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00:01 | So hello, this is Sergeant waves race uh with a particular emphasis in |
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00:11 | exploration. So this slide that you're at now is the introductory slide for |
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00:19 | first lesson, but since time is , we have asked the class to |
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00:25 | this material in advance. And so been done since we're not going to |
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00:31 | that in this course. Uh we're not gonna cover it in |
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00:37 | I'm gonna stop this um uh stop sharing this. And I'm gonna um |
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00:49 | up the the next lecture. Okay. And now I'm gonna uh |
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01:14 | share the on my screen. So this emergency notification uh slide is |
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01:34 | in here at the request of Of H. Uh and of course |
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01:39 | not, there's no alerts going on now we've known about it. But |
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01:44 | they do is they ask you students the ta to update your emergency contact |
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01:51 | . So make a note to yourself do that. So then let's begin |
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01:58 | the actual course. So here is program for the next few weeks. |
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02:07 | can see, we're starting off very , and then we're uh we're getting |
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02:13 | uh standard rate theory and wave And then there's a lot of complications |
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02:20 | . And of course in in hydrocarbon reflections are really important. So we |
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02:25 | a lot about those here. And at the end of this part, |
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02:28 | here, everything above this arrow is standard waves and race although that are |
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02:38 | topics which are normally ignored in an course like this, but I think |
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02:45 | important because we're gonna make a lot approximations and assumptions up here, which |
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02:50 | not true. And so we're gonna to go back and do a better |
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02:56 | to have more advanced thinking to to to address these issues. So, |
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03:07 | the way, can you all see uh my mouse on a zoom or |
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03:12 | I need to? So they always for lesson objectives. And uh |
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03:21 | uh, here's the list at this here, you might you probably |
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03:30 | you know about vectors and matrices. might not know about tensors, but |
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03:33 | will you probably think, you know stress and strain and you know about |
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03:38 | law. You might not know about concept of compliance is, and you |
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03:44 | not know exactly what uh stiffness is . Of course, these are english |
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03:50 | , you know, the english But of course, there's a technical |
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03:54 | , meaning for that, which we're use here in this post. So |
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03:59 | , and we're gonna mention uh an octopi briefly, and then take it |
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04:04 | again at the end of the So the last distance. So the |
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04:09 | question is, what is that And you probably have a good |
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04:13 | I'm just gonna state that it's a property of a material that makes it |
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04:19 | when you apply a stress. And it makes it recover when that stress |
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04:25 | removed. If you push on. at my uh my hat here, |
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04:30 | gonna push on before me, but doesn't recover. So uh that's not |
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04:35 | elastic behavior. That's a more complicated behavior, which we're not going to |
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04:43 | about in this course. And so what you can say is that electricity |
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04:49 | what makes spring springing Now. This one of the major topics of classical |
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04:56 | in the 19th century uh over 100 ago. However, the last important |
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05:03 | was a guy named E. Love. I'm not trying all those |
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05:08 | uh stand for, but the name might be familiar to you because we |
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05:15 | waves in geophysics which he invented called waves. But you see, he |
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05:20 | this book in 1928 and uh it's the best and most complete um exposition |
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05:32 | elasticity in the literature almost 100 years now, isn't it? You can |
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05:38 | uh it's been reprinted of course and can buy it on amazon for a |
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05:43 | bucks. And so that that might fun. Might be worth your while |
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05:47 | . So the next question is, are seismic waves? So there are |
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05:51 | of stress and strain inside the rocks they're approximately elastic. And so you |
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05:56 | see these pictures here, and you here uh the waves are going up |
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06:02 | you can see if you look closely that near this black zone, the |
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06:08 | squares are compressed together a little And so this is a compression wave |
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06:13 | up and here is a shear wave up but as it moves up, |
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06:17 | sideways and I'm sure you know these facts but you're gonna find out in |
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06:22 | course that these ideas are seriously who simplified in the material that you've talked |
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06:30 | before. Now in these pictures there's surface waves anywhere. Um if there's |
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06:37 | nearby surface then that affects everything. so they're all of course in our |
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06:44 | , there always is nearby surface. put our instruments on the surface, |
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06:49 | ? And even if the waves are from the other side of the earth |
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06:52 | spend most of the time in the deep interior of the earth when they |
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06:58 | our instruments, they know about the . And when we record them, |
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07:03 | don't record the incoming waves. No don't. We record the incoming waves |
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07:09 | the interaction with the free service. all together in in the data. |
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07:13 | so that's why we have to understand . Okay, so um let's first |
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07:19 | sure that you understand the mathematical vectors and matrices. So here's a |
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07:25 | quiz. This is uh not for but uh Stephanie tell us the answer |
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07:32 | the question. The length of this is what uh It's b Yes. |
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07:40 | good. Uh The length of this is what? Okay. Now why |
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07:45 | it that we uh we look at take a look at this. Now |
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07:51 | gonna go backwards and you see we we write it differently. So what's |
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07:56 | going on there? Yeah, well actually there is a real technical |
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08:04 | but it's not going to uh it out that for some applications uh it's |
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08:11 | smarter to write, I write it a column victor rather than as |
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08:16 | Um and for some applications for some of science, that's really crucial. |
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08:23 | for us, we're gonna be doing you said, we're gonna be switching |
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08:26 | and forth. Okay, Now, about this, what's the length of |
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08:30 | sector as well? Yeah. Uh uh how about this? The length |
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08:37 | a vector is given by this little , Are you familiar with that? |
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08:43 | the dot product? It's good for . So uh Stephanie graduated from a |
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08:51 | institution with a bachelor's degree in geophysics two years ago and she was exposed |
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08:58 | this kind of stuff when she was undergraduate and she still remembers good for |
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09:03 | . And so um uh that's the product. Now, I want to |
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09:08 | at this point and uh tell you the screen, you're looking at, |
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09:18 | you see this surrounding part here which uh surrounding the image that looks like |
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09:25 | uh that this um image is part a a digital course. And what |
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09:35 | says here is we're looking at page of 15 and there's some controls here |
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09:40 | I can tell you that that is , you can buy this course from |
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09:46 | scg uh if you do that, you shouldn't do that because the version |
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09:53 | had was several years old and of I'm continually improving and revising it and |
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09:58 | have not kept up, so don't that, but they wanted this set |
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10:03 | to be um so that a student take this without any instructor uh in |
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10:10 | room or online completely by himself, in his um uh in his office |
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10:18 | Beijing maybe and doing it. And there's a lot of support for that |
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10:25 | of learning, which you can see of it here, for example, |
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10:30 | down here, uh there's a little for a narration. So since this |
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10:35 | designed for people who english for whom is not the first language, uh |
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10:41 | have uh feature in the ScG version they click on that, then they |
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10:50 | scrolling down here, they see a uh written words which tell exactly what |
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10:58 | instructor is saying. And the generation recorded pre pre recorded by an |
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11:05 | not an expert, by a non , but he has a good voice |
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11:11 | that's by an actor. And so the actor says is repeated in a |
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11:17 | uh window here. So the student see the words as well as hear |
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11:24 | and that helps a lot of people english is for whom english is a |
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11:28 | language and then there's this other controls you see and you'll see more of |
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11:35 | ? Uh a lot in this course particular right here. So this looks |
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11:43 | it's a link, right? It like uh since it's underlined it looks |
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11:48 | it's an active link and it is the scG. Uh It is in |
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11:54 | S E. G. Version. you click that and it takes you |
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11:59 | a definition of the mathematical dot So if you don't remember uh there |
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12:05 | is. So uh we'll talk more these um um support capabilities in the |
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12:15 | program because they all have uh analog the materials that we're using today. |
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12:26 | . Mathematics so much Rebecca now we the deal with mathematics and mathematics. |
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12:33 | matrix is a rectangular array of symbols or expressions. So individual elements |
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12:40 | called the individual items of Carlin's So here's an element here. Here's |
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12:44 | element here. And this is an of a three by two Matrix which |
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12:48 | three columns and two roads. So uh let's see if you know |
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12:57 | to do matrix algebra. So what's sum of these two matrices? Didn't |
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13:06 | you DD as in David. And did you do that? I know |
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13:10 | you did that. You added this to this one and got 13. |
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13:15 | added this one to this one and six adding up the elements easy as |
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13:21 | . But it's deceptively easy because now the product of these? Uh |
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13:32 | Okay. Uh so so this is complicated and this is where uh people |
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13:42 | fail. So that's why uh we're about that now and so I'm gonna |
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13:49 | this opportunity to uh to quit this lecture and I'm going to um um |
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14:00 | sharing this screen will come back to and I'm going to share another |
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14:32 | I'm gonna have to browse for Sorry about that. She had this |
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14:38 | to hand kids. Mhm. So is what I call math 101. |
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15:09 | , so thanks for that, let I'll share my screen. Okay, |
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15:31 | everybody can see that. So this what I call math 101. And |
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15:36 | the pdf equivalent of this file is the blackboard for your identification. So |
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15:44 | let's uh let's go through this. and you see we're not going to |
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15:52 | all of mathematics. Uh we're but gonna do that, I'm gonna remind |
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15:58 | now of the kinds of mathematics is will be using over and over again |
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16:05 | this course because you will understand that stress is a matrix actually, it's |
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16:13 | special kind of matrix called a And strain is also a tensor. |
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16:18 | before we get into those things let's sure you understand what the basic concepts |
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16:24 | . So now uh the screen asked question is uh is that a |
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16:32 | And the answer is no, it's an arrow. And uh it's an |
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16:39 | feature of Microsoft. So sometime when in word or PowerPoint or something like |
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16:47 | , try typing dash dash, right bracket and it will magically turn into |
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16:57 | ash dash, right bracket, no in the train. And so who |
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17:02 | what other tricks are like that in point inward. But this one I |
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17:07 | uh stumbled. So now the question , is this event And uh |
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17:13 | it has the form of a But it's notation. It's it's it's |
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17:21 | . Now, if it's a notation uh, three apples, seven oranges |
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17:24 | four plums, then it's not a . It's just a shopping list, |
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17:29 | ? Or maybe and you can make laundry list. Uh that looks just |
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17:33 | the important thing is to don't assume the notation means unless you're confident. |
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17:39 | . But if it's notation for three in direction one and then also seven |
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17:45 | in direction two and four units in three, then its effect. Now |
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17:51 | have to uh agree what we mean direction one and direction to it. |
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17:56 | that's called a court system. And most often that is three orthogonal |
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18:01 | Sometimes we just have to and we the orientation and we agree where the |
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18:07 | the origin is. Okay, so we have we can have a unit |
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18:13 | be this long in the one this long in the two directions, |
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18:17 | long in the three directions. And we label the directions of the numbers |
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18:24 | sometimes the letters and so on. the order of directions corresponds to the |
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18:31 | hand rule. Okay, so let see. And the right hand |
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18:35 | So if if if this were alive would take you to uh description of |
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18:40 | right hand and by the way, those descriptions words uh things like this |
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18:48 | in another file which you have on blackboard called the glossary. Okay, |
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18:55 | uh the right hand rule is like uh if you take your right hand |
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19:00 | , index finger 12 and three. , so left hand you get a |
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19:07 | result, 12 and three. C is point of difference. So uh |
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19:13 | really easy to make mistakes uh in systems if you don't observe the right |
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19:21 | rule. So uh even if you're handed use the right hand rule, |
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19:27 | you won't you won't this is the time to be creative. You all |
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19:33 | , stick to the standards. Now lots of coordinate systems and sometimes uh |
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19:40 | directions are different at different points in system, for example, a spherical |
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19:46 | system as radius, uh a polar and has with or a cylindrical system |
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19:57 | uh linked along the cylindrical axis and and as much length. And so |
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20:05 | you can imagine it's easiest to think it in a quarter system. Uh |
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20:14 | here here's a cylindrical cross section of cylindrical system. Here's the here's the |
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20:21 | axis along the axis of the price . And you can see that the |
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20:25 | uh looking at this point, the angle and the radius sectors point in |
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20:31 | direction is that you are over the radius vector will be pointing in |
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20:34 | very different direction. So that's all potentially um potentially confusing, but we're |
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20:45 | set up a mathematical machinery so it takes care of itself. You don't |
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20:50 | to worry about. There are sometimes directions are not even orthogonal. Why |
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21:00 | you want to have uh important system looks like that, you know, |
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21:05 | it on this? Well, it out that there are important reasons for |
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21:12 | instances where that is really useful. example. Uh Do you know what |
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21:17 | you know the mineral calcite? And you seen uh usually at this point |
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21:23 | bring with me crystal of calcite, has natural angles like this. Have |
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21:31 | seen large crystals of calcite? And a rahm bic solid with uh directions |
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21:43 | this. Uh You can imagine appointed of uh of calcite pointing in here |
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21:49 | all my fingers are along the edges the calcite. So what that means |
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21:55 | if you go 33 cell units in direction and seven cell units in this |
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22:02 | and 17 cell directions in this movement be again at uh the corner of |
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22:11 | uh cell inside that calcite crystal. that's a useful thing in that |
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22:18 | Although in this case in this course won't be using things like that. |
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22:23 | gonna be using Cartesian coordinate system. . Okay. We're gonna use these |
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22:30 | notations and Stephanie has already encountered some these. And so uh huh What |
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22:39 | thrown when we use all these different for these records? Okay, |
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22:48 | intuitively, you can say um uh you're so Stephanie, when you're explaining |
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22:58 | geophysics to your baby, Not but in a couple of years, |
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23:03 | is gonna say, what do you ? Mommy. And uh you're gonna |
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23:06 | explaining your physics to your kid. kids can understand words like this vector |
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23:14 | a quantity that has both magnitude and . You know, like position or |
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23:19 | or forced by contrast. A scalar magnitude, no uh direction, like |
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23:25 | and temperature. And those are things kids think about. And so in |
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23:30 | five years she'll be asking you these . So you already told us about |
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23:38 | category in therapy and here's in here's a picture of pythagoras himself. |
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23:47 | was a Greek who lived about 2500 ago founded a school of mathematics and |
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23:53 | he taught things like vectors and And um I do not know if |
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24:00 | made any other inventions besides this but this is a non trivial uh |
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24:07 | . And the direction is given by ratios of these various components to the |
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24:14 | or an equivalent other set. You imagine describing the direction in terms of |
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24:20 | non dimensional um combination. So of the vector has a magnitude one with |
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24:30 | the unit vector. And then we a special symbol for that. And |
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24:36 | the inter vectors in the three quarter will are like this, see these |
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24:41 | parents here. And so uh we answered this one and this one and |
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24:50 | one. And uh now when you those two vectors together, Stephanie, |
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24:56 | was like this where you have these vectors with these these components and these |
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25:01 | and you just added them together. put this one up here, uh |
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25:07 | to nose, just like no, gonna put the green one up |
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25:11 | uh tail to nose. And uh answer was exactly what you said. |
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25:18 | you could think of it, you like this. And of course it |
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25:23 | matter whether you do the Greenman of one first and the green one second |
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25:28 | vice versa. It doesn't matter. so here's the little formula for uh |
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25:35 | the some of those. And that's she told us. Now here's where |
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25:42 | It gets uh complicated and where we to drop out of elasticity to come |
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25:50 | math 101 here is the definition of of the doctor. So it says |
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25:56 | , there's two ways to multiply the product is a scalar and uh defined |
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26:03 | that and you can see that if is equal to X. Then we |
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26:08 | here X one X one and X , Y two X two X two |
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26:13 | X three X three. And so is the square of the link. |
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26:19 | and so here is a really important of notation. It says whenever indexes |
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26:29 | , it means that we sum over liars so we might should have a |
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26:34 | sign here but we don't need it we look here and in this combination |
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26:40 | see an index repeated. So um guy who invented that idea that wherever |
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26:48 | have um an expression involving matrix if you have a repeated index, |
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26:58 | means you've got to some or it's if there was an I and |
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27:03 | J. Then you were talking about IJ component of uh this is 22 |
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27:11 | eyes. So that guy was Albert . So here we are following in |
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27:17 | footsteps of Albert Einstein over 100 years , it was about exactly 100 years |
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27:24 | when Einstein became famous because he predicted famous equations that uh gravitational bodies been |
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27:38 | . So we like those by a body I advance and of course that |
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27:47 | completely unheard of at the time, he was well respected. So they |
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27:53 | the next year there was gonna be uh so uh there's gonna be a |
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28:02 | eclipse, wood or in front of sun, so that what that meant |
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28:09 | you could see a star behind the , close to the sun because the |
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28:16 | was blacked out, but you had be in the right place to do |
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28:19 | and you to be there, you to be in the south atlantic |
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28:22 | So the World Navy sent an expedition the south atlantic ocean to observe this |
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28:28 | and to observe the position of the behind the eclipsed moon actually behind the |
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28:36 | sun. And sure enough they found , it appeared in a different direction |
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28:41 | of being in that direction. It a little bit like this and it |
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28:45 | and it was exactly the amount which had predicted. So he immediately became |
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28:54 | and before that he was just a with frizzy hair and after that he |
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28:59 | famous, there was an interesting point he made two mistakes in his calculation |
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29:03 | that And one mistake cancel out the estate. So if he if he |
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29:10 | published uh mistaken uh thing that he have been so famous, but |
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29:17 | he he he would have corrected it of course, but everybody would have |
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29:23 | he's fudging it, but the fact he nailed it right off and it's |
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29:29 | a small, just a small deviation that, he nailed it right after |
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29:34 | became things and he also invented this that whatever and injections repeated it means |
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29:39 | some of the world values. now we're not gonna show it |
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29:45 | But another way to write this thing is in terms of the length of |
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29:50 | . Length of Y times the coastline the angle between the vectors. |
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29:55 | now here's the other way to make uh to multiply two vectors together or |
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30:02 | by the way, this way is way that we talked about Stephanie was |
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30:09 | quite sure. And I'm I'm supposing she would be even less sure about |
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30:14 | cross buttock because that's more complicated. here's the definition, the eye, |
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30:21 | component of the cross product between two X and Y is given by this |
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30:27 | . Uh you can see it's got that the J component of X and |
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30:32 | cake component of Y. And then funny symbol here, which I'll explain |
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30:36 | a minute and you can see that were implied here that we're summing over |
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30:40 | and J. Something over over J K. Here's J and K are |
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30:47 | , I has left over here uncared . And here we are talking about |
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30:52 | United. So what is uh the I J K. Well, it's |
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30:59 | if any of these two indices are same and it's one if the if |
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31:05 | list of I J K is 123 231 or 312 minus one otherwise. |
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31:12 | if you look at this list this is a sort of natural order |
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31:16 | if you take this one and running to the end so that gives you |
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31:22 | . If you take this to and it around to the end gives |
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31:25 | So these are called even permutations of . And if you do something else |
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31:32 | them it's uh odd permutation. So in symbols then uh for all for |
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31:42 | 3 um components, uh you see looks like this, it's it's |
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31:49 | I can never remember my this I always have to work it out |
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31:53 | detail. And furthermore, it can shown that we're not going to show |
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31:58 | here. It can be shown that is equivalent to X. Y. |
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32:03 | signed dated uh all multiplying times the vector in the Z direction where Z |
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32:10 | perfected X. Y. So those concepts which we will use later in |
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32:15 | point. So I already answered this . And so so much for |
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32:23 | So mathematics, let's go back to vector mathematically in a vector, which |
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32:30 | to different components if you make a choice support system, according to the |
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32:35 | transformation rule. So let's think let's that this is what a mathematician, |
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32:40 | say right here. So let's examine that means. Uh Remember we can |
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32:45 | any coordinate system that we wish if talking about position. The position of |
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32:53 | corner of my, of my So uh we can define that in |
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33:02 | of any coordinate system we like and completely arbitrary and the laptop does not |
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33:08 | , it knows where its corner right. Uh So we want to |
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33:13 | this uh set up our mathematics so we don't uh commit ourselves to a |
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33:23 | an arbitrary choice of things. So let's talk about this uh in three |
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33:31 | . And here's our coordinate system you can see X one here and |
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33:35 | three here. And can you see two is pointing out of the screen |
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33:40 | is it pointing into the screen? uh Stephanie tell me whether X two |
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33:46 | pointing out of the screen or into screen if we have a right handed |
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33:50 | system or you're guessing so hold up right hand and point it in the |
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34:00 | one direction, point your index finger the X one direction and then uh |
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34:07 | uh your your middle finger in the two direction and it is uh is |
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34:14 | three now pointing out or down? . Okay. So now turn now |
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34:19 | your hand over sir. Excuse pointed ? Yeah. Okay. Yeah yeah |
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34:30 | right pointing out right, and that's way you have to figure it out |
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34:34 | your right hand every time you just answer that question. So now this |
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34:43 | is sort of a logical order system we set up here. Um but |
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34:49 | the physics doesn't care. So we to set up the physics so that |
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34:54 | independent of our choice of of a system. So let's consider this two |
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35:03 | vector. Uh Consider the the position shown here and the length of the |
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35:08 | is this like we said now consider choice of foreign system rotate. So |
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35:15 | can see that it's it's the same . I'm gonna go back and |
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35:20 | You see it's the same vector but we've got a different quarter system and |
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35:25 | gonna call it now in this quarter , we're gonna call it vector |
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35:29 | Prime with elements vector elements, it's prime and X two prime. And |
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35:38 | uh you see the new voting system the same as the old program system |
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35:46 | . We could have changed the the origin too. But uh we didn't |
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35:51 | that in this example. It's only further more than the second reporting system |
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35:56 | also orthogonal, right by our So that only the only difference is |
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36:02 | rotation by a positive rotation angle. . So if we hold up our |
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36:09 | like this with our Z. Pointing of the screen uh uh counterclockwise angles |
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36:19 | positive. So here are the two coordinates written out with their components. |
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36:30 | you obviously you want to know what the uh relationship between these sets of |
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36:35 | and so you can work it out your high school knowledge of plane geometry |
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36:43 | this is what you find that uh prime is given by this former ex |
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36:48 | minister platform. So there are those transformer questions. These these may be |
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37:00 | in the following way for the ice is equal to the sum here. |
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37:05 | showing the some explicitly of the repeated . Uh and then so we have |
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37:16 | matrix here uh with J component It is right down here. Um |
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37:25 | eyes count the rows and J count column. So this right here is |
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37:37 | rule for transformation with rotation like So quantity to a mathematician and quantity |
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37:43 | a vector if it transforms for it to another court system like this, |
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37:52 | we talked about. So that array called the matrix and we denote it |
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37:59 | their Children. If it has two we say that it has ranked two |
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38:06 | we can work through opponents like If each index counts up to |
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38:11 | we say that it is a dimension . So uh one we showed there |
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38:19 | right two dimensions. So uh Normally gonna be um looking at matrices which |
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38:33 | the same rank and dimensions either two 2 or three. Now it's easy |
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38:44 | imagine matrices of rank three or more you can't write them on the |
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38:51 | All right, so we have, we have three indices here, it's |
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38:55 | to imagine that that's possible. But third index uh here would would count |
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39:02 | depth, can you see that here depth one and here is depth to |
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39:06 | all the so but that's hard to on the screen, and I can |
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39:12 | you that we're going to see before day is over. We're going to |
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39:17 | matrices with rank four. And so I can't write that at all. |
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39:25 | either in a two dimensional screen or three dimensional uh picture like this, |
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39:31 | four dimensions. So you can imagine four dimensional that uh matrices are ranked |
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39:40 | are gonna cause special problems for Okay, so uh now this one |
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