Welcome to our lecture series Math 3120, Transition to Advanced Mathematics for students at Southern Utah University. Throughout this lecture series, I'll be your professor. My name is Dr. Andrew Missildine. Like I mentioned a moment ago, at Southern Utah University, this class is called Math 3120. It has the name of Transition to Advanced Mathematics. As this is the very first video in our lecture series, I think it deserves us to explain, what does it mean to transition to advanced mathematics? I'm assuming if you're watching this video or participating in this lecture series, you probably don't know a lot about advanced notions in mathematics, like logic and set theory and what have you. Your past mathematical experiences have probably been very computationally focused, that is to say, things like calculus, pre-calculus, algebra, trigonometry, things like that. In that setting where computation is king, a math problem looks like the following. We have a problem. We follow some pre-established process, which we call an algorithm, to solve the problem. This process involves some level of mathematical computations, maybe we add numbers, we take the logarithm of something, we take the derivative or integral of a function, what have you. And then we present our solution like, yay, the area under the curve is 17 pi, what have you. This process I just described here is a pretty good template for solving most mathematical problems that you've probably seen before. But I should mention that that perspective of what this is what mathematics is, is actually quite limited. It turns out it's really just the tip of the iceberg when it comes to mathematical reasoning. This is because mathematics is actually more than just numbers and computations, which people erroneously corral mathematics into being, oh, mathematics is about numbers, it's about computations. There's much more to it. Mathematics is more than just a list of strategies for solving difficult problems. Instead, I like to think of mathematics as a lens by which we view the natural human and digital worlds. Mathematics is a philosophy of how we humans interact with the world around us. Mathematics teaches us to look at the world and with two perspectives in mind. We look at the world in a quantitative way and in a logical way. And those are the two tools that mathematics brings that other scientific or philosophical disciplines don't offer. We solve problems using quantitative and logical methods. Now, I'm assuming that the viewer of these lecture videos is what I would call a proto-mathematician. That is, you have some mathematical training and you want to become a professional mathematician someday. I mean, that could be like, that could be a math professor or a statistician, an actuary, a math educator or many other data scientists, many other disciplines as well. But many things would fall under the mathematician umbrella. But at this moment, you're watching this video because we're a proto-mathematician who desires to transition into more advanced mathematics. And as such, it's necessary for you to learn how one communicates inside of the mathematical sciences. After all, by saying mathematics is a lens through which we view the world, we must infer also that mathematics has its own culture, its own history and its own traditions, which it does, it has all of those things. Now, the mathematical culture, I should mention, it of course is going to intersect with the ambient cultures of the human mathematicians who practice it. Humans have culture and it can be sometimes very diverse when you compare different people together. And as such, that informs how people practice mathematicians. There are noticeable subcultures among the mathematical practitioners, some to do with their own personal heritage and some to do with just best practices where they come from. There are many mathematical standards which one could discuss in this lecture series. And I don't want to get into all of like the social science type of things, mostly just focusing from a professional point of view. This is how the typical mathematician approach is something, but be aware of that. And when I say like this is the typical mathematician, there is of course some diversity in there as well. And these are all things that the uninitiated need to know as they transition to advanced mathematics. Perhaps I'm taking a long way to describe this, but to be clear, this transition that I'm keep on talking about means that one has to learn how to communicate inside of the mathematical sciences. Communication involves reading and writing mathematics. It also means hearing and speaking mathematics. That's how we communicate and how do you do those four things with mathematics? There are many perspectives, many aspects that we could explore as we try to understand how communication is unique in the mathematical fields. And that's the main topic of this lecture series, learning how to communicate. So while this class is called Math 3120, we might as well call it Communication 3120. Now, I actually don't know if that's a course number at SUU, if it is, this is not. I'm not really trying to change the name here, but the point is Math 3120 is how do mathematicians communicate with each other? That's what the point of this class is, okay? And so what you're gonna see in each lecture is that each lecture will be broken up basically into three videos. The first video is gonna introduce some type of mathematical or what we could say quantitative content, some new content that's important for the mathematical disciplines. This very first one is gonna be about definitions. We'll talk about that in just a second. The second video in every lecture will then focus on some type of logical content. And then the third video will then focus on communication itself, focusing on communication skills or nuances about communicated mathematics. In particular, they'll try to be related to the topics we learned in the previous two videos of some kind. You'll see that of course as you watch through lecture one. So the main, the first thing to discuss in this transition to advanced mathematics is the ever important meaning of a mathematical definition. Okay? And mathematics, it is of the utmost importance that our language be precise and unambiguous. If we are to craft sound and logical, valid proofs, then we can only do this if we're certain of what the heck we're talking about, right? We can only infer truthfulness from a theorem. If we know what the theorem says, if we can read the proof and be like, oh yeah, that's correct. If we don't understand what we're saying, then no one can infer any truth whatsoever. So it has to be that our language, our communication is clear. Now ambiguity exists when a term can be reasonably interpreted in more than one way. For example, take the word bank for a moment, bank, B-A-N-K in the English language, of course. Now a bank could describe a financial institution, you know, a place where you deposit or withdraw money, but a bank could also refer to the side of a river, all right? So if I just say what does bank mean by itself, you don't know, because it could mean one of two things possibly more, but at least means two things. Consider the following word, W-I-N-D. What is this word? Now, some of you might have said, this is wind, you know, the thing that blows through the sky, but some of you might have also said it was wind, right? Is it a noun or is it a verb? You know, how do you know how to even pronounce the word? You have to know what its meaning is just to even know how to say it. Now, in natural language like English, which we're speaking in right now, context generally is the tool to avoid these ambiguities. Like, you don't just say the word wind all by itself, you typically are going to put in a sentence, like the wind is blowing hard today or I have to wind up my key. I don't know, whatever people wind up, something. But context is then king. Context gives it meaning. Now the word wind is an example of a homograph. It has the same spelling, two different words with the same spelling, two different pronunciation. So if I gave you the pronunciation, you probably could figure out what it was. But then you look at a homonym like bank where the spelling and the pronunciation are identical. And so the only way to tell the difference between these words is by the context. So in mathematics, it's important we realize that context plays a role just like it does in natural language to avoid ambiguities, all right? Context, you know, in mathematical language, sometimes is needed to avoid vagueness. And it's really, really important that we understand the context we use when it comes to mathematical writings. And to give us context, it often is important, often imperative that we define terms in a mathematical sense. Now a definition is a statement that enumerates properties that the defined term possesses. Just to be clear with the definition, because I'm trying to define the word definition right now. I know it's a little bit weird. We can only infer, you know, from this definition, we can only infer from a definition the enumerated properties given by the definition, no more, no less. And so I put on the screen right now two examples of very common early definitions that students would commonly see in a transition to advanced mathematics class. The definition of what does it mean for an integer to be even? And what does it mean for an integer to be odd? Honestly, many of you probably have seen these definitions before. You know, back in primary school, we talked about even and odd numbers, but let's give it a formal definition to avoid any chance of enumeration, excuse me, avoid any chance of enumerations on a big deal. No chance of ambiguity, that's the word I was looking for here. We say that an integer in is an even number if there exists some other number A such that in equals two times A, all right? So an even number, if we read it again, is an integer such that that number N equals two A for some other integer A. That's what an even number is. If I say, oh, this number is even, then you gain this property and nothing else. So if I say this is an arbitrary even number, the only thing we know is that it is divisible by two. That's the only thing we know about it. I guess we should also mention that for an even number, it does have to be an integer. We don't talk about even rational numbers. We don't talk about even real numbers because when you think of the divisibility for rational numbers and real numbers, everything's divisible by two in a manner, speaking, because if you take the number five, you could factor us two times five halves if you look at the rational number system. So built into the definition, we are referring to as an integer, AKA a whole number, which could be positive, negative, or zero. And when we factor the integer, the two factors themselves have to be integers. One of them is two and one is some other integer. We don't care what that is along the integer. That gives us the definition of an even integer. Now, with this definition in hand, we always know what an even integer means when we talk to each other. An even integer means this, nothing else, okay? And even if I call it an even number, the definition has built into it that it has to be an integer. We can't talk about an even rational number like we said before, that would be nonsense. Nonsense, just so you know, has a mathematical meaning, nonsense is when someone uses a mathematical definition incorrectly to say something like, oh, two plus two I is an even complex number, is an example of nonsense, because describing whether an integer is even or not, it has to be an integer. A number can only be even if it's an integer. Other number systems don't have it. It's nonsense to use a definition incorrectly. We have to make sure we avoid ambiguity, but we also have to make sure we avoid nonsense. Now, conversely, an integer we say is odd if the integer n can be written in the form n equals two a plus one for some integer a, okay? So that's what it means to be odd. In particular, an odd number is just one more than an even number. And you've seen examples of this before, right? The numbers four, six, 12, and 20 are all examples of even numbers, and I don't think you need much convincing, but I will just check the definition to be sure. Four is even because it's two times two. I wanna point out here that when we talk about this sum integer a, this integer could be two itself. It doesn't have to be different. We don't know anything about a other than it's an integer, but there is an integer so that four is two times that integer, so four is even. Six is likewise even because it's two times three. 12 is even because it's two times six. 20 is even because it is two times 10. And that's all it means to be even. You're two times a number. Conversely, the numbers one, five, 31, and negative three are all examples of odd numbers. Why is that? Because I can write each of those numbers as an even number plus one. One equals two times zero plus one. Zero is an even number because it's, sorry, two times zero is an even number there. You have five. Five is equal to two times two plus one, right? 31 is equal to two times 15. That gives you 30 plus one. Now you have to be a little bit careful with the negative number here. Look at the definition. Negative three is an even number, excuse me, it's an odd number because it's, because it is positive two times negative two plus one, like so. Negative two, you take two times negative two. That's an even number. It's negative four plus one. It's negative three. That makes it an odd number. And that's all it is to determine whether we have an even number or an odd number. When it comes to a definition, we cannot infer any property from the definition except for those which are explicitly stated in the definition or things that we can logically conclude from the definition, but that gets into the logic side. We'll talk about that later. For example, well, let's say something about this. We'll say one thing. For example, suppose we assume that in is an even number, then we know there exists some integer such that n equals two eight. That's the definition. We can assume that. This is an acceptable inference because it follows by definition. This is a statement you see a lot when written in mathematics by definition. So we make an inference where we'd say something like, oh, n is even, n is even. So n equals two A for some A. You'd say something like that. That follows by definition. That it's not a really deep inference there because it follows directly from the definition. The fact that we could say more from that using logical implications is one of the benefits of logic. Can you say more than that? Well, with the definition, we can't say anything more. We need some logic to get beyond that. Now, on the other hand, with the same even number n here, so let's say that n is even. Therefore, n equals two A. We cannot infer, we can't infer basically anything else. We can't infer that, oh, n equals three B for some B. We don't know that. I mean, it could be true, but it might not be true. I mean, for example, take the number six. Six is equal to two times three, so it's an even number. Six is also equal to three times two. So that is true. That applies for six here, but if we take, for example, the number 10, 10 is two times five, so it's an even number, but on the other hand, 10 does not equal three times B for any integer B. There's no way to do, there's no divisor of three for the number 10, something like that. So this is what I mean by we can only infer what the definition gives us. The definition of an even number gives you that you're divisible by two. It doesn't give you anything else. Now it could be true, but you need more information than what's provided to actually go with it. So you have to be very careful that we only can say what is given by the definition. This holds by definition. Let's consider another definition for a moment. We say that two integers have the same parity if those two integers are both even or they're both odd. And in this situation, we use the word or here, we're saying that at least one of these two things have to be true, they have to both be even or they both be odd. Now you might be like, at least, can they both, how can both happen, right? Can they, a number can't be even and odd, can it? Well, at the current moment, we actually don't know that because for a number to be even, there just has to be some other integer so that the number N equals two A. But to be odd, we just have to require that there's some number two B plus one that happens. Could there be different integers maybe even the same integer, right? Is there some integer A and B so that these can happen simultaneously? Maybe, maybe not. I mean, we probably believe that an integer is either even or odd, but at the current moment, we have improvement such a statement, we allow the possibility that both could happen simultaneously. So when we talk about the word or here, we're saying at least one of these two things is true. And so you're the same parity if you're both even or you're both odd, all right? So the numbers eight and two have the same parity because they're both even numbers. Now, if two numbers are not, if they don't have the same parity, then we say they have opposite parity. That would mean that they're not both even or they're not both odd, which would have to mean that one of them is even, one of them is odd. So take for example, eight and three. Eight and three, they have opposite parity. Eight is an even number, it is not odd and three is an odd number, it is not even. So they have opposite parity. Let's look at one more definition before we end this video. So one thing I should mention about definitions is that when we look at a definition, typically they'll be marked off with some type of header of some kind of title and typically be like definition. So you know that the next paragraph is gonna define something or multiple terms. It typically have a numbering system of some kind. This is quite common mathematics. Now the numbering system we have is actually got three numbers here. The first two numbers represent the chapter and section that we are drawing from from the textbook. Now our lecture series is loosely based upon the book of proof which is an open source textbook about proof writing. I say loosely because, well, for many reasons we won't, we'll sometimes use examples from the book, sometimes we'll use examples from other places. We definitely won't follow the order of the book because after all, this is the first video and we're in section 4.2. This numbering is just to help you know where in the book these things are located and then there's a number that tell you in this lecture we're at item six. So there was like 4.2.1, 4.2.2 and this will include definitions, examples, theorems, et cetera. So look for these numbers from the future. You can reference a theorem or a definition like, oh, by definition 4.2.6, such and such. Another convention we often use with definitions is that the word being defined often will be written in bold. Some people like to use italics to do it. That's perfectly good. You should put something when you're putting a definition there. So it's very clear that this is what we're defining so that people can find it later on if they need to check the definition. This is gonna be another definition about integers. Why do we keep on defining stuff about integers? Well, I'm assuming that by watching this video, we don't necessarily have a background in advanced mathematics. So we have to use a mathematical topic that I can assume the audience knows so that we can talk about things like definitions. So we're using the notion of integers. We know what integers are. Otherwise, we probably wouldn't be watching this video. So it's pretty safe to make some definitions about integers. It's a good starting place. So let A and B be integers. And be aware that when I say this, let A and B be integers. The only thing I can assume about A and B is that they are integers. I can't assume anything else about them. For example, I can't assume that A and B are different integers. They could be the same one. All we can assume is that they're integers because that's what was stated. And so we can only get what the definition of integers implies. So we say that the integer A divides the integer B if there exists some other integer N such that A, N equals B. And that means that A divides B. And this is often denoted using the following notation to try to abbreviate it. We say that A line here B means A divides B. And that's how you read it, A divides B. And so let's see some examples of this. Note that three divides six. And this is true because six is equal to three times two. Two is an integer. And so there is some number, some integer two in this case so that three times two gives us six. This gives us that three divides six. Some other definitions we throw out there. If A divides B, then we say that A is a divisor of B. Therefore, three is a divisor of six. Also, if A divides B, we say that B is a multiple of A. So since three divides six, that means six is a multiple of three. It's also true that two divides six. So two is a divisor of six. And hence six is a multiple of two. And so putting these terms together, we can talk about divisibility of integers. It's also true that four divides 20. That is four is the divisor of 20. And that's because 20 is equal to four times five. There exists an integer so that four times that integer gives you 20. That makes 20 a multiple of four. Finally though, we gotta give you a counter example here. Here's an example where it's not divisible. Seven does not divide 12. There is no, there is no integer such that 12 is equal to seven times that integer. Now, proving that such an integer doesn't exist can be problematic. And at this moment, I'm gonna kinda take it for granted. I think many of you probably will believe that seven does not divide 12 in the integer sense, but how does one actually prove that? I mean, what if you just have to choose a really big number that you've never thought of before and it'll work? We will get to that. Be patient, young grasshopper. We will get to that in the future. All right? Now, one other thing I wanna mention before we close this video on definitions is that we talked about even number just a moment ago, right? We said that N is an even number. This happens if and only if you often abbreviate that statement if and only if you see it a lot. If and only if you'll abbreviate this as IFF. It's not a typo. N is even if and only if N is equal to two A for some integer A. That's the definition. That's why we get that. So being even is exactly the meaning that you can factor in as two times some other integer. But when you look at this, by definition, this is the same thing as saying that two divides N. So it turns out even means you have two as a divisor. That's the same thing as this. So in retrospect, we defined even numbers first and then we defined divisibility. But we could have done the other way around. We could have defined divisibility and then defined even to be using this divisibility notion. This gives us an example of what we call an equivalent definition. That is you could have defined the same topic, the same concept in two different ways. And even though the definitions look different, it turns out that everything that's true about one is true about the other as well. These two definitions of a number being even, that is this statement and this statement are equivalents to each other. And so when one has an equivalent definition, it doesn't matter which definition you use. And so which one do you choose as your definition of even? Like we use this one, which is a little bit more elementary because we hadn't yet introduced the notion of divisibility. That was a choice that we had to make. It was a convention that was made by the mathematical communicator. And that's something that we need to be aware of. It could be that as you're trying to write it's like a mathematical textbook or a mathematical paper, you choose one definition over another, even though we're equivalent because one is more convenient for the other. Maybe it's logically closer to the topic you're searching for. There's a lot of reasons why one can use equivalent definitions and we'll explore many of these in the future.


tired of it. It's very draining. A lot of the work that I do, I'm very honest. As a scholar, I'm in the place of, and I have to piggyback on my homie Ronnie, where I said, I'm talking to the people that want to hear what I'm saying, right? Because that's where I am, even with my scholarship, right? Because now it's saying, hey, I need some other voices to make the superiority, and it shouldn't be the black professor, the black researcher always talking about the inequality that African American students are facing in mathematics. I'm tired.  And so as educators, how are you voicing this? Because we need allies. We need allies. And it takes people in those powerful positions and powerful structures to now say, now this is a priority for me, right? Because I see other folks suffering. And so this is a conversation that I have with a lot of colleagues. We're tired of talking about it. And so now we need other voices to come to the table and say this is a problem as well. I don't know if I want to take that. What do people in power do? What can they do? There's a great teacher in New York City that some folks probably know. Can we hear this? Yeah. There's a teacher in New York City that some of you may know, Jose Vilsen, who has a great saying about this that stuck with me for a while. He says, you know, in math we often tell our students to show your work. It's kind of our refrain. You do the problem, show your work, show me how you thought about it. He said, you know, as math educators, this is do the work. Like we need to think of that as a refrain in exactly the same way that we need to always be pushing ourselves to say, like, where does this fit? Like Kim, I was saying, like equity needs to be everywhere. It needs to be in spaces like this where we center equity, where we have a specific time set aside and it needs to be the lens through which we look at every other decision we make and every conversation we have and every time we talk about a student or talk about an initiative in the school. And I think for white folks, the work means that you're trying to make those changes, but you hold this tension in your head that you're like constantly approaching that with this enormous sense of humility, right? You're like, you're trying to make change. You're trying to do right by students and you're recognizing that like we don't know, right? And we still have to learn and we still have to be open to the idea that even though we're putting in the work, we're still coming up short, that if we just go about business the way we have and we, you know, teach and we see who learns, then we reproduce the exact systems that already exist. And if instead we take our job as to root around and find that brilliance in every student and we approach that with the humility that every student that we don't surface brilliance in in our class is something that we need to work on and we need to get better at the next class, the next week, the next year, that's the work. And I think that's a really hard tension to hold in your head and it's a really hard like humility to come to, especially with the way we've constructed mathematics that this is thing that you're just good at and we're privileged into and that's who we are and so we need to constantly be pushing ourselves to step back and have that. Sorry, I just love that song. I'd like to also say that having people with influence and they don't all end up being white because I've had to struggle with a lot of administrators who looked exactly like me but had a different agenda, but I'd have to say collaborating, have active curiosity, like seek out opportunities to go, am I walking with this concept as a strand through everything that I'm doing? Like you said, Kimon. It's essential that we practice it until our muscle memory is like this is natural for me. And yes, it's hard, anything you start is hard, but it's about time for the people with influence to recognize that it's time to change your own paradigm of comfortable. If you're still comfortable then you're not changing your environment. And that curiosity helps to inspire curiosity in others. I literally had to go have a meeting with the principal to change curriculum because I wasn't teaching from the textbook. At the beginning of every school year I teach a video from the BBC called The Story of One, which is the history of numbers. I had to prove that that was mathematically related. The history of numbers includes the whole globe. So every student in my classroom, the Muslims and the Egyptians and the Africans and the British, all of those students see themselves in that video, but I had to prove to an administrator that a video about ones and zeros was mathematically related. And I had to take two white female colleagues with me to advocate for me. Thank you. But I could teach from a textbook that was five years too old. Shay or Andrea, do you want to take this question? No? Okay. Before we jump to the next question, I just want to highlight one thing that you said, Dylan. Dylan shouted out to Jose Wilson. Some of you has read his amazing book. This is not a test. It's dope. It is a really good book. And talking about like what we can do, I think just the fact that you all are here in this room, I mean, give yourselves a round of applause for being here, right? Showing up, right? Just showing up is a big part of it, right? And being in those spaces. And for those of you who want to continue that conversation, Jose and the Edu Color Group, E-D-U-C-O-L-O-R does a monthly chat, and they do some amazing stuff just online so you can sort of get connected to that work, right? Okay. So here's another question from Julie Wright. Julie asks, my school's whole teaching staff is white and Asian American. 25% of our students are neither white nor Asian American. My district's math teachers are overwhelmingly white. We rarely acknowledge this directly to students. Should we? And if so, what's your guidance? Sorry, voice, right? Julie teaches in a school in which only a small fraction of her students, about a quarter of her students are white and Asian American, yet the majority of the teachers are white and Asian American. Is that right? 75% are white or Asian American. Okay. So I think what we're getting to is that there's just a disconnect. There's an unproportionality. Is that right? Okay. So should we talk about this? And what's your guidance? Is that right? Okay. I'm sorry, I'm not following the question. Again, what's the... Is Julie here? Julie, you want to read? Yeah. Did she clarify? Yeah, I'm just... Do you want me to clarify? Yeah, yeah, yeah. Shout out. I know what that feels like, or what that felt like to me. I wish somebody acknowledged it, and I'm not acknowledging it to them. But I love your guidance on whether and how... So I'll just say this. I did an ethnographic study with African American high school boys. And the biggest thing that they talked about for me, and I'm gonna just say it like this, they said, black people don't do math because I don't see it. So I just let that sit with you right there. And so no matter how much we talk about it, where's the visual representation? Because that becomes critical for success. That becomes critical for me even thinking that that is attainable, right? It wasn't until one of my boys Carter, he said, I didn't know black people did math until they brought a black mathematician in as a guest speaker to my class. And he talked. And for the first time, he told me about the history of mathematics. He told me that African Americans had major contributions in mathematics. And he said it opened up a whole new world for me. But I'm pretty much out the math game now. He said, Kim, I'm not gonna I'm not doing well in it. But his speech was in powerful. And I wish I had that when I was in like, Kinder or first grade. And so now part of it, and his statement was wrapped into, I don't see folks that look like me that does math. And what happens is in his particular community, it played out because that was reinforced in his home life where his parents also said, Yeah, as long as you get a C in math, I'm good. Right. And so it was that in school, what happens when I come from school, didn't disrupt that any kind of way. And so part of it. And so if he's situated in your school, right, it's going to continue to foster the notion that we don't do this because I don't see any teachers that even represent that. Or you're not even bringing in a space for us to have conversations about that.  Right. And so how are you even having conversations with the students that you have, right, in terms of showing African American mathematicians, right, that becomes very important. So for me, visual representation is huge. And see, I would have I'm saying you said and going to see hidden figures is amazing. It's excellent. But have you had real tangible mathematicians in that school space would be my question. And so how are you providing access? Because we talk about access, right? I need somebody I can touch feel Oh, Teddy's a real statistician. Right. I have a friend that's an African American and he's a statistician for the NBA.  So it's bigger than just having a conversation. What are you actually doing that's making it material for kids? The materiality piece of it? Well, what I want to address this question, Julie, he says it so well. We need to be conscious that we're functioning in the last 40, 50 years of just getting to equality. And so that was such a struggle and it still is it's ongoing. It has not ended. But equality is not equity. So equality was the power for us to even have the conversation. But equity is now just like King was that is now it's about doing what we said we were going to do. So it's actionizing the philosophy that they wouldn't let us talk about before. So equity and equality are not the same things. Equity is action. So Peg has a really good question. Peg Kegel, am I pronouncing that correctly? Yeah, where are you, Peg? Yeah, there you are.  And you're asking for advice. Is that okay? Yeah, right? Mm hmm. So I yeah, I'm gonna front this by saying that I want to jump up and down that question. There's there's there are a lot of scholars who don't use the word ally anymore, just because it's the word that's been coopted. But but it also there's an aura of missionary sort of altruism associated with being an ally versus you see yourself in that walk together. You see the struggle is all of us together in this and not that you choose to help, right? I'll just put that out there. I think there's this fear that could be paralyzing where we there's there's this fear that can be paralyzing where we feel like we get this conceptually. But we're not sure what a first step is. And article that I think should be mandatory reading for educators right now is Rachelle Gutierrez again. It's called creative and subordination. I forget the subtitle. And I'm happy to share a link. But it's this really great piece that gets that like we work in these institutions where we can feel like we don't have any agency and actually making a change, right? And like navigating this line of how we support students within these structures that just recreate a lot of the inequities that we want to work against. And she has some really, really both harsh words for the way education is set up right now and some really great ways to think about steps that we can take, even in institutions that feel like they are unchangeable to do that advocacy for students and create situations where those students can shine and activities for folks in power to take on to practice that type of empathy and practice that type of thinking. So I would say check that out and think about what of the strategies that she offers there. Can we engage within schools to make small changes right away? I also just want to add and you make a brilliant point. I also want to say the way for me you can de-center your whiteness is co-constructing knowledge with your students. Because what happens when you do that is you're no longer, you're no longer the main act. You're no longer the feature attraction. It's not about you. You are actually empowering your students to create knowledge together in a space together, right? Because oftentimes we as educators and we as teachers go in and it's all us. I say if I go into a classroom and the teacher doing 90% of the talking, that's not a place where I want to be. Right?  Whoever it is, your nest is being centered when you are the one engaging in ideas, showing kids how to think, what to think, what to do, how to go about this, solving this formula instead of providing multiple ways of knowing. Right? And so you de-center whiteness when you really start co-constructing knowledge with your students. And that's a skill that takes years to learn. But it's something that I think is a beginning concept. So that's moving away from that teacher-centered classroom to more of that student-centered space. And then your whiteness becomes de-centered. Yeah, I just want to echo what you're saying, Kimai, is just like we know that in order to be a really good math teacher or to be a math teacher that your students empathize with is to show that you're learning to position yourself as a learner. Right? Position yourself as a learner also about these issues. Right? You know, many of us have no idea. We've never taught black history in school. And so you're learning along with them. Your students have stuff to teach you that you never learned. Right? I think for me, you know, being Chinese-American, I walk this fine line as well. You know, I am a person of color, but I'm extremely privileged in the math space. Right? And I've learned to navigate that and somehow sometimes takes advantage of my privilege, but also just be a learner. And it's always been a learning. Like, I'm just starting to learn about, like, LGBTQIA history in the United States. This is something I never learned, and I had to really immerse myself and get, connect with other community. And so that's the other part is none of us does this work alone.  Right? It's really isolating. So find compadres. Find a crew. Find your people. Right? Find what keep I'm going to steal your word. Find your cipher. Right? Anyone else want to take this question? So we're down to our last five minutes. And I wanted question three for us to think about action steps we can take as lovers of math to make math more equitable. And you have to say this in something that's tweetable, something short and sweet. So why don't we start? Andrea's got the mic. You want to start? Okay, something tweetable. I would say I want to teach mathematics and do mathematics in a way that's very personal. So I want my students to be empowered to do mathematics themselves. And I want to present myself as a real human being who does mathematics. So I don't want to strip away any part of me in order to fit in with some norm that might be might have been set for me. Can't tweet that. That was a lot of characters. Sorry. I don't use Twitter. Okay. I'm thinking. I don't tweet. I guess I'm still thinking. You were not supposed to hear those other comments. I guess to make math more equitable, what I would say is if if if you're the only one who gets it, you really don't get it. That's what I would say. My tweet has a citation to Danny Bernard Martin who is who I've been reading on this and is is my my recommended read to everyone in this room for your next steps in your education. But you know paraphrasing all of his work, do the work to question what mathematics is until you can surface brilliance in every student. Be conscious that the students who walk into your classroom are living in the path that was made for them and not one that they make. Let them lead because equality is not equity. I guess I'm ready. No, I'd say math equity means relinquishing control. I know we shouldn't drop them. I got to follow that. I think I think remember every kid we work with could be your kid. So I think we have we have one more question that's popped up on Twitter. Yeah. Oh, my apology. You are absolutely right. I'm getting a technological privilege. Does anybody here have a question? Anyone have a question? Please raise your hand. I'll come by when we voice the question. Yeah. Oh, we have one there. Yeah. 90% 5% of you kids are white and the boarding program we have a lot of Chinese students and Asian students and I had a BC Cal class this semester and almost everyone was Chinese or white man. So going forward when they get to the college and they have other people of color, how could I like what could I tell them so that they're welcome and prepare them so that when they see someone else, they don't just kind of disregard. I don't feel like I'm talking so much. That becomes a critical question because and I'll go back to the study I did with my brothers in the R one space in the STEM space. The problem became nobody want they were last to be picked when any time they were form formations of groups. So my African Americans talked about how Asian and white males did not want to work with them via vis-a-vis lab space lab projects. Anything just did not want to work with them because they feel that they didn't have enough intellectual prowess. Right. So that becomes a reality. If I were you man, I honestly, day one, start talking about and showing up the statistics of who does math in the United States and I'd have an honest conversation about that because what that's going to allow you to do is going to first of all allow you to have conversation of how are these people in that space even feeling about it because I never even want to make assumptions with students about thinking that they won't be welcoming. There may be brothers in your class that are extremely welcoming but you may not know that because you haven't facilitated a discussion to even find that out. Right. And so I think showing statistics, you know, I'm very forthright when my teachers come to my math methods class. I literally show them who's participating in STEM and it's broken down by race and miles like oh my God because the hard data hits home because we don't talk about it. Right. As a nation we say it but rarely do we put math and now we talk about math work. Right. I literally show the math associated with and people are astonished. We got to do something. So I would I would welcome you to even have that conversation and see where that leads you and see what the what the folks in your classroom space is actually thinking and then that sets you up to how to further engage them. I think that's our time. I apologize. Thank you very much. So before you roll just quick shout out to the math equity workshop group. Those of you who have deeper questions about this we're just a representation of some people but there's a lot of great teacher leaders here a lot of great math equity folk many of them are sitting right here in the front. You've got some questions to ask come on up. Cool. Have a great Friday y'all. Peace.


Right, it's happened. I've finally ascended. I went an entire week without using the internet whatsoever not the internet I actually ended up not using Computers or phones or anything like that either And you know, I think if I wanted to do a good clickbait video, you know I'd probably say something like and it changed my life But the weird thing is it actually didn't change my life In fact my life weirdly enough was I don't want to say it was definitely wasn't exactly the same But you realize exactly how little you need the internet when you go without it for all this time I mean I've talked about in other videos mainly when talking about like social media and like cell phones and how much I hate them You know, there's this tendency There's this habit that people get you know when they're addicted to cell phones where if they go like a second of boredom Or even if they imagine they that they might become bored. They just immediately pull out that phone Scroll through the social media feed whatever. So there's this tendency, you know, psychologically what happens there is You know, what once you start doing that and you you see all this like pseudo information you see all this junk on social media And you sort of feel like you're being informed or you're being bettered or you're learning more about your friends by doing this kind of stuff But when you just go go cold turkey without it You sort of realize like exactly how untrue that is and I think that was definitely my My case when I went a week without the internet. And so anyway, how did this happen now first off? A week or so before this week happened. I sort of felt like okay I've been using the internet in maybe antisocial ways in the sense that You know, I was using it, you know, I don't have a social media account or anything, but I do have email I get lots and lots of emails So responding to emails, of course I you know barely respond to five percent of my emails but it still takes up a lot of time to deal with them and you know, there's I guess a lot of enjoyment in that and in the same way I have, you know, RSS feeds for all these things that I subscribe to and Sometimes you just get too many like you always feel like okay. Well great this new article's up here I better look at that or this guy posted this And it's sort of exact sometimes it ends up being exactly the same as social media just without the upcoming and stuff like that So, you know, I was definitely like, okay, I've been using this stuff too much I'm nearerotically updating my RSS feeds too much. So I'm just gonna go without it Okay, we're just gonna go without it for a week. See how we can do, okay? And now first of it should be obvious that once I you know, I turned off the internet I woke up the next day and obviously that was the most productive day of my life It was probably the most productive day of my life like before 10 o'clock in the morning because I got up, you know, you do like the thing is when you have that ability to be bored Okay, this is another video. I got to do like why being bored is good Okay, but when you have that when you have boredom sort of Lurking around you you actually end up doing all the stuff that you've know you've had to do for a long time One of the first things I did is I actually cleaned out Actually, I washed up my master bathroom, which I hadn't really been keeping up with like when you have a well there's a lot of sort of Minerals that get in the bottom of like bathtubs and stuff like that.  I ended up fixing my car replacing the valve on my spare tire Finally getting a code reader for my car Changing the oil just like all these things like all at once that most of which, you know I've been putting off for like at least a month like I've needed an oil change for at least a month or so Maybe around there if to be safe So all this kind of stuff I ended up doing it, you know immediately now what I'm not trying to say Like the in not using the internet like its main benefits was not actually the fact that I got way more stuff done Although it definitely is true that I got more way more stuff done. You also like spend your time very differently Like I read I probably read two or three books I usually read no books a week, but I probably read two or three books or at least parts of some other ones And I did that and also like for example I had long phone conversations with people because you know if I'm just sitting at the house if I'm You know the thing is so on the internet.  Okay in real life if you're bored You do one thing you have to do and if you get bored of that you go to another okay? Like you know, I started running ethernet wires in my attic Which I've actually been meaning to do for like a full year And when I got bored of that I'd do something else or if I was bored of doing all this kind of work I'd call a friend and talk for an hour So which is something that that is just not in my not something you usually do nowadays I actually got a new phone by the way. Well a new phone. I had this phone that really had really bad audio We're talking about landlines here. Of course, you know, we're boomers.  Anyway, I'm just yapping about the things I did in my life Okay, it's not too important But just suffice it to say that it got lots of stuff done and you just spend your time differently Okay, like when I went out on the town usually when I go out on the on the town You know, I pretty much do what I need to do and come back But you know in this case, you know, I talked to random people There's an old lady in a shop. I ended up talking to talking with her with like for like 90 minutes or something like that You know just all the all this kind of stuff that I guess normally you wouldn't do Your time becomes more important when it's not to burden by trivial things Okay, and not just more important But you have so much more of it that you could actually do things that are a little more depthy, you know If that if that makes any sense so And the weirdest thing is even though like I mean I mean the thing is like the temptation of using the internet occurred to me several times Oh, and let me tell you well, I'll get to you know, what it was like when I first turned on the internet again But you know, I was never really that tempted like you're always like, okay Well, I could use the internet and sort of cheat here But I have so much so many better things to do and it just sort of recalibrates your your desires to that So I don't know it's just it's just different and in time and like that week that one week felt like it was like months In terms of the things I got done like every single day was like a week basically So, you know a long time Anyway, so eventually the weekend did and I was like, okay Well, now I can use the internet again, you know by my own Valuation so I decided to turn it on and just sort of I mean, of course And in the back of the month in the back of my mind for all this time I was like, oh dude something terrible on the internet could have happened like I could have My websites are down or like someone could have hacked my bank account Or you know all this kind of stuff that it is just just crazy thoughts So I logged on and I synced my emails and saw the week's worth of emails that I haven't seen at all There was only one that was important that I missed and that was no big deal.  It feels so good. I can do all this kind of stuff that I've been wanting to do But really the feel like like that my first impression was wow, I haven't looked at the screen for like a week It's actually really painful to look at the screen. My boomer eyes were squinting. I was just like what? It look it literally hurts to look at a screen. You don't realize it until you haven't done it for a while because you know, I didn't even look at a Sell I may have looked at no I did you look at a cell phone screen for like a split second I had to turn my cell phone on and off at one point, you know, you guys know I usually don't I basically keep my cell phone off all the time But um, there's one situation where I had to of course I had airplane mode on it.  Yeah, like it was just not like the guilty pleasure. I thought it was gonna be it's almost as if like I've been deconditioned from using the internet now I still have I Don't know like I feel sort of bad I did even record videos over that period like not on the computer, but like with my phone And like I don't know I barely feel motivated to upload them because like the internet seems like some fake world now Now I literally feel like a guy who's just talking to his phone in the woods. Is that what I am? I don't know. Anyway, so Yeah, so of course, I always highly recommend people to reduce the dependence on technology and this is probably a good example I Like the thing is I I sort of thought that I'd be really bored when I stopped using the internet and I'm sure you will be bored To some degree But the issue is like boredom is the thing that drives you on to do things that are important Okay, so, you know, if you're in this constant state of pseudo Stimulation like you're never gonna get anything done. That's just the fact of the matter Now at this point, I don't I don't know if I'll be like, okay I'm just gonna go without the internet for like a month or something like that and go from there Because I sort of feel like okay now I can really easily constrain my internet usage like it's not something It's too difficult So I might be able going forward like in the past couple days where I have been using internet now I don't know. It's just I haven't been using it that much. Frankly. It's not been like I'm returning to some kind of drug All right. So anyway, that's about it You don't have to you don't have to get to 10 minutes on YouTube anymore Even though my YouTube reflexes are almost there, but so that's about it. I encourage you to do it try it out Stop using social media. That's about it. See you guys next time


وأقول في القرآن ما جاءت به آياته فهو الكاريم المنزال وأقول قال الله جل جلاله والمصطف الهدي ولا أتأواله الحمد لله رب العالمين له الحمد الحسن والثاناء الجميل وشدوى الله إلا الله وحده لا شريك له وشدوى النسيدنا ونبينا محمد صلى الله عليه وعليه وعلى آله وأصحابه والتابعين لهم بحسان إلى يوم الدين أما بعد إن شاء الله تعالى في هذا الفيديو سأتحدث عن مظاهر دنوو الهما سأعطي بعض الأطفال في which low aspiration enters شيء في which low aspiration is found in دنوو الهما low aspiration يأخذ مظاهرة عدة it goes into many forms in our life the first one inshallah that i want to mention here is having low aspiration when it comes to seeking knowledge and it happens in five ways having low aspiration when it comes to seeking knowledge it occurs in five ways the first is ملا يطلب العلم مع الصلاة a person who doesn't seek knowledge at all he will not go forward in seeking knowledge he does not exert the effort to learning the necessary knowledge that he or she should know he's happy to be ignorant he doesn't want to exert what is needed from from them in gaining knowledge even though he has the physical and the mental ability he doesn't want to seek it that's دنو الهما في طالب العلم that's the first form the second form is من يطلب العلم a person who will seek knowledge in seeking knowledge because he knows the value of knowledge the second way in which low aspiration occurs is a person who will seek knowledge and it is not because that this person understands the virtue of knowledge and if they are not doing it to raise ignorance from themselves or to purify themselves or to spread that knowledge no the highest aspiration here is is to attain that certificate and to get that job after it oh when i when i get that certificate i'm going to be the إمام of that big message in my local area الله أكبر always wanted that all he sees seeking knowledge is that certificate it will guarantee me a job and i'm going to live a life of happiness and that is what many people are even if they you look at the academic sciences that they learn they'll learn medicine and what and they do it for money so i can get money a good job they don't do it because they want to help the umma they don't have that aspiration very low aspiration the third is a person who seek knowledge but his reason is he can be higher than his friends and his colleagues where you marry a Sufaha he wants to put down the ignorant ones and say yeah you don't know anything when are you going to start studying exactly i'm better than your he wants to he wants to point out the scholars and put the scholars in difficult situations and ask them difficult questions and ويصرف وجوها الناس إلي wants to turn the people to him he wants to be the the focus العلامة محمد الخضر حسين رحمه الله he said something very powerful he said ويطبه لينال به رزقة أو ينافس فيه قرينة حتى إذا أدرك وظيفة أو أنس من نفسه الفوز على القرين أمسك عنانه ثانية وتحن عن الطلب جانبة وإنما ترفع الأوطان رأسها وتبرز في مظاهر عزتها بهم أولئك الذين يقبلون على العلم بجد وثبات ولا ينقطعون عنه إلا أن ينقطعوا عن الحياة يقول رحمه الله they have not given knowledge يقول لم يقبض حق العلم they have not fulfilled the rights of knowledge and they haven't understood the honor of knowledge which people the one who is seeking knowledge for risk what am i going to eat what am i going to drink or he's learning it so he can compete with others and belittle the people and put the scholars in in a situation and show the people that he knows better than the scholars those people haven't understood the honor of knowledge the fourth way in which low aspiration in knowledge occurs is بعض المتلسبين للعلم a group of people who attribute themselves to knowledge as soon as they become you know grounded maybe in some sciences or they might learn some issues related uh to the religion or they might gain some knowledge here or there opens his tongue straight away to the people of knowledge and he understood a miss alla hero there and he maybe studied a chapter or two of knowledge and then he now goes out and he critiques the scholars those were grounded who are older and more senior and more great noble than him he opens his tongue at them insults them belittles them this is a sign of low aspiration as well ولذلك at the poet he said this line of poetry what it means is that that's two years of age he completed two years of age is now moving on to the third year and um they called it is because the mother when she gave birth to it the camel gave birth to this child she abandoned it in breastfeeding and she had another child that she was breastfeeding with it and so she gave milk to that other camel so ابنولبون is a male that's finished two years and moving on to the third year in the zakat the person he doesn't give the ابنولبون unless he has first he has to give the ابنولبون goes first if you don't have it then okay ابنولبون is accepted from you he's saying the poet here is saying and the camel that's three years and he's two years and now going to the third year can it really come into a challenge and fight with البازل القائس he can't the the camel that's eight years and again that's can it really go at it no so the poet here which is جرير he's saying that i am here the and the other people are ابنولبون so today a lot of students are just ابنولبون they are ابنولبون compared to the scholars who are the uh poet his statement here which is the بزل القائس scholar is senior than you more knowledgeable the more grounded than you and also another poet he said إذا تلاق الفيول was دحمت فكيف حال البعوذ في الوسط وماذا سيبقى في مسكيطة تكون في حالة حيث أن أبقى أنموز يجب أن يدخل right sometimes you find a student of knowledge he will get into the discussion of great scholars talking and it's not open for you if you come here or crush you it's low aspiration the one who has high aspiration أبو this is if he can knowledge he's a an objective in life is his goal is to benefit others that is what he wants and he also wants to respect the scholars and when i say respect the scholars i don't mean that you just take everything that they say that whatever the scholar says take it without questioning him no what we mean is look in the evidences look at the statements observe research without a doubt but respect the scholar for his اجتاعدة he's reached and he says he's he got it right or wrong and if the prophet said that the scholars are either between one or two rewards how are you here to shame him and name call him you see and what we have to also understand brothers and sisters is that الفطر السليمة the pure فطرة والنفوص الزاكية and the honorable souls will never ever result to those type of actions to speak about a person of in a very despicable way and what we have to understand is that people will never accept what you say if you're opening your tongue to the people of knowledge and it's a sign of low aspiration that all you want to become known for is that you open your tongue on the people of knowledge that you open your tongue on the people of knowledge و لذلك at the poet he said من قدر و إن كنت مشارن إليه بالتعظيم فالكبير والعظيم يعني don't ever put a person who's high in the eyes of the people don't try to put them low because the truth of the matter really is that you're the one that's going to drop you're the one that's going to drop not the person who is you're saying something about the fifth and final point in regards to having low aspiration in seeking knowledge is that which takes place between the students مِن تَحَسُدِن وَتَغَيُر وَتَنَارْفُسٍ غَيْرَ شَرِفٌ the enmity and the grudges and the feelings that happen between students of knowledge knowledge really is meant to be the thing that brings about lights brings about harmony brings about good it's not what brings about hate and grudge and enmity the poet said الراجل اللبيب زيادة ونقيصة للأحمق الطياشي مثل النهاري يزيد أبصار الورا نورا ويُعشي أعينا الخفاشي the poet said العلمو knowledge للراجل اللبيبي knowledge is to the one who is smart a good thing the smart person knowledge brings good for them it pushes them to that which is good it benefits them لكن الاحمق the demoted one الطياش that jumps on everything knowledge is actually destruction for him when knowledge comes to him he doesn't he doesn't but he becomes arrogant and he starts to insult everybody إن شاء الله تعالى i'm going to stop there for this episode anything which i have said that was wrong or incorrect is from me and shaitan and allah and his messenger are both free from it سبحانك اللهم وبحمده أشد الله إلهي لله استغفروا كواتوب إليه السلام عليكم ورحمة الله وبركاته how can you do a two second action right now that will give you a share of the reward of everything we're doing on this youtube channel simple like this video and click subscribe why it will allow youtube to recommend our videos to other users and imagine the huge amount of reward that could be waiting for you on the day of judgment if you did that with a sincere intention of spreading the dean of allah you'll be rewarded for every single person who benefits from one of our videos as a result of your like or subscribe that's an easy two second action that you definitely don't want to miss out on do it now click like and subscribe and don't forget to make that intention


It is impossible to imagine theoretical physics without the Kroneka delta. You will encounter this relatively simple yet powerful tensor practically in all fields of theoretical physics. For example, it is used to write long expressions more compactly and to simplify complicated expressions. In combination with the Levi-Civitta tensor, the two tensors are very powerful. That's why it's worth understanding how the Kroneka delta works. Kroneka delta ij is a small Greek letter delta which yields either 1 or 0 depending on which values its two indices i and j take on. The maximal value of an index corresponds to the considered dimension, so in three-dimensional space i and j run from 1 to 3. Kroneka delta is equal to 1 if i and j are equal and Kroneka delta is 0 if i and j are not equal. Let's make some examples. Delta 11 is equal to 1 because the indices are equal. Delta 23 is 0 because the indices are not equal. A times delta 33 is equal to A because the indices are equal, so the Kroneka delta is 1 and delta 23 times delta 22 is equal to 0 because the first delta yields 0 and the second delta yields 1, but 0 times 1 is still 0. In order to represent an expression like this or an expression like this compactly, we agree on the following summation convention rule. We omit the sum sign, but keep in mind that if two equal indices appear in an expression, then we sum over that index. Another advantage of the sum convention in addition to compactness is formal commutativity. For example, you may write down the expression like this in a different order as you wish for example like this. This might help you see what can be shortened or simplified further, but be careful. There are exceptions for example with a differential operator delta j which acts on a successor. You can't move something before the derivative that is supposed to be differentiated, so you should be careful with operators in index notation. Now let's learn some important rules using Kroneka delta. Rule number one, indices i, j may be interchanged. Delta i, j is equal to delta j, i. Why is that? According to the definition, if the indices i and j are equal, then delta i, j is equal to 1, but then also delta j, i is equal to 1. And if the indices are unequal, you have delta i, j is equal to 0, and delta j, i is equal to 0 as well. So you can see Kroneka delta is symmetric. Rule number two, if the product of two or more Kroneka deltas contains a summation index, like in this case j, then the product can be shortened such that the summation index j disappears. Delta i, j times delta j, k is equal to delta i, k. So instead of writing two deltas, you can just write delta i, k. We say the summation index j is contracted. Let's do an example to understand it better. Consider delta k, m times delta m, n. The summation index here is m, so you can eliminate it by contracting it and you get delta k, n. Next example, delta i, j times delta k, j times delta i, n. Here you have two summation indices i and j. So in principle, you can eliminate both of them. From the first rule, you know that Kroneka delta is symmetric. So you can swap k and j in delta k, j and then contract the index j. You get delta i, k times delta i, n. And then you contract the summation index i. The simplified result is delta k, n. Remember that the contraction order is not important here. You could have contracted i first instead of contracting j first. In both cases, you get the same result, delta k, n. So which wave simplification you take does not matter. Rule number three. If the index in a, j also occurs in delta j, k, then the Kroneka delta disappears and the factor a, j gets the other index k. a, j times delta j, k is equal to a, k. This rule is basically another case of index contraction. This rule tells you that you can also contract summation indices that don't have to be carried by Kroneka delta. Let's make another example. Consider gamma, j, m, k times delta, n, k. The summation index is k, so you can eliminate it. The result is gamma, j, m, n. Rule number four. If j runs from 1 to n, then delta, j, j is equal to n. Why is that? According to the summation convention, the summation is carried out over j here. So delta, j, j is equal to delta 11 plus delta 22 and so on, up to n. And each Kroneka delta yields 1 because the index values are equal. So 1 plus 1 plus 1 and so on results in n. If, for example, like in our case, j runs from 1 to 3, then delta, j, j will be equal to 3. Consider a three-dimensional vector v with the components x, y, and z. You can represent this vector v in an orthonormal basis as follows. Here, ex, ey, and ez are three basis vectors which are orthogonal to each other and normalized. In this case, they span an orthogonal three-dimensional coordinate system. The Kroneka delta needs vectors written in index notation. Here we do not denote the vector components with different letters x, y, z, but we choose one letter, here the letter v, and then number the vector components and the basis vectors consecutively. The vector components are then called v1, v2, and v3. One of the advantages of index notation is that this way you will never run out of letters for the vector components. Just imagine a 50-dimensional vector. They are not even that many letters to give each component a unique letter. Another advantage of the index notation is that by numbering the vector components in this way, you can use the sum sign to represent the basis expansion more compactly. It becomes even more compact if we omit the big sum sign according to the summation convention. Look how compact the vector v can be represented in the basis. v is equal to vj times ej. Here is you know with sum over index j. Whether you call the index j, i, or k, or any other letter, is of course up to you. Now that you know how a vector is represented in index notation, we can analogously write the scalar product of two vectors a and b in index notation. For this we use the just learned index notation of a vector. aiei, scalar product bj, ej. In index notation you may sort the factors as you like. This is the advantage of index notation where the commutative law applies. Let's take advantage of that and put parentheses around the basis vectors to emphasize their importance in introducing the chronica delta. The basis vectors ei and ej are orthonormal. Recall what the property of being orthonormal means for two vectors. Their scalar product yields either 1 if i and j are equal, or it yields 0 if i and j are not equal. Doesn't this property look familiar to you? The scalar product of two orthonormal vectors behaves exactly like chronica delta. Therefore replace the scalar product of two basis vectors with a chronica delta. So a scalar product b is equal to ai times bj times delta ij. If you remember the third rule, you can contract the summation index j if you want. So the scalar product becomes ai times bi. And you get exactly the definition of the scalar product, where the vector components are summed component-wise. We can write out the double summation over i and j just for practice. In other words, we have to go through all possible combinations of the indices i and j. So we get a1 b1 delta11 plus a1 b2 delta12 and so on. Because of the definition of chronica delta, only three components of 9 in total are not 0, where i equals j. So you may omit all summands with unequal indices. Using the definition of chronica delta, you get the scalar product you are familiar with.


This is stock number 9J324A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used Jeep headquarters. Today we are checking out this super clean 2017 Jeep Grand Cherokee Trailhawk. This vehicle has the 5.7 liter V8 Hemi motor. From this HD video you will be able to tell that this Jeep is extremely clean all the way around. Bright white is the color. We shoot all of our videos in 1080p so if you have HD capabilities on your computer tablet or smartphone device turn them on right now because it's like you're right here looking at the vehicle with me and definitely your best way to check out the Jeep before you actually get here. Comes with the painted and polished aluminum rims, Goodyear Wrangler all-terrain Adventure 26560R18 tires and these tires look really new. I would say probably about 80 to 90 percent of the tread left on those front tires here. Didn't see any dents or dings on the front fender. Has the LED running lights, the projector lamp headlamps, the LED fog lamps, the front bumper parking sensors. You also have your adaptive cruise control. I believe it's a radar there as well. Passengers side fender is in excellent condition too and the passenger side rim no scuffs or scrapes on that. As you go down this side of the Jeep you can see just how clean the body is, how reflective and mirror-like that paint is. We take these HD videos so if you are far away or even if you're close by and just cannot make the trip down but you're still interested in purchasing the vehicle you can still see the vehicle hear the vehicle and have confidence in what you're looking at before even get here. Very very clean down this side I didn't see any dents or dings. This back rim is in excellent condition as well and the back tires have just as much tread as the front tires. As we come to the back of the vehicle rear bumpers in really nice shape it does have the dual rear exhaust full towing package which includes a receiver hitch four pin and seven pin wiring. LED tail lamps, the rear gate is in excellent condition. This is a power lift gate so you just press the button and that comes up for you. Back here the storage area is in excellent shape. You do get a factory subwoofer and a full-size spare tire with the jack tools right there. You also have the sunshade right here which is a really nice feature that comes open like so. So if you got groceries and stuff keep the sun off of them. Also kind of a safety measure keeps your stuff valuables out of sight out of mine type deal. Press this button it'll beep at you three times and then it'll come on down. You can see that gate shuts nice and smoothly and solidly. As we go down this side of the Jeep just as clean as I pass on your side no dents or dings on the rear quarter. This back rim is in excellent condition for full disclosure and down this side just as clean as that pass on your side no dents no dings very very nice. It does have blind spot monitoring heated mirrors and built-in directional mirrors. Inside the trailhawk package gives you the black leather and suede interior. You get the trailhawk stitched into the backrest and there are no scuffs or scrapes or rips or tears on those seats. It does have driver's side lumbar, factory floor mats throughout, power windows power locks, power mirrors, memory driver's seat and power fold-in mirrors as well. Auto headlamps you do have the telescopic and tilt steering wheel to power and as we hop inside the vehicle you can see that this one has 16,779 miles. You get the 7-inch TFT display, you get your compass, outside temperature display as well as the digital speedometer and you can set those up however you want to read different things such as the radio station or miles to empty or whatnot. Here's your cruise control as well as your adaptive cruise control right there. Bluetooth and information center controls your paddle shifters and then there are audio controls on the back of the steering wheel. This one comes with the 8.43C touchscreen radio, has the factory navigation you can also do your heated and cooled seat buttons right there as well as your heated steering wheel. You have AM FM and Sirius XM radio capabilities. This is also where your backup camera shows up.   You do get the, this is the quadrilift system so you can put it up and down and it'll actually raise and lower the vehicle. Your snow sand, auto mud and rock Quadra Track 2 system with the neutral setting for flat towing as well as a four-wheel drive low and a crawl select setting right there as well. Eight speed automatic transmission, passenger carpeting and seat are in excellent condition no rips or tears on that. This vehicle has never been smoked in, smells very clean, does have the power vista that's what it looks like when it's opened up all the way to dual pane and then this front piece does open up like a normal sunroof.  You do get the 911 and assist system in the mirror which are features of the vehicle that you never have to pay a subscription for. We'll take a quick look at the back seats and then we will check out under the hood. Back seats are just as clean as those front seats no rips no tears back here it does have the latch child safety system like I said very very clean no rips or tears and these seats do fold down almost completely flat for extra storage which is really nice if you got a haul a bunch of extra stuff. These seats are heated on the outboard side and have two USB ports as well back here and factory floor mats and carpeting. We'll take a quick look under the hood I would personally like to thank you for checking out the video today hopefully from this HD video. You will have been able to tell just how clean this Jeep is all the way around the inside and out under the hood. I was in line it does have the 5.7 liter V8 Hemi motor engine base very clean runs very smooth this vehicle has been fully safety and inspected by a service shop has a fresh oil and filter change all the fluids have been checked and topped off The vehicle has been gone through mechanically 100% and is 100% ready to go to see more pictures of this Jeep or one of our other 450 new and used cars trucks SUVs minivans wranglers half tons three quarter tons one tons you name it we got to go to our website www.summitauto.com full pictures and descriptions of every single vehicle from two locations all at summitauto.com and if you'd like to check out more HD videos you can go to youtube.com www.summitauto.com remember to like subscribe and share on this video and all the videos that you see there in fact in a second you will see a link to subscribe to our youtube channel on your left it links more SUV and inventory videos like this one on your right if you've not been to our website on the bottom link to this vehicle on our website click those check us out and we really look forward to helping you with this super clean 2017 Jeep Grand Cherokee Trill Hawk Hemi thanks again