're just going to get multiplied by an integral that is zero. So we can just delete all of that and all of that. So what we've discovered in fact is a very simple fact, which is that the coefficient that we were after, the first of our unknown constants is not unknown anymore. We know now that doing the, let's just finish the story. So if we go from naught to two pi of just the x, as I've taken the constant out in front, it's equal to naught to two pi of whatever the periodic function we're supposed to be trying to reconstruct is from naught to two pi. And we could note that the integral of just one from naught to two pi is two pi. So if we rearrange it a little bit more, we've just got a0 is equal to one over pi, the integral of the function we're trying to analyze from naught to two pi. So look, we've managed to get one of the things that we were after. And that should give us some hope, but it did rely on this cute trick that doing an integral over one complete cycle of any sine or cos will always vanish to zero. Doesn't look like we're going to be able to use quite the same trick again, right? So if we come up back up to our statement here written in huge writing across the screen, we've managed now to identify what that should be. But how are we going to get at the others? There's still an infinity of a constants unknown and an infinity of b constants that we don't know. We'd like to do somehow the same trick where everything except the one we're currently asking about vanishes because it gets multiplied by zero. What could we do? How could we change this function? What could we do to it? So that, for example, cos of 37x, which has the constant in front a30, a subscript 37, is the only one that will not vanish away. Well, at the moment, cos of 37x, when we integrated over one complete cycle, it does vanish away because cos of 37x will spend as much time above the axis as below the axis to put it that way. How could we put a stop to that? How could we put a stop to that? Well, the most basic way to take a function that was integrating to zero because it was spending as much time positive as negative would be to make it have to be positive. How could we take our particular cos function or sine function that we're looking at and force it to be positive? The most basic way to make anything positive is just to square it. So we could try squaring our cos 37x, which is that we just imagined we're trying to get at that particular constant. And so if we squared it, what would that mean? Let's try multiplying the whole function by a particular, let's say, cos term and see what that gets us. So I will copy this one more time, at least one more time, paste it in. And the idea we're now exploring is that it might be helpful to cause one of the terms in our sum to become squared because then when we do an integral it won't be able to vanish anymore. And maybe the others still will, let's find out. So I'm saying I won't take cos of 37 because that seems very specific. I'll write cos of mx and m can be 37 if you want to imagine it that way. So that's going to be, of course, cos of mx multiplied by that whole sum. So we're going to get a0 cos of mx and then all our cos terms are going to be multiplied by this extra cos of mx and all our sines will get multiplied by cos of mx. Now, looking over on the right, on the left hand side and remembering that f of x has been provided to us as some kind of equation or diagram like my brown line here, I could just provide you that. I could just draw this brown line really carefully and say that's the function. But if you give it to you and then you could use a computer to integrate it based on my diagram or as closely as you could approximate it. Sure, whatever that thing is, if I want to, I can perform the integral of over one complete cycle of that thing just like we did before but now with the cos of mx and it will come out, again, it will come out to something or other. It might come out to zero, but it'll come out to something. So again, I've already used the symbol c so I don't know, let's call that... What would be a nice symbol for it? I don't know, e for some reason. So that integral will come out to some value but what happens when I perform that integral on the right hand side? That's where we got our miracle to happen last time and everything vanished except the one term we were interested in. Well, now what about that term? What about the a0? That's now multiplied by cos of mx. When we integrate over one complete cycle, it will vanish because, remember, m is just some integer, e.g. 37, and so it will vanish away. But what about these other terms? So what happens when I do an integral from 0 to 2 pi of cos of mx times cos of nx where these are two integers that are not equal to each other? So I understand that m is not equal to m. Well, that's a puzzle. That's a mathematical question that's reasonably gonna need you to stare at it and think about it. Now, the answer I can tell you is that it will be 0 but the proof that it is 0 is actually one of the exercises that goes along with this course so I won't get into that. But I will just state it. So it's exactly what we might hope that wherever we have a mismatch between the particular cos term that we're focusing on, the one that's mx and any other cos term when we integrate it will vanish. How about the other set of things we're gonna get here which is cos of mx times sin of nx. How about that one? And now I'm interested in that case for absolutely any values of cos of, excuse me, any integer values of m and n including when they are equal to each other when they're both 37 because it's gonna be the sin term 37 but I'm trying to actually discover a subscript 37 which is the cos term so I would like that all to vanish and it does. So that is a part of the as I say part of the exercises is to prove that. Now it's not really a Fourier series it's vital for Fourier series we need that in the analysis we're doing right now but it isn't about creating a Fourier series it's simply a maths problem that could be asked in I suppose an A-level maths exam it's about that sort of level to go ahead and show that these things are true but they are. All we need for the moment is for me to just assert that these things are true and that you would find that you could prove it with enough effort or maybe it's not much effort I don't want to give it away I will say this it's a good moment for me to actually whiz all the way back to the beginning when we were talking about even and odd functions. Is there anything interesting about even and odd functions when it comes to thinking about what happens if we integrate them? So let's look at the odd function in particular if we perform an integral from any value let's say minus k to any value k let me try and spot the right point for that so that's just about here if it's an odd function the integral must be zero and I guess you can see why it's that for every little bit of area that there is let's choose this bit for example for every little bit of area that there is on the one side there is exactly the equal and opposite sign or area with equal magnitude and opposite sign on the other side and so the integral from minus k to k of an odd function is zero and the reason I'm mentioning that right now is that if you were to try and prove this little remark I have written down here about integrals over one complete cycle of causes and sign it can be helpful to notice whether or not you're dealing with an odd function now 0 to 2 pi is not integrating from minus something to positive something but remember these integrals only need to be over one complete cycle it doesn't matter if we're integrating from 0 to 2 pi or for example minus pi to pi because all of these functions are periodic with at least the period 2 pi and it may be that it completes several cycles within that but that's still a period with 2 pi we can go if we want we could rewrite these as being from minus pi minus pi to pi and they would necessarily come to the same thing so if we can spot any cases where there's an odd function going on then that may be a shortcut to have waffled enough figuring out both of these is an exercise all we need is that it is true so what we see here is that as before all the terms in our integral are going to vanish so let's write it out nope try harder to write it out here we go here's my assertion now based on admittedly a piece of maths I'm just claiming and I'm challenging you to do it if I do the integral could be from minus pi to pi I'll keep on writing from 0 to 2 pi because that's what I started writing and I don't want to mysteriously change halfway through this thing it's equal to some number when we're given the final form of our periodic function that where we would like to break up but what is it equal to when we do think about the series expression it's equal to a whole bunch of zeros and then just the term a m integral of cos mx and squared I'll put the square there dx only the term where we had cos of mx multiplied by cos of nx but they were equal so we can just write them both as m only that term survives and what is it equal to well that's an easy integral to do why do we know that well let's just draw a quick sketch just because I realize that otherwise might sound a bit mysterious if we looked at something squared it can only do this this is cos squared doesn't really matter how many complete cycles it fits into two pi you can see that I hope you can see that the area under this is the same as the area over it up to the fact that my sketch looks horrible so the brown and the orange areas are the same and so the area must be half of the total height which is one times the width which is two pi so it must be must integrate to pi that's the intuitive reason but it's pretty easy to just turn the handle and just follow the rules of integration and just do it anyway regardless of how that integral is done that is the answer to it and so now we are in a really strong position to write down any of these coefficients changing back to my more sensible color blue we're saying now that if you want to know any of the coefficients a m which are the coefficients of the cos terms we can get those just by doing one over pi times the integral from 0 to 2 pi of cos of mx fx dx so you give me that fx function I will now be able to tell how much of each cos is inside it and the same without sort of just saying all the same things again unsurprisingly the same is going to be true of the bm terms when we just go ahead and we multiply through what a sine mx term so we can say this straight away and you can verify it but it's exactly the same steps as an argument that we just went through so what we've done we've done it okay so what have we done that the in fact let's go and get the the first one we managed to discover was the constant a0 term have a look at what that looks like when we put it with its friends so if I move this down a bit maybe make it a little bit smaller there'll be room to pop in the the one we previously discovered but what we can see and this is the point why we realize it was helpful to put in a factor of 1 over 2 it's because now the a0 expression looks extremely similar to the am expression and in fact they are the same expression we realize because if we put m is equal to 0 in then we would have if we put m is equal to 0 into the general expression for m here which we derived only for m for the ones that were cos like terms so they were m is 1, 2, 3, 4 but if we put m is equal to 0 in here then we would have cos of 0 which just vanishes it's just 1 and that is the line above so in fact we don't even have to write down three different lines here we only have to write down the two lines and we understand that these expressions are four this one is for m is greater than equal to 0 and this one is for m is greater than equal to 1 because there is no b0 number so these do the job of capturing for us what all those unknown initially unknown constants are so the statement we're able to make is this if a periodic function can be described as as a sum of signs and causes with some amount of each one then this is how much of each of those terms there should be now in the next lecture we're going to go ahead and we're sort of a little bit out of time to do it now but it's it's it's quite easy so it's almost tempting to do it now we're going to go ahead and put in the triangular wave which is the one that we were looking at in the maths software and we could see it starting to work for us we're just going to finish that story do it properly now put in the triangular function as our target function work out these integrals in that case and show that indeed what we should be doing is just adding together cos terms we don't need the sine terms only the ones that have odd values of m and the a0 excuse me the a constants are actually going to drop off with the square of that m factor which is what I was of course just typing in because I already knew the answer in order to make our little maths exercise work so now we're going to finish that story but more to the point we'll be able to do any periodic function that we want and we will therefore be able to just create these sums of signs and causes the Fourier series for any problem that we're interested in good I'm just going to round off now by a little exercise that is in the notes that go with this course and it invites us to think about because we've been playing with a lot of integrals we've been multiplying by signs and causes and doing integrals and all sorts of things and it may seem that this is just a one-off trick that happens to be the thing we should do to solve this puzzle of the Fourier series but it's just you just have to memorize oh that's the maths trick that lets you get to the answer for the Fourier series but it's a bit more general than that and one way to emphasize that is to think that the things we've been doing today are actually very similar in spirit to another kind of maths not involving functions that you've already met and the way to stress that is maybe to make a table and just compare very quickly so let's have a table with just two columns and um nope doesn't want to go straight that's annoying that table with two columns there we go and on the one side we're going to have the the Fourier series tricks that we've just been talking about so I'll just put Fourier series on the one side and the thing I want to compare with is actually just vectors vectors as in good old three-dimensional vectors that are made out of i, j and k components right so we've played with that that kind of thing so what on earth have Fourier series which are all about taking periodic functions and expanding them how are they like vectors well I'll show you so in the one case the general object is some function f of x which is periodic so I'll just write down the periodic the the rule that shows that the thing is periodic in the other case the general thing is some general I'll just write general vector v in both cases those are the the sophisticated thing that we might want to build up now in the case of Fourier series we argue that this general periodic function can be built out of sine nx where n is some integer and cos nx components in in the vector case we can build up our general vector with appropriate amounts of i, j and k Cartesian unit vectors so in both cases we have something basic and we can use it just by adding terms together we can make the basic thing represent the complex thing in vectors we say any amount of i, j and k and k the correct sumo any amount the correct amounts of i, j and k added together can give us any vector that we want in three-dimensional space the Fourier series claim is with the a correct amounts of the sine and cos functions we can build any periodic function so it's that what's similar here is that we're building a general complicated thing out of simple building blocks and we can even push the analogy a little bit more when we want to work out how much of our general vector lies in the let's say Cartesian i direction what do we do well you might remember this so if you look at my general expression for a vector it's got vx times i and v y times j and vz times k if I want the vx part of that how would I get it so I would write that vx for example is equal to I would do the dot product between vector direction that I'm interested in the i component and the general object and that will just extract for me the interesting bit because the other two parts will just go to zero and the trick we were just doing with our integrals is very much like that what we're basically saying there is if we want for example the am component which is how much of a particular cos term is there but we don't do a dot product because that's just and that makes sense in the context of vectors but we do the analogous thing we do an integral over one cycle of our function and again we multiply the building block component which instead of being Cartesian i vector it's now that cos term I'm sorry and I should have an m there we multiply that by the full complex object and we do the interval so in the place of doing the dot product between the building block of interest and the general vector we do the product between the periodic function the building block periodic function which is in this case a simple cos and the general entity and we do an integral because that allows us to look over the full range of the function there's no analogous thing needed in the case of the vectors because the dot product is essentially automatically doing that for us there is a well there is okay there's one more thing to say just to and this is a very deep connection actually so this is a complete orthonormal set of entities mathematically but I'll just highlight one more feature to just try and convince you that there's a strong analogy here even though one thing is vectors which are just these sort of simpler objects and the other is functions if we do the dot product between two different building blocks i.j for example is going to be zero and if we do the equivalent of our dot product for our building block functions so we say cos of mx times cos of nx or indeed of a sine but let's stick with cos for our example and we do the integral over the full cycle that is also zero so the building blocks have this relationship with each other that they come to zero when you do the correct operation in vectors case it's the dot product and in the Fourier series case it is the the integral over one cycle and this is the this is the generalization of concept of being at right angles for vectors this is true because the i.j and k are at right angles to each other it doesn't quite make sense to say two functions are at right angles to each other you know they're not geometric objects but they do have this property that when you combine the two different functions and do the integral they come out at zero so the word orthogonal is used in both cases it's an orthogonal set of building blocks well okay that's enough hammer it's not very important to grasp that it's just a nice observation that we really met the same kind of ideas from two very different angles now when we've learned about vectors and now we're learning about Fourier series but with that thought I'm going to end this lecture and in the next lecture we'll actually build some Fourier series we will can reconstruct properly now our triangular wave and look at some other interesting ones as well thanks a lot for listening and remember that the notes if you would like to look at them are available online various places but if you have any trouble getting hold of them just go to simonb.info thanks a lot


Hey guys welcome back to come again John wise here Shannon and I are going to share this review of the DC universe Classics DC Comics 75 years superpowers green lanterns light 5-pack so this was available a few years ago during DC's 75th anniversary and This was a Walmart exclusive as you can tell here by the sticker And this was an awesome pack because this gave us some figures that we didn't have yet in the DC universe Classics line it gave us to Maury It gave us a green lantern version of Sinestro, which I believe at the time We only had the black and yellow or the purple and yellow costume version Of course we had a repaint of the Hal Jordan figure that we already had such as this one here The only difference is between this one and this one in the five pack. However, as you'll see has The white streaks in the hair So that was just before he became parallax, which is really cool. We also have a John Stewart that's In his alternate up costume there and of course fan-favored guy Gardner so This is where Shannon's gonna go ahead and take over and guys so please forgive the shrewdness of this portion of the video I'll be reviewing the packaging pretty much via photos since John didn't really go over that I received my five pack from my brother-in-law and sister-in-law a few years ago for Christmas And I have to say that I spent quite a while just examining the packaging Before I even open it up, of course the DC Universe Classics 20 line was completely sculpted By four horsemen studios and produced by Mattel. They were first available in 2008 and this one as John said was a Walmart exclusive. So let's go ahead and go over the packaging Shall we I really like the backdrop that Mattel used for this five pack very green lantern Esk the style in which they pose these guys in the packaging is very iconic with Hal Jordan Posing as if he's charging from his power battery and the other four green lanterns Looking as if they're flying in different poses each of which pose directly next to their power battery on the left side You see a nice little collage of different DC Universe characters posing in green light Superman at the top Joker and Wonder Woman Sinestro flash cyborg dark side Jade Catwoman ultra human night Martian man hunter Hank Henshaw the cyborg Superman and Blue beetle then down below we see that it was produced by Mattel and that some pieces may be choking hazard On the right hand side. We have more DC Universe characters in green lanterns light Batman at the top Lex Luzer how Jordan gigantic firestorm I'm not entirely sure who that is with the beard But then we have Vixen dead man the question specter is a ton of another character I can't really tell and then finally two-faced with of course the Mattel logo in the lower corner as We move on to the back We see each figure posing next to a green lantern comic book with their own bios down below each DC Universe logo at the top followed immediately by the green lantern oath in brightest day in blackest night No evil shall escape my sight let those who worship evils might be where my power Green lanterns light so on the left hand side We have John stir standing next to an issue of the green lantern corpse comic book The issue shown is said to be the 1986 collector's item premier issue of the green lantern corpse But it's actually green lantern volume 2 Number 201 with story by Steve Engelhardt. I do have this issue in my personal collection So if you would like me to do a comic book review on it at some point Then please let me know in the comments below for the sake of timing I'm gonna refrain from reading the bios for each character and just move right along to the next one below John We have Guy Gardner who is also presented next to an issue of the green lantern comics Which is yet another one that I have in my personal collection from February 1986 Green Lantern volume 2 number 197 the explosive Confrontation you demanded Green Lantern versus Green Lantern as you can see we have Guy Gardner and John Stuart at odds About ready to fight with a depowered how Jordan between them saying up guys Maybe we could talk to us over and once again if you'd like to see a review of that issue Or any others presented on here in the future. Let me know in the comments below Moving across we have Green Lantern Sinestro standing next to a copy of Green Lantern Emerald Dawn number six written by Keith Giffin and Gerard Jones Which retold the origin of how Jordan as a Green Lantern for the new continuity Established by crisis on infinite Earths Unfortunately, I actually don't have this issue, but I do plan on buying it in the future So if you want me to review that six-part story and compare it to the original Silver Age story Then leave me a comment below and I'll fast-track that Moving on up We have Tomah Ray next to a copy of Green Lantern Sinestro core secret files and origins number one and then in the center how Jordan next to a copy of Green Lantern volume 3 number One from June 1990 which apparently is the return of how Jordan as a Green Lantern After being away from the core for an extended period of time and he's joined on the cover by Guy Gardner and John Stuart This issue was written by Gerard Jones So now guys, let's go ahead and take a look at the figures inside As you can see here it comes with Tomah Ray, Sinestro, how Jordan, John Stuart and Guy Gardner Now these guys all pretty much have the same sculpt with the exception of their heads and Guy Gardner Guy His arms and legs are pretty much the same sculpt, but as you'll notice he is a little bit taller And he's got this Rubbery plastic vest going on. All right guys honestly They could have taken a little bit more time on this sculpt of Guy Gardner The body is fine In my opinion. The only thing that even remotely makes it look like Guy Is the hairstyle? They just they kind of made his face too square and blockish His chin kind of protruding out. He actually looks more like Willem Dafoe than Guy Gardner The vest is kind of a rubbery plastic very similar to many of the old-school capes that we got from the Superman and Batman lines of the late 90s. It looks like they may have just slid it over the plastic body Before they put the arms on. It's not glued in place Doesn't appear to be and I don't think No, you can't really take it off either unless you want to remove the figures arms and Maybe risk tearing it. It is just a plain black plain black underneath It is not the same torso sculpt as the other ones considering Guy's neck piece has more of a Sailor's turtleneck style going for it The paint job is pretty good. I would say that the forest green shade Does go well with Guy Gardner I like the little white stripe He's got going down the side and The stud the silver colored studs and buckle on his jacket His head Is on kind of a ball joint can look up Slightly down about that far turn all the way around The collar does not hinder it at all His arms can go up to about there Down up about there Back down single joint at the elbow Shoulder swivel then you got the wrist swivel waist Can't turn all the way around Legs can go up that far Back that far Knee single joint thigh swivel Can go out to the little split motion and you've got an ankle rocker No side tilt just forward and backward and Guy Gardner like the others did come with See if I can get a power battery Now just like in the previous review This powder this power battery is solid on the inside. No lens or anything like that Still very nice and I'm glad they changed the color to be more of a metallic green for this one as opposed to the lime green of the Hal Jordan Avenue sir power battery now let's go ahead and take a look at John Stewart all right guys here. We are John's with John Stewart. This is one of his I think is second Design for his uniform. It's got the thin green The land massive lantern symbol in the middle the white around the edges Very reminiscent of parallax And then it comes down very narrowly down the abs into the trunks Now with this John Stewart since they were going this route with the figure I Don't think they should have gave him the Military style cut since this is obviously the 1970s and and Maybe early 80s version of the costume. I Think they should have really gave him the Short Afro that he had while during the time that he wore this costume. I Could be mistaken in fact, let me check on that real all right guys So I wasn't really able to find this exact costume I was able to find one similar to it from Green Lantern Corps of the 1970s However, it had the smaller Green Lantern symbol in the center, but it was narrow just like this But in that one he did have the short Afro rather than the military cut, but overall not bad see if I can get it to focus. There we go. I Really think they should have taken that extra step and given him the bright green eyes Like the John Stewart from The animated series, but still not bad He does have the same articulation as Guy Gardner His head is a little stiffer though It doesn't seem to go up or down as much as Guy Gardner's did And it does turn all the way around It probably doesn't help though that These have been sitting in a box because I have nowhere to display him right now So that could have stiffened up the articulation But his arm does go up to there Turn around goes all the way above his head. He does have Bicep Swivel a single elbow joint wrist articulation ab crunch waist swivel Which actually sure if I Yeah, Guy Gardner did have the ad crunch as well. It's just Not as articulating because of the vest waist swivel Leg go up out to there Go for up to there Back to there thigh swivel single knee joint and ankle rocker and The detail on his ring very nice This is another figure. I don't mind that they did the forest green look for him For this costume now if it were one of the newer costumes for John Stewart I would have liked to see more of an emerald green slime green style look Or paint job But that one's not bad and it comes with the exact same lantern as Guy Gardner In fact since we took a look at his ring Let's take a look at Guy's ring and his ring if you notice Not very well-designed. It just kind of Looks like a paint blob really let's move him out of the way and then we'll take a look at Sinestro all right guys, so here we have Sinestro as you can see he comes with the exact same power battery as the others So we'll go ahead and get that out of the way Can see his widow peak here they did a pretty good job with that Even though it is slightly crooked the mustache They really should have painted a thinner style mustache on him Because even in the 70s 80s He still had a very slim mustache almost a French mustache really and his head his forehead is bigger Than the others really and And that's to be expected because that's how he was in the comic and the paint job on his face Very nice. He does have the bright green eyes that I wish they would have done on John Stewart He does have all the exact same Articulation as the other one, so I'm gonna refrain from going over that but I would like would have liked to see him maybe in a possibly a brighter costume not exactly Emerald or slime green But not as dark and muted as forest green either. Maybe like a happy medium in Between there. I think that would have looked really good on this Sinestro And now we'll take a look at Tomar Ray Again power battery exact same. Oh I keep forgetting. Why don't you guys keep reminding me? We have Sinestro's powering it is slightly better than John Stewart's I mean guy gardeners But not my not by much So before I get on forget on this guy, let's take a look at his powering. It's pretty much exactly the same Little bit better detail a little bit better paint job Not bad And again, he has the exact same articulation exact same body sculpt just different head And his head does go up and down. It's stiff right now His fin here it's made out of more of a more pliable rubbery plastic Which is really nice pointed ears Nice color Green Lantern mask Sorry guys. I got the windows open.  They could have maybe done his beak a different shade Maybe a lighter shade of the skin tone I think that would have looked a little bit better instead. It just the same paint job I Understand they had to do it that way to keep prices down, but I think even adding just a little bit Different shade it wouldn't brought prices up that much. I mean look You got a slight darker shade here at the bottom of the fin Overall, not a bad figure. They did change that paint the inside of his mouth a darker shade kind of a maroonish and The detail they put into on his fin It's very nice So there's Tom Array and now let's move on to the final figure in the set. All right guys here We have how Jordan? You will notice exact same articulation exact same body sculpt exact same battery Exact same power ring The only thing different is the head His head does rock up and down side to side like the others you can't see he has the graying on his temples Free parallax however, I guess you could say he was Infected by parallax at this time It just hadn't made its presence known yet again with this how Jordan I would have liked to see maybe maybe the same style green as I Wanted for a sinestro Not as dark as this forest green, but also not as bright as Emerald or slime green either just like a happy medium. This is a More veteran more seasoned how Jordan He's slowly becoming parallax, you know, they would have just made him look a lot better if they did a Slightly brighter green. Yeah, let's go ahead and compare him with the how Jordan from the DC universe classics Abin sir Green Lantern two-pack and you will see pretty much the same exact head sculpt except This how Jordan Sides of his hair are Darkened Also, if you'll notice the skin tone on the face Faces This one's plain looking this one actually looks more leathery really shinier and The different paint job. This was how's first costume so it doesn't come over the shoulders like this how it does got the black outline to the Land to the lantern symbol This one though, I would have liked to see a brighter green done on him I can understand why they would do the forest cream because that was the shade of green that they used back then For him though, I would have liked maybe just a little bit Lighter shade. All right guys here. We are as you can see They their power batteries do fit rather nicely in their hands They can hold them without work risk of them falling out overall. I wasn't that impressed with this line Guy Gardner would probably be the exception. I I don't really like when toy lines use the exact same sculpt Across the board on their action figures because they might as well just give us one figure with four different heads overall, I'd probably rank this maybe a seven out of ten And again, they look at how Guy Gardner just kind of towers over all the others I don't really care for that. It makes him it makes him look too hulkish really He should be about the same size as the others. He actually looks more like a class to a classic cartoon villain in this toy line Not really impressed on his own. He is an okay figure But when he's lined up next to the other figures in the line Not really that great. So there you have it guys the DC universe classics DC conics 75 years of Superpower Green Lanterns light Sinestro Tomar Ray Guy Gardner John Stewart and Hal Jordan. I'm Shannon for comic NTV. Take care Hey guys, if you liked this video make sure you smash that subscribe button and click on the little bell to receive notifications on all our upcoming videos Hit the like button make sure and leave us a comment so we know how you felt about this video And don't forget to share with your family and friends until next time. I'm Shannon for comic N. We're all geek culture collides


This switch is called a transmit receive or TR switch. It's a combination of spark gaps and transmission lines tuned by stubs to present an easy path for RF energy in one direction, a hard path in another. One spark gap is located in a stub of the transmission line between the transmitter and the antenna. The other is in the stub of the line between antenna and receiver. The system is normally tuned to block off the line from the transmitter and leave the line open to the receiver, like this. The strong RF pulse from the transmitter causes the gaps to spark, changing the values of the tuning stubs, making the path from transmitter to antenna the easy one and at the same time blocking the path from transmitter to receiver. After the transmitter fires and shuts down, the gap stops sparking and the path from antenna to receiver is again open. Now when the echo returns, the normal action of the lines sends all the energy to the receiver. Once the echo signal returns, it must be picked up and amplified by a suitable receiver. Basically, a radar receiver is the same as the average radio receiver. Both are super heterodines. The parts and even circuits are a good deal alike. To understand radio, you'll find the mechanics of the radar receiver pretty easy to grasp. In some sets, when the weak signal enters the receiver, it's amplified immediately. But in all cases, its high frequency has to be lowered before we can work with it any further. It gets lowered in the mixer. Another frequency generated in the local oscillator is mixed with it, and the result is a lower intermediate frequency, or IF signal. But this has to be amplified still further. It may go through as many as seven stages of IF amplification before being fed into the detector. The detector filters out the intermediate frequency, and what's left is like the square pulse that originally triggered the transmitter. But this isn't strong enough yet to register on the indicator, so a final buildup takes place in the video amplifier. The signal is now ready for the indicator. We'll take up the whole subject of indicators in the next film of this series. What you've seen and heard today is a very simple explanation of the different components of any radar set. An illustration of what takes place inside a timer, transmitter, and antenna when they send a pulse of RF energy into space, and in the antenna, receiver, and indicator, when the weak returning echo is picked up. In a single second, this cycle of interconnected actions can occur 5,000 times. So you see, the simplest radar is still a complicated piece of equipment. That's why you've got to know what makes each unit tick.


Welcome back. It is past Christmas. I hope you had a good Christmas or a good holiday break. We're not 2020 yet But I wanted to show you all the submissions of my 100 frame animation Contest which includes emissions from the Academy and everybody else that submitted their work online So it's going to be destroy and first date and the winner will be selected in January So for now, these are just all the submissions that I've received and I hope you liked them as much as I did So let's watch them. Has anyone ever told you how beautiful my eyes are? Hello, that's a good first date


Hey guys, somewhere in this video, I'm going to give you instructions on how to win 15 different postcards. Hello guys, I am at the 2019 Enigmatic Nomadix Van Build Fest. And today we have a very special video. This is my good deed. I've been wanting to help out for a long time and I can finally do it. Alright, let me give you a quick rundown on the good deed that we are doing. We have a guy named Joe who you'll meet at some point. Joe is 71 years old. Joe has a very leaky RV roof behind me. The rig is at least 30 years old. I was in touch with a company called Liquid Rubber. I said, guys, here's this cool event. It's a good deed. We can pay it forward using your products. So Liquid Rubber has agreed to send down some very valuable products. Joe, could you tell me a little bit about yourself and a little bit about your RV? Well, I've been a full-timer since 82. My house got ripped off and flinted the first time I worked overtime in five years. I lived in my van for three years, worked about a year in the shop. And I haven't had an RV since. You ever do anything to your roof as far as painting it or anything? Well, I've had people try to patch it and stuff and I haven't had any success. It's been an ongoing issue with me. And it's been leaking, right? Yes, badly. And I wouldn't say it's on the top of your to-do list to get that fixed up. It's not like you had plans to do this, right? Well, I was hoping I could, but I didn't think you guys coated the roofs here. Well, thanks to Liquid Rubber, who sent all this stuff down here for free. We're able to help you out and give you a roof patch job in the neighborhood of, if you took it to a shop, $2,000 to $4,000 for the parts and labor. Like one place at $16,000 or something? Yep, there you go. So guys, what we're going to do today, we're going to give Joe a brand new roof with the help of Chris over here from Switzerland and his dog Zeus.  Step one, we're going to use the Liquid Rubber Smart, the RV Smart Cleaner to clean the surface. It's odorless, non-toxic. Step two, we're going to use the Liquid Rubber Seam Tape. This is a patch over areas that are leaking or have a seam that could potentially leak. I see a seam right here. Potential leak. Step three, we're going to use the Primer. And then step four, we're going to use the RV Roof Coating. And that is waterproof, solar reflective, no mixing required, no solvents or VOCs. So again, thank you guys so much for agreeing to send me down some very expensive products to do a good deed, to pay it forward, to make sure that Joe no longer has a leaky roof on his RV. It is going to rain tomorrow here out in the desert. So without further ado, are you ready Chris? Yeah, sure. Right, let's get to work. Now guys, we have a special guest here helping us out. We've got Jimmy from Almost Happy Camper. Hey, how you doing? I just want to show you guys, can you put some fresh stuff right here? I just want to show you guys how good of cleaner this stuff is. Look at this. These rags were white when they started. Yeah. But look at the difference of the roof. It's just incredible. Ready to use, don't have to dilute. Let me get out of the way so you guys can see the full difference here. Look at that. That is good cleaner. Step two, we're going to take this liquid rubber seam tape. And we're going to look for some weak areas in here. Maybe we'll patch over those little patch spots here. We're just going to put this wherever we see fit. So we found ourselves a pretty good size hole here. It looks like it's an exterior thing that holds the lights on. So that's not dripping inside his rig. But just to be on the safe side, we're going to put this patch down right on top. A little overlay. Go right over it. Use this to flatten it down. We were doing the exterior edge here for Joe. Joe had some serious leaks going on. This is step two.  He had a lot of cracks. He had a lot of cracks around here.  It's a little hard with the nitrile gloves. So we took our gloves off. But as you can see, Chris from Germany. Sorry, Chris from Switzerland. Never Germany. It's pretty easy. You just put it on and you slap it down. You cut it. Maybe Chris can show us how you slap it and just kind of press it into place. There you go. All right, ladies and gentlemen. Joe's roof is taped from front to back side to side. Around some of the vent openings. We did a double layer in the front. We have Chris helping me out with a final piece there. We actually took that off the second roll. We used two rollers. Total of 100 feet for a 30 foot RV. Step three. Apply some primer. Finally finished with the primer. It is a little challenging to work with if you step in it. So just watch where you step because the name liquid rubber, man. They are not playing. They are not joking. It literally is liquid rubber. You can see on my hands here is very sticky. But we're going to let this dry up here in the Arizona sun. And then the final step will be to use that giant five gallon bucket of the RV roof coating. All right, here's how to win. All you need to do is tell me your experience using Elastomeric paint. On the roof of your RV school bus or van. And or send Joe your well wishes during this difficult time. I really hope to see him at the next van build. All right, guys. So a couple hours later after a carne sada burrito. And a little powerade. We're back here. You hear that? It's sticky. Now the primer ends up sticky when I called liquid rubber this morning just to verify the process. The steps, et cetera. They said, well, maybe you want to do half with the primer. And then put the RV top coat, the roof coating on top of that. And then do the other half. And I didn't quite understand why. I just thought to myself, let's do the whole thing. Now I know why. It's very sticky. If you step on it, it might pull it up. So I'm just going to, from this ladder, I'm just going to take a nice roller and just roll it out. Then when it's dry, I will stand on it and then complete the rest of it. This is what the new roof is now looking like after maybe three coats. This is the end of day one. We got a lot accomplished for Joe and his older RV with absolutely zero roof reflectivity whatsoever. So we're going to continue on tomorrow and put another two to three layers on top. We got the crew. We finished Joe's roof with a big thank you to Chris, who's a huge help.  Tamara is right over my shoulder. We helped out Eileen. Jamie got one. So Joe came back yesterday. It was ecstatic. So you said this thing has been leaking ever since you got it? Yes. The roof has been bad for a while. Yeah. This is going to be a relief. Good. A big relief. Good. So we got a couple inches the other day and you didn't notice any... I didn't notice any dampness on the walls on the inside. And normally you would. Normally I would. There we go. Well, so I mean just a testament to the products they worked. Thanks again to the liquid rubber for supplying all the paint. I was able to pay it forward, able to do a good deed for Joe. He had no idea we were going to hook him up and fix his entire 30 foot RV roof. 31 foot. 31 foot. So bring it in Joe. Alright man. Good to meet you man. You guys have been great. Yeah. Right on. Well thanks. And thanks again to Chris. Adventures of Pi and Zeus on Instagram. I'll put a link down below. Alright guys. We were on the final paint job. I did donate some extra to steampunksteve to help someone out. That will be on his channel. We have a large bus. What I don't know how many bus rigs have we done here. We did Joe's. We gave Eileen some. We got Tamara now with a fully white roof. She has a little bit more of the liquid rubber RV roof coating to put on. But as you can see, is it already cooler to the touch? Next to the white and you can feel the difference. Yep. So it will. That's why people put this stuff on the roof. So anyways, thanks again to liquid rubber for sponsoring this video. I'm sending a giant pallet of goodies to do good deeds and share the product with people who can definitely use it. Anyways, that wraps up this video. Don't forget to hit the subscribe button. Hit the like button. It's just a really fun day. Well, fun week doing good deeds for people. We'll catch you guys in the next video.


Holy Christmas Mm-hmm. That is Let me show you what it is. Don't show them our icebox of shame. Oh, it's our icebox of Christmas cheer holiday moments And chocolate fudge coated So much Christmas cheer. Mm-hmm. I'm gonna my super store. It's our little pieces of candy cane And dark chocolate fudge Good morning, sir weasel Sir weaselot and The weiner weiner you weren't in yesterday's video. There were very upset masses of people Yeah, they weren't an uproar uproar They needed their weiner It was terrible it was terrible What do you have to say to the people? Well, so we got over here. Well, so you made an appearance yesterday. You still haven't told us Any new war stories? Good people are waiting. He's been very silent lately. People aren't believing your stories, man They don't believe you a commander-in-chief What if I roll over and show my belly? Does that help? Diesel Lord of the weasels All of them every single one And the woosles So we keep Hearing a little bit more about grandpa's situation Since he got COVID-19, so I don't really know what's going on with grandpa I haven't talked to him in a very long time His kids my mom and her sisters and brothers have been able to talk to him But for the most part he's just there all by himself and My dad was on a trip down to Georgia and then to Ontario back to Georgia It was gonna be a bit of a longer one and he did a u-turn Picked up a different load And it's taking that load back home dad is coming home to be with mom that's her dad and It's my last grandparent on either side he's got to start eating gaining strength They've said that they've done everything they feel they can and that You know they want to keep them comfortable, so We'll see we'll see what happens right now. I'm taking my pickup out for a drive Clearing my head a little bit and I'm gonna go get a truck wash because I always feel better when my when my truck is clean At least on the outside if it's not gonna be clean on the inside might as well be clean on the outside On a little bit of a happier note and a more positive note. It's going to be a great week weather wise here in Steinbeck Supposed to be above zero so above the freezing point. I'm kind of excited about this happens in December sometimes December sort of like It's the beginning of winter right because winter starts on the 21st of December So sometimes we have a really cold December sometimes we have a pretty warm mild December Usually if it's a mild December like it is right now that means January is going to kick our butts So my favorite car wash that I go to is a super splash here at Shell There's also a really nice car wash over at timber falls Behind me there. That's where Britt likes to take her vehicle. They're both really good I mean there's more car washes in town, too But I like this one because they have people that will shammy my car for me and so does the timber falls one So both of them are equally good. I just I'm a creature of habit So I've always come here Even though their car wash attacked me the one time you remember that It was here where they just installed a new system. I forget what it's called We'll see it when we go in but a new automatic system because first they wash your vehicle manually And then they wash it with an automatic wash and then they shammy it for you And then you can leave and they got this new automatic system And I guess some of the settings weren't quite right yet and it kept hitting the front of my truck Whack whack remember it was on the vlog, right? Barely did any damage and I well it did scratch the front of my truck and they were on the ball immediately Got me on got my email got my information and immediately paid to have it repaired So they made the situation right right a ways no questions asked. They saw the evidence. It was on on video I mean you can't dispute that and They fixed it and I've still been coming here ever since I Was like this thing attacked me once but I still come here.  Okay, I Find something I like and I go with it. I know that it's not always gonna be perfect But you know as long as they make it, right? I'll keep going. They're not gonna lose me Right creature of habit That's why this whole like lifestyle change that I'm in right now is so like messing not messing with my head But it's it's a big deal for me because I really don't like to have big changes in my life when I find something that works That I like to do I'll stick with it. I'll be your guy You can count on me But as life changes and I need different things in my life and I need to be around home more I have to make certain changes and I have to find a new rhythm Right because I had a rhythm going and I have to find a new rhythm that works for me works for the family Works for home life works for work life Got to find a new work-life balance because everything was the same for the last like 10 years or so and But I'm a creature of habit, you know Change doesn't always sit well with me So that that goes right down to my car washes. I will keep going to the same place for my car wash If they wouldn't have made it right though They would have lost my business honestly And I probably would have talked about it on the vlogs and they may have lost more business through that They don't even know who my who I am or I don't think they watch my videos at all I mean But since they made it right right away, so I mean they're good business people They're honest business people and they have integrity. They wanted to fix it They went out of their way not even knowing that I had an audience or anything online And that's what I like to see good honest business owners. So Why go somewhere else? I mean the other car wash Brit has the same experiences there She loves going there. I love that car wash, too Hey as well fuel it up first, right? I think I get some kind of discount on the car wash if I fuel her up first guess we'll have to find out The suspense No, I don't think we do get a discount when we fuel first, but we do get air miles It's not a total loss. I need to fuel anyways 72 smackeronies Yikes Oh, I even my trucks upset about that like hey, what huh you imagine I was 72 liters like I Can't imagine paying fuel prices that they do in other parts of the world. Oh, man, there would be there'd be all kinds of All kinds of spank going on if that would happen here people be on the streets At least I'd hope they had probably not no they'd just take it Fuel price here is I believe 97 cents a liter Believe that's what I paid so I can look at the receipt 98 9 so 99 cents per liter that makes sense because it was $72 for just over 72 liters Hey makes sense. All right car wash time truck wash Truck wash. Let's be let's be realistic here Exterior you can also go for the interior off there to the left.  Hello Good and Todd. Oh, thank you So you gotta do the yellow them don't do that. They won't like that All right, all right, all right Wait, where's all the attendants usually there's people here. I guess she's the only one today.  That's a that's a pretty cool Especially a man of trouble where everything freezes right away, right?  This is the thing that attacked me this laser wash But all right, and I always turn my vehicle off in here A lot of people leave their vehicles idling which I don't understand because it's a very enclosed space and you have Attendance working in here like washing vehicles behind us there and chamoising them ahead of us here And I can't imagine the exhaust that would build up in here with usually they have three or four cars in here At the same time and a lot of them like almost all of them leave it idling Must get very I get lightheaded when I'm in an enclosed space with a lot of vehicles idling all the time with no ventilation But you turn your vehicle off in the car wash if you can I mean I mean that's just I Think it's a nice thing to do. I don't know. We should be nice people. We should try to be nice. I Love getting car washes though. It's like it's that getting a massage It's just I thoroughly enjoy it The whole process. Oh Here comes the cotton candy and smells like bubblegum, too Smells great And then the unicorn poop which also smells like bubblegum It's always such a great experience, you know, there she goes That feels good now the name of the game is to get home without getting the truck dirty again Since we've been having such warm weather, that's a challenge, especially seeing all the traffic out there. What in the world?  We should be okay It's not too far of a drive It's always such a bummer when you go through the the whole process of the glorious truck wash and You get behind someone who just sprays you with all the filth on the road Like this guy beside me. He's getting pretty aggressive. He wants to go fast. Oh, he's turning. Oh, thank God I wouldn't mind having one of those. I want a side-by-side. That'd be a lot of fun out here I could make good use of that This is for all the Dodge owners in my audience Still got love for you And the Fords These were obviously traded in for new Dodges. This is a Dodge and Stein back here Like these trucks are massive like that's a 1500. It looks like a 2500 I understand why people want Dodge, you know, it's a tough-looking massive truck. This is a 2500. This thing's a beast That thing is a honking beast. That's the big horn I was talking about, right? I don't like it necessarily just in white, but I like this color here this 1500 I love that dark red color. That's the same color as my truck in American truck simulator Very close to it. Anyways And then here at the back you got a 3500 Chevy Silverado and look at that. It's like the same size as this 1500 Dodge See that? So I get it. I get it. I understand why you guys like the trucks I'm just on team Chevy. Ah, I'm sorry I don't know what to tell you Not gonna change my mind on that one We have magically transferred into the wife mobile It too needs a wash I'm gonna take you to the other car wash now her car wash. You want to know why? Because we got a coupon So I'm filling this one up with gas right away too and we're gonna go through the timber falls car wash This is the one where it actually touches your vehicle That's why I don't like it as much because the other car wash that I went through before with my truck is touchless but this one's got those like The spinny things but it does a great job They both do a good job and before you go through the spinny things They wash off all the dirt by hand so that it doesn't scratch your vehicle But my worry is that it'll always either like break off my antenna Or scratch my paint if they leave some dirt on there and then it scratches it So far, I haven't had any problems here, but we have a coupon So I figure we may as well. I got the time today and uh, both of our vehicles need to wash badly. It's been About a month. That's way too long to have dirty vehicles. I like to have clean vehicles but the thing about that is My vehicles are never clean It's right. They're hand sanitizer here stanks. I remember that. Oh man, that's nasty What do they put in there?  This is my wife's it smells like flowers. Oh stinky feet Or flower.  Wow timber falls Do something about that sanitizer man. Whoo. Oh now I smell like flowers What a day And to top it all off their uh, their car wash right now Is down so I can't show you this car wash today Uh, there's they should have it up tomorrow or the next day But we need to car wash right now. So we're going to go back over to my Super splash on the other side of town. Just get it washed there It smells like flowers in here. I don't know what's going on, but this strange lady crawled in my car. Oh, what's my wife? Oh Didn't didn't recognize her with her muzzle on Mine's a muzzle. They want to make sure I don't bite anybody. That's why they make me wear it when I go inside It's a problem. Okay. It's not that I can help it I got attacked by a gang of skunks in Ontario Now I have to wear a muzzle That's why you stink. I'm pretty sure what that that's what that's all about anyway So, uh, we stopped here at uh, the drug store again for drugs Not even the fun kind The legal kind. Yeah prescriptions and stuff. So we got our prescriptions now. We got to go get some groceries Uh, what else do we have planned? I'm pretty sure that as far as our plans went for the day I'm gonna make mini meatloafs in a muffin tin try saying that 10 times as many meatloafs in muffloin muffin tin Mufflin Mini mini meatloaf in a muffin tin. All right, mini meatloaf in a muffin tin mini meatloaf in a muffin tin Mini meatloaf in a muffin tin You guys can skip this part. Okay. Good. I think we're okay I was so scared driving it all the way over here across town I thought for sure it was gonna get all dirty again, and I just washed it This is a real canadian superstore Or canadian walmart And there's another dodge Right on cue Hello Look Zavina's all in his cage or safe so that you don't trample him. Oh, but we still got franko here Careful big guys. See this guy doesn't get under foot as much him He'll go directly underneath the big dogs just asking to get stepped on And then he'll blame them for stepping on him This guy here here's a war veteran though. He'll just fight him off. I'll bite you.  They are our primary competition for the year 2021 There are our arch nemesis is But friends too, but friends too. Yes Uh, we they challenged us to a christmas decoration challenge 2021 so for next christmas And they just made a video of their whole christmas display this year go to my facebook page linked down below uh facebook.com slash official trucker josh And uh, you'll be able to see it there It's on my posts there. Go check it out and they did an amazing job. It's all handmade hand painted They did everything themselves We have a lot to uh, we got a lot of work to do but I think that our christmas display is actually pretty awesome I think I like our colors better. I'm just gonna be honest christmas colors. Troy and Angie are more Unique and out there, which is good because then we're not exactly the same So uh, you guys and his audience will all have a chance to Have a say in who's is the best next year? Yeah But we're not going to tell you what we have planned Because they're watching it's gonna be intense. It's going to be the greatest christmas display ever Right ever We're gonna make a special video yet of our christmas display this year Just so that you guys know what our display is because both of us love christmas so much Uh them and us so all of us. I mean not both of us all of us So we're gonna show you our christmas display this year yet Not in this video, but in one to come very soon yet before christmas And then you can maybe we won't Okay, the wife The boss over there It was her idea to show you I was like, let's just leave them in suspense We'll show you our christmas display this year yet. We have probably i'm saying close to what 10,000 lights on the house right now In the house fence in the backyard front yard Have I showed them the backyard yet? I'll give you a little sneak preview of the backyard I don't know if I'm going to show you the backyard yet But uh, we have the backyard done and everything Uh, it doesn't pick it up very well and well Yeah, that's kind of it doesn't really show a lot, but the front of the house We literally have people stopping in front of the house to gaze at it Every night So we'll show you real soon And then next year it'll be them versus us and us versus them A christmas challenge. I hope you guys uh, stick around until then subscribe If you haven't already hit the like button to this video It does a lot more for me than you think and the most you can do for me Is share this video with your friends if you think that your friends and family would enjoy our content as well Let them know about us spread the word And I'll talk to you later spread the joy the christmas cheer I just want to show you the food because i'm really hungry right now.


Probiotics have become commonplace in many products we consume, like yogurt, certain drinks and supplements, because they are thought to have numerous health benefits. But to understand how exactly these products work, we first need to understand the concept of a microbiome. A microbiome is a community of microorganisms such as bacteria, fungi and even viruses that inhabit a specific environment, communicate with each other, share resources and exhibit a particular collective action based on their surroundings. Many of these communities of microorganisms exist within other organisms such as humans, but they can also exist in the world around us. Let's start with the humans first. There are multiple microbiomes within the human body, including on the skin, in the gastrointestinal tract, lungs and even the belly button. Healthy microbiomes are critical because each of these collections of microbes ensures our body functions normally. For example, the bacteria in the skin microbiome use a process called quorum sensing to communicate with each other and form a biofilm that helps defend us from harmful infections and other skin conditions. The most well-known microbiome in the human body, however, is the community of microbes that live in our digestive tracts, the gut microbiome. Here the microbes primarily help us break down the food that we eat into nutrients that can be absorbed by our body. More recently, scientists have discovered that the gut microbiome also plays critical roles from a very young age, like helping to develop the immune system. If any of these communities are disrupted or become unbalanced, it can have significant health consequences, which we see with antibiotics.  In the gut, this can lead to abdominal pain, fatigue and disruption of blood sugar regulation. Probiotics, which are a collection of the good bacteria, can help restore some of the gut microbiome that is lost due to antibiotic treatment, disease or other factors. While microbiomes in the body are the most well known because of their direct impact on our health, microbiomes outside of the body are just as important. For example, there is a unique microbiome that exists around deep-sea hydrothermal vents, areas of the seafloor that spew scalding water and chemicals that would be toxic to most living organisms. This community of microbes is able to convert many of these toxic chemicals into organic materials that can be used by other organisms, keeping the ecosystem around these vents healthy. So the next time you see probiotics or your deep-sea diving, don't forget about the microbiomes that are playing a huge role in not only your health, but also that of the environment around you.