So this lecture is going to be about social institutions, what they are and about how they fit into a functionalist and a conflict point of view. Let's pretend for a moment that the worst has happened, apocalypse has occurred. And we have succeeded. We've killed off all the zombies. And the question is now what? How do we build a new society, a post-apocalypse society? Well, in order to do this, sociologists say that we are going to have to answer some basic questions. And these questions are what sociologists call social institutions. Now I want to be clear, you should think about these questions as kind of like DNA. These questions are answered differently by a lot of different people creating what some estimate to be over 7,000 different ethnicities. In some ways it's 7,000 different combinations of answers to these questions. So these questions are what all societies have in common and these questions have to be answered by societies in order to establish and maintain a society. And of course, if we're building from scratch, the first thing that we've got to figure out is how do we make decisions together? We've got to decide what to do, how to set things up, how to get things done, how to work together. And in order to do that, we have to establish some rules or at least some practices that will help us know what to do and how to make those decisions. And the answer to the question, how do we make decisions, sociologists call governance. So governance is a social institution that is based on how we make decisions. Each one of these social institutions has a question that has to be answered. And of course, societies are all over the place and how they make these, how they answer this question. Everything from one guy at the top gets to decide everything to we all have to talk to each other until we all agree in a kind of consensus and a whole bunch of other things in between. The next question that we got to answer is how do we distribute goods and services? So we need food, we need shelter, we need clothing, we need to be protected from the elements. We need to have people make roads, we need to have people provide transportation, provide manufacturing and so forth. And as we become more diversified and more complex a society, we have all kinds of goods and services that we need or want and they have to be distributed in some way. We call this distribution of goods and services or the answer to this question of how we distribute economy. So an economy is basically the method by which we distribute goods and services. And of course you can imagine that there are lots of answers to this question as well. In capitalism, most goods and services are distributed by who can afford it.  But there are other societies that distribute goods and services on the basis of need or on the basis of family status and so forth. So there are lots and lots of different answers to how we distribute goods and services. And of course if we're going to maintain our society and sustain it over generations then we have to figure out what to do with the kids because we have to make babies in order to survive and once we make them we have to have ways of taking care of them. Now this social institution is called Families, but I don't want you to have just a single family type in your brain when you think about this. Families can include everything from the entire village raises the kid to extended families, to fictive kin. All the different family types that we've talked about before are just any kind of way in which a particular group or society protects its children. So the word families is sort of misleading. We do not necessarily mean specific family types based around biology. That's just one answer to this question of how do we carry on from generation to generation. And then if we're going to meet up with other societies, exchange with them, people start moving around as the population regrows and so forth. You have immigration and of course you have to teach the kids who come up. So we have to teach our ways to newcomers in order to pass that information on from generation to generation. And again we call that education and in the same way that you don't want to think families in just the way we answer the question you don't want to think just formal education at this point. The answer of how do we teach our ways to the newcomers can be anything from mentoring, to sitting around a campfire, to telling stories, to having rituals that people go through. There are all sorts of ways in which we educate newcomers. And newcomers again can represent people who are immigrating into a group or they can be newcomers in the sense of new people as in children. This last social institution that sociologists have talked about traditionally may not be quite as obvious as the others. The question that gets answered is what is greater than ourselves? And this may not seem as obvious to the survival of the group as these other questions are, but it certainly seems that this is something that all groups have in common. And when you think about those sociological ways in which groups identify themselves and understand themselves, you understand that there is a need to know who the other is or who the out group is. And for a long time in human history that did not involve other human beings because human beings live very far apart from each other and didn't encounter each other often. I should say groups of human beings didn't encounter other groups very often. So the answer to this question has been, well, has been called by most sociologists as religion, but I don't want to mislead you too much. I've called it higher order because the answer to this question can include the climate, it can include nature, it can include things that are outside the control of the group all the way up to things, you know, deities and understandings of gods or who God is and so forth. So again, this doesn't seem like it's a survival question, but it is a question that every society so far has had to answer. And sociologists speculate that the reason that they, there's a need to answer that is that it helps develop group cohesion and group identity. So governance, economy, families, education, higher order, these all constitute what sociologists call social institutions. Now I want you to distinguish between the idea of an institution and a group. An institution is an abstract concept. It is not an entity that you can see or something that happens in the material space whereas a group is, you know, a group of people. You can see that they're a group and there's some ways in which you can know by observing that that group exists. Social institutions are observed sideways. They're not observed directly. So to give you an idea, governance is a social institution, but Congress is a group. So even though Congress is acting upon the question of how do we make decisions, governance is the answer which would be democracy or representative democracy or whatever, but Congress would not be the governance, Congress would be the group that performs the social institution. So I hope that that's clear. You need to distinguish between what is a group and what is a social institution and the distinction is that a social institution is abstract where a group is more concrete. So we're going to talk about social institutions. The next question that often comes up, especially because this is where conflict theory and functionalism break with each other, is what is power? And we've talked about this before in the semester, but just to remind you, a good definition of power is strength over time. Strength in that it is somebody who tells somebody else what to do and they do it. And over time meaning that it can be sustained that as, you know, that when this particular group entity or person has power, they are able to repeat this process and have people repeatedly obey them and do what they ask of them, not just a one-time, a one-off. We don't think of people as being powerful and they get their wages once. We think of people as being powerful if they get their way consistently over a period of time and then they stop getting their way over a period of time and we start thinking of them as less powerful. So time is a very important element to this. If we're going to talk about power, then we have to start talking about the power elite. The power elite is a concept by C. Wright Mills. He wrote a book called Power Elite. This is where this comes from.  They play together. They go to the same schools, grew up in the same neighborhoods, go to the same church, belong to the same clubs. You know, they basically know each other, have known each other's families for a long time, belong to the same organizations and so forth. What he's saying is that this is no different than any other social group. This is no different than your inner circle, your closest people to you. It's just that when the powerful do this, when a group of people whose decisions influence other people's lives do this, it can become a situation in which that power is multiplied. So when he looked at how these social groups played together, he identified Mills, identified three particular centers of power in which the power within these entities become interchangeable with each other. And he talked, this was written in the 1950s, and he talked about political, military and economic elite. And what he means by interchangeable is that as the power in one of these areas grows, then the power in the other two areas grow. And one of the reasons that they're interchangeable is because there's an exchangeability of roles at this elite level. So oftentimes it's talked about as leadership skill or as people who have good standing in the community, that kind of thing. So it's taking a look not so much at their history, their education or their abilities or skills. It's taking a look at where their positions are within society and within the social group of the power elite. And what this means is that if you are in politics, it's not unusual for you to also end up being in business. And if you're in business, it's not unusual for you to end up part of the political system. And if you're in the military, it's not unusual for you to retire and become part of the economic system or go into politics. And so this exchangeability of roles means that if your experience was in one of these spheres, it opens doors for you in the other two spheres. And I mean this is even true of political and economic going into the military. Most of the time if you're part of this elite, you come in as an officer. And I know that at least one situation where the guy who was the first head of the CIA was made a general in the Air Force without ever rising up through the ranks. But usually it goes the other way. Usually you have people like Dwight D. Eisenhower or even our first president, George Washington, who the only office they ever held politically was the presidency that they were deemed as qualified for that job because they were generals. And their military record allowed them to get in. And you also have a lot of people who come out of politics and go into economics or vice versa and especially within the same realm of regulatory industry. So you have people who go into politics and write laws about particular industries and then once they're done with that and they've served their terms, they go into the industry that they wrote a law about and become experts on the loopholes in those laws. So this exchangeability of roles at the elite level happens all the time, sometimes to tragic ends. We saw this with Katrina, the head of FEMA, the Federal Emergency Management Administration, was a friend of President Bush's and he had no background in emergency management whatsoever.  He was doing things like calling for meetings the next day instead of doing things in a timely manner. And within three days of all of this falling apart, he became the person that was fired and somebody else was brought in. But by that time the damage was done and quite a few people may have lived that died after Katrina because of that mismanagement. But most of the time it doesn't have those kind of dire consequences. And we know that a lot of money is being exchanged and there's a lot of conflict of interest and a lot of expansion of power for a few people. But it doesn't have the dire consequences that Brownie being the head of FEMA during Katrina had. So this occurrence is not that unusual. Nowadays we would probably add some other spheres into this. If Seawriting Mills were writing now, we probably would add media because there are a lot of people who come and go in these spheres into the media. And you might also add sports because you now have athletes who pursue political careers or athletes who come out in the middle, you know, people who come out in the military and pursue athletic careers and people who are commentators on television shows, that kind of thing. So there are, you know, celebrities and sports and media are probably expanded circles of his power elite. But the basic thing that Mills was getting at was that this is a logical progression that occurs when a group of people interact. People interact with each other and are close to each other within social circles. And it doesn't require an evil plan somewhere. There's not necessarily an evil cartel making this happen. Basically this is what happens when you and beings hang out together and create groups together. But, you know, you and I would, if we worked somewhere and knew that our boss was looking for an employee and we knew a friend who was looking for a job, we had to hook them up and recommend them to our bosses and so forth. And, you know, somebody more qualified might not get the job, but the ripple effect probably wouldn't go beyond that. The problem is that when those who are part of the power elite do this, the ripple effect of this goes way beyond their group. So Mills's publication of this and his observation of this mechanism is part of what helped create the need for affirmative action. Because we came to understand that it wasn't the prejudices of the people who were making these decisions that were keeping other groups out, minority groups out and keeping minority power out. It wasn't a matter of attitude. It was a matter of doing what most people do in their social groups. So it needed to be broke up. There needed to be a conscious effort to break that logical progression and ensure inclusion of other groups in a conscious way. And to use the terminology in an affirmative, meaning a positive way. A couple of things to mention at this point. One is that it's pretty obvious that political military economic also parallels Eisenhower's idea of the military industrial complex. And of course this is created as a situation and unintended consequence of a push for more and more military intervention or military S intervention. And there are of course other industrial complex that exists when the political and economic spheres become interchangeable with each other and create these power situations which have unintended consequences. And another concept that we haven't talked about before is interlocking directorates. Interlocking directorates are when people serve on the board of directors for more than one corporation. And this is perfectly okay to do if those corporations don't have any thing in common. If they're not in the same industry, they're not competitors with each other, are they're not part of the down chain. So the way that goods and services are distributed and manufactured is that you generally have a vertical line between the manufacturer down to the customer. And that vertical line involves companies that are competing with each other to produce things, to manufacture things. And then you have companies that are competing with each other to distribute those things, usually trucking or shipping companies. And then you have retail outlets that are selling these things to the ultimate customer. So this chain that exists in economics can be monopolized. If a company owns the means of production and they own the means of distribution and they own the retail shop, then they can have an economic advantage over companies that they're competing with at all three of those levels. And this of course is why software companies like Apple and Microsoft get in trouble all the time with antitrust laws, because they very often buy up some of their distributors and own this chain in a way that makes it very difficult for smaller companies to compete with them. But another way to accomplish this is to put people on the board of directors for all three of these different types of companies within a chain. And of course, you know, you can't, if you sat in a board meeting at one place, you can't forget or not know what was said at that board meeting when you sat in a board meeting in another place. And these interlocking directorates can create that vertical monopoly. And there are a lot of laws against horizontal interlocking directorates, horizontal monopolies, so you can't serve on a board of directors for two companies that are supposed to be competing against each other, because your service of that would give them an unfair monopolizing of the market. But a lot of times the chains are not within one country, they're multinational. So consider for a minute that you might have a manufacturing company in Mexico and a trucking or shipping company out of Canada and a retail company out of the United States. And you basically would have somebody who sat in the board of directors of all three, there's certainly no rules about whether or not an American can serve on a board of directors in another country. And when it becomes international, it becomes very hard to break it up over antitrust because there are no international laws to appeal to and you lose jurisdiction. So this is a way that a lot of companies sort of make their positions more solid by having these interlocking directorates. And they generally are suggestive of an amalgamation of power and creating unfair advantages in the market. But they wouldn't exist if the power elite didn't exist because these people who were tapped to do, to sit on multiple boards are usually part of this power elite. So the last functionalist idea that I want to talk about, and this also can be considered a conflict idea because it helps locate where the power is, is something that sociologist Robert Merton talked about, and that is manifest versus latent functions. So the word manifest means visible and the word latent means hidden. So we're talking about the explicit function that an organization or a group has. This is a declaration on their part. You know, most organizations nowadays have something called a mission statement. Let's take an example of the College of Southern Nevada. You can see the mission statement for CSN up on the inner campus communication systems, little televisions all around. It shows up every once in a while. It's on their website. It's in all of their literature. They're pretty direct about it. You know, it's basically the same thing that would be said at most colleges, has to do with, you know, making sure that a diverse population is educated and prepared for the world outside and qualified and the jobs that they're going to do and so forth. And, you know, this is, of course, one of the purposes that CSN exists, but it probably isn't the only purpose that it's serving. There are purposes that CSN and other educational institutions serve that benefit either society as a whole or a power elite group. These benefits sometimes are positive. I'm not trying to imply that they're only negative, but oftentimes they are negative. And as such, they're not talked about directly. So they can be hidden functions. One of the ways to figure out where they're at is to follow the money. So who's benefiting? Who's profiting from an organization? Can give you an idea of what purposes they're serving? Taking CSN, one of the functions of college and, in fact, of most education in the United States nowadays is to teach you how to obey. To be a good worker bee, you are faced with all kinds of authorities, both administrative and within the classroom. And these authorities you're expected to listen to and obey as part of getting your degree. And this puts you in the habit of listening to authority and obeying authority, which puts you in a habit that a lot of people in the power elite like. Because whether you work for them directly or you merely vote politically on a local or national level, you oftentimes do things because you're told to by them, and directly or indirectly. And as such, that benefits them. And that's probably one of the reasons why education persists and why colleges like the College of Southern Nevada persists. Has as much to do with this kind of function as it does with educating and training and so forth. So the thing about latent functions is that they oftentimes are very powerful, not just because they're hidden, but because they're the thing that actually people would lose money or time or effort or power if they went away. And so these latent functions are oftentimes the reason why you can make perfectly logical arguments that in order to fulfill the manifest function, things need to change. But if those changes would create problems with the latent function, then there is a huge resistance to that change. So this has been a discussion of some functionalist ideas. These ideas of course have a conflict theory take on them, but you should be able to understand after this lecture what a social institution is and how power works within society and specifically how manifest and latent functions work.


यि लक्र जगंखा सफ्व दी आन्रक्तोंते तुंतऻ केगी मे थी मेन है Broken たo N parka Ji ॐकि रसत장님 की आतो संद चा सunning ऑाफotionaffง!! कुषा चातदे में आपत खाल ही ओओवह लाँत बन।त रडा दाअequछगुं। त globally do journey or hard expedition tutorials जDB। तो समझ९ारी छवनों शरह कbooks, these videos i turn yourself in this kind of things i try to make the other people a lot of interesting videos and in these videos they are the only teenagers especially boys and that why they do like this many people when they are going will come after the abilities they will clap for all time in the fourth generation they really started to courageous they did ha ha ha even in the Poland they said they must achieve great-great thing ڈین ڈین ڈین ڈین ڈین ڈี ڈین ڈین ڈین ڈ dez git the next building that mashing. ڈ romance ڈ Oracle j installed to the city. ڈ carriers ڈn Потому Juara boo pe pohhmal come into town. ڈen ڈین ڈین ڈین ڈین ڈین ڈین ڈین ڈین ڈین ڈین ڈین ڈین ڈ 이� my cats wirez all over Yogyah. ڈین ڈ domain plane sam bhaze slimring me down. BRUNU TensorFlow%산ู ▦  impaton phabas  NBC disappears geben soak pya änder  canonical avía changing a cool �翻邊 జੰ અ distractions pulled ons last month in a place, where fried blows had struck her. I made my own video. Many people had Breaking promises. Got 3Lakh views from this video. They expect me to achieve 100,000 views from this video. So many people like That. That's a blessing. Do people watch eating videos in a better name than最دEER? Jie 도 gym retreatment the poor people". ॐ 푝 records. ॐ jo ॐ jCon achive twice.  groundbreaking zombie పట మలిలాయ황 .ichtlich కాతాది ˈɉ ən ɟ ɟ ɚ ɦ ɸ ʀˈd ɐ ɇ ɟ  bonded up ɒ ʃ ɸ ɕ ɱ ɫ ɫ ɦ ɔ ɛ ɪ ɒ !  तो अब ब्रोठी थी ज Freund completely open. आप उछ definitely sh lengths k怪 yo mo hat write विन यो वेग अपक लाना वैत हैंA a turn त penniyon7 Powder नब उकिष और मैं भो फोरत गे five या जो Matter of Man मैं नहये देख formulू भी अगरे गा pull


or knee pain when you bend, run, or go up and down stairs? How about hip or low back pain when you're simply walking or getting up from a chair? Aloha and welcome to Movement Matters. I'm your host Christine Linders, physical therapist and board certified orthopedic clinical specialist. My show is designed to bring you not only the most effective physical therapy tips, but also holistic information to help you achieve total body wellness. Today, I'm going to teach you simple tips and strategies to correct how you move when you walk to decrease your pain right now. Also, you will learn quick and easy home stretches and exercises for each of these areas to end your pain. You might not know that how you point your foot when walking affects the joints above at your hip, your knee, and your low back.  Here's why. So when you stand and your feet are symmetrically pointed forward, something that's called the chain reaction happens efficiently when you walk.  I know that sounds like a complicated concept, but it's really important if you want to avoid plantar fasciitis, Achilles tendonitis, knee arthritis, and hip arthritis, which can also lead to low back pain. Let me give you an example. So I know a woman who is 77 years old right now, actually 78, and she walks with a very crooked knee and it's identifiable. And one of the things she suffered with since she was, I think in her 50s, was painful bunions on her foot. And this one was the worst one. So to avoid the painful bunion, she tended to walk with her foot out. Not that that helped, but it kind of was what the body said to do. When you walk with your foot out, your arch collapses too much. Your shin bone rotates too much.  So one bone rotating inward too much while the other one has to go straight because you're moving forward, even though your foot is out, creates wear and tear at the meniscus of your knee, creates wear and tear of the cartilage of your knee and leads to painful knee arthritis. So this same person was having so much pain going up and down stairs. So I suggested that she turn her foot from here with the crooked knee, say here to align that so that the shin bone wouldn't have to dive in so much, creating that twisting at her knee. And of course, she had so much less knee pain. Another problem with walking with your foot turned out is plantar fasciitis. Plantar fasciitis is something that I have had so many complaints of lately. And it's sad because with the shutdown, retirement, people gaining weight from being at home, everybody's trying to be more active now to lose weight, to be more healthy to get moving. And I'm seeing a lot more injuries from walking, which is something that we do on a regular basis.  And then your arch has to collapse more. And then the tissue on the bottom of your foot gets wrung out and twisted. And you get this painful, painful condition I've had it in your arch, in your heel. And how you know it's plantar fasciitis is you get out of bed in the morning after a nice restful sleep, and you step down and your heel comes down and you go, and you hobble your way down to the bathroom in the morning. That's plantar fasciitis. So I have a video right now to show you that will explain one of the best things to do for plantar fasciitis to get yourself out of pain right now. I know rolling your arch over a ball is exhausting, right? Even I want to yawn watching that. But anyway, it's actually quite very soothing. So the fascia gets these little micro tears in it and it forms scar tissue. And if you were to rub your thumb on the fascia on a healthy foot, it would feel kind of smooth. But if you would run your thumb on the arch on the fascia of an unhealthy foot or one that has fasciitis, you'll feel these little bumps. And that's because it's scar tissue forming over those micro tears. Now what the ball does is it irons are the scar tissue, so it becomes smooth again. So a ball like I use is about the size of a lacrosse ball, a tennis ball. I use a golf ball. I've had plantar fasciitis so many times I have extremely high arches and I used to run Ironman triathlons and things like that. So my feet are very mobile and I'm doing a long distance running or was. So golf ball comes with me everywhere I go, but that's one of the fastest way to get a plantar fascia pain right now. Now let's look at video number two, which is the next thing you want to do. Okay, to stretch your calves, it's really important to put one foot directly in front of the other pointing straight and not turned out.  You want to make sure that you don't fan your foot out. You want to look down to make sure it's in a straight line.  Then you want to make sure that you bend the back knee to stretch the other big calf muscle that doesn't cross your knee joint. You shouldn't feel any pain during this stretch. Just a pulling back here on the first stretch and a little bit lower. Enjoy your walk. So why I emphasize keeping your foot straight is again when your foot is turned out like it's almost natural for everybody to do when they stretch their calf, the arch will roll in and you end up stretching your fascia more than the tight calf muscle. So you want to make sure that foot is pointed directly straight. So you direct the stretch into the calf muscle so that you can help the fascia not be so strained from having a tight calf. So I have a question. Thanks so much for sending this in. Can foot pain be caused by back problems? The answer to that is yes. And I know because I experienced it when I herniated two discs when I was living in New York City.  So my foot was moving in and out like this when I walked in. I didn't even know it. I knew that my leg was getting weak because it was getting skinnier on that one side. And I had a lot of numbness, but I didn't realize how my foot was doing. I didn't feel off balance at all until I started having pain. So your nerve can get compressed in your back from a disc injury or another back injury that will cause irritation in that nerve.  So that definitely is true. And also, should someone complete at least one course of physical therapy before considering surgery? In my opinion, yes. There's exceptions to that rule if you've had a back injury, however. So you do need to go see your doctor and that's who would refer you to physical therapy. But most doctors now are sending people to physical therapy because they're having good results in getting rid of their pain and also getting that education and core awareness and how to bend and move and stretch so that you can maximize the outcomes of your surgery if you actually need it in the end. So that's a great question. So now you've stretched your calves, you've rolled your arch out and that's something you need to do every day a couple times a day.  And in the morning, when you wake up, you'll need to move your ankle up and down like this because the reason why it hurts in the morning is as soon as you sleep and you sleep like this.   I always say 20 times and all my patients come in and tell me, wow, that's so fantastic. I feel so much better to get in the morning, get up in the morning. And also it's important, especially here in Hawaii where none of us wear any kind of shoes in the house. If you have plantar fasciitis, you need to get a pair of house slippers or shoes or something that has an arch support. I did a previous show where I mentioned Vionics or Scots where they have the arch support. You can see it on the inside. I can even show you, I'm wearing one right now. See this dome right here? You have to have arch support there while you're in your house walking on your hard floor. My patient came in yesterday and said that she finally got house slippers. She doesn't even walk down the hall to the bathroom in the morning without the slippers and her foot is 80% better. So take that to heart and get some house slippers that you can wear inside until you get rid of your painful foot pain. Now let's go to video number three. To strengthen the intrinsic muscles in your feet if you're suffering from plantar fasciitis, you want to curl your toes or curl your toes into a towel or curl your toes into the sand. Here in Hawaii, we have endless stretches of beaches. So a great thing to do is walk and curl your toes around the sand or simply just stand and curl your toes in the sand. But when you are walking in the sand, you want to curl and stretch. I think I'm going to get taken out by a wave here. I was up on the North Shore and earlier in the in the morning the waves were breaking and there were some surfers out there and they were coming up the beach though with the waves. So that was a little funny to do in that video. But yeah walk in the deep soft sand and curl your toes into the sand. Strengthen the intrinsic muscles that run between your toes and in your arch here in the foot and that will help you recover faster from your plantar fasciitis. Okay, let's go on to knee pain. So let's say you have knee pain. It could be on the inside. It could be on the outside. It could happen when you're walking up or downstairs running. It could happen when you go to get up from a chair or get up from the toilet. You could have just torn your meniscus doing aggressive household remodeling and gardening. Whatever it is let's look at video number four to see what you need to do to get pain-free. If you have knee pain all of a sudden or if you have osteoarthritis and you're getting too much wear and tear or you even have a meniscal injury that you just suffered because you were gardening, the first course of action is of course ice your knee if it hurts. Then what you want to do is massage your knee. There's a lot of tissues around the knee. I like sports cream but you just start on the sides. You massage your IT band. You massage your quad.  You massage around the kneecap. I like to take my fingers on either side of the kneecap and go around the patella tendon here. You don't want to push down on the kneecap and grind in there because that can cause pain if you have arthritis. So you massage your knee the back. That's your first course of action. The next thing you want to do is you want to re-educate the quad muscle because if you have swelling it will shut down the inside one. So you want a tense. You press down. You hold it. You can hold it for five seconds, 15 seconds, 30 seconds and relax. If it's too painful to push down you could take a rolled towel put it under your knee and then push down like that even lifting here as long as it's pain-free. And the next thing that you want to work on is restoring your alignment. You want to walk with your foot straight. You want to make sure your knee is diving in and you want to make sure that your hip is not weak which is I'll show you in the next video. Okay so I was stumbling over my words there a little bit. I'm one of those one-take wonders when I do these videos. You want to make sure that your knee is not diving in. That's that crooked knee syndrome I mentioned in the beginning of the show. It's very important that your knee stays over your foot and does not dive in. I forgot that very critical knot in there. Okay so you're going to do that to help decrease your knee pain right now but also if you remember me talking from the beginning about keeping your foot straight now you also have to start working on the alignment of your leg when you walk, when you open downstairs, when you go to get out from a chair.  It can be very subtle so even my patients sometimes have a hard time noticing it. Other times they look down and say oh my oh my gosh I never knew I was working walking with my foot out.  If you have stairs in your home and you go to put that foot up look and see is it turned out. I guarantee you it will be if you're standing and it's turned out. Turn it straight and start to improve your alignment. My patient yesterday came in and told me I'd seen him last year and now he's back. He has very severe osteoarthritis in his right knee and his right hip and he's waiting for a joint replacement. But he told me yesterday because I said he was doing so much better. He said you know what I remembered what you said about turning my foot straight and I'm doing it so much more now and it feels so much better and he said he'll never forget when he met me last I think February or March. The first thing I said to him was turn your foot straight. He had terrible pain radiating down his thigh from his hip into his knee and he said it was a miraculous how much that helped with his knee pain. Now he'd been walking like that for years so he got the wear and tear leading to end stage osteoarthritis in that leg yet when he still turns it straight now even though he's going to get a joint replacement surgery he still feels better. So alignment is so important. I want everybody to really grasp that concept. Let's look at video number five. One of the most critical things to keep in mind when it comes to knee health is alignment and your knee relies heavily on the alignment from the hip and the alignment from the foot below.   Well you can see my foot but the foot is rolling in the arch is collapsing. The other reason is the hip is not strong enough because for various reasons and so the hip is dropping like this when you walk and you can see when my hip drops this hip my knee rolls in here so if I'm standing on this leg and my hip is stable I can do a squat and have alignment perfect. If my hip drops my knee goes in behind the other one so it's very important to strengthen the hip and it's very important to strengthen the foot. The number one reason for over pronation at the foot when your arch collapses is tight calves which I showed you earlier so stretch your tight calves. If your arch is weak use strength in your arch and if your hip is weak then you do these next exercises. Okay so let's go straight into video number six to see the next exercises for your hip. There are many ways to strengthen a weak hip. Here are just two of my favorites. It gets the gluteus medius which is the muscle that's responsible for keeping your hip and your pelvis stable when you're on one leg.    You don't lift because it uses a different muscle you want to use the one in the back. You slide your legs apart for 10 times that keeps your pelvis stable, works your gluteus medius, do not let your knee unlock because that could hurt your knee.  Keep your foot straight though because remember we're working on alignment you're retraining your big brain how to hold your body you'll do 10 then you switch to the other side you keep squeezing your buns you keep pressing your knees down and you do three sets of 10 an easier one that works the rotators is to put the band above your knees sometimes it's a better place to start you keep your spine and your spine again and you open up your knees that works the rotators you can also do this one in sitting it's just to help strengthen your hips you can also open up your knees and lift your bottom that's a bridge that gets the big gluteus maximus so these are just a bunch of exercises to start working your hip muscles to stop hurting your knee so those are just a few exercise there's many that you can do but I wanted to give you the ones I think will help you right now to end your pain right now so I have another question how do I know when to use heat or cold to treat muscle pain that's a little bit of a tricky one to be general about so let's say you wake up in the morning and your neck is stiff and you feel very tense after a hard day at work that's a great time to use heat your muscles have been tight like white knuckling it your knuckles get white they don't have blood flow and you put heat on to bring blood flow to the muscles so it can just relax and go ah when you don't want to use heat is let's say you just had a car accident you had a bad whiplash or you were running and you pulled your hamstring or your calf muscle that's an acute injury that is over you're going to have an inflammation response you don't want to put heat on that because that fuels the fire of an inflammation response that you'd want to use ice for for a muscle injury I hope that's helpful I know it's hard to generalize okay so let's get into the hip now we kind of breached on how to strengthen the hip and how to turn your foot to help your knee but now what if you have hypersitis from running or you just have hypersitis and it's painful to lay on it at night because that bone there is just pressing into the bed for hours or you have hip osteoarthritis or you have IT band syndrome which is pain down the outside of your leg from running or walking the hip is kind of the next joint up that kinetic chain from the foot to the knee then you get the hip but it's the interface between the back so now if you have hip pain or you have hip pain that's kind of towards the back it actually could be your back that's hurting I met a gentleman at the beach yesterday I was getting some food and I was talking to him about my show and that sort of thing I mentioned it and he said oh I need to talk to you I've got some hip pain or back pain when he said I want to lose 40 pounds I've been trying to walk and my back hurts me so much that now when I bend over to set up for work I don't like it just it just kills me it's like a week and when I walk it hurts it's just really sore so one of the things I found out in his history that he had had a hip replacement on each leg but the side that his back was hurting got complicated it got infected and he had to have it revised and I asked him that after I watched him walk and I could see that he was twisting a little bit when he walked he didn't have hip range of motion on the left side so I want him to be able to walk to lose weight because he's trying to be active to lose the weight like the rest of us are trying to be active now during all this crazy six months of shutdown and everything like that changes so I gave him a couple stretches and let's look at video number seven to see what I told him to do all right to stretch those tight hip flexors because you're doing your walking program to lose weight and you want to get rid of your hip and back pain this is what you do you go to your pretty beach you get in a lunge position try not to get slashed by a wave and you bend you want to make sure you suck your stomach in and you can raise your arms over your head it's like your stomach in and stretch you're going to stretch the right side not this side I shouldn't switch legs try not to fall here so you're going to stretch this leg you bend you keep your tummy in and you stretch you don't extend your back but you want to get this muscle also it challenges your balance so you want to make sure you do that before and after you do your walking program and enjoy sorry about the blur on that one I think there was so much moving going on the background but I think you get the gist so also another example was one of my patients who I was working on her shoulder a couple weeks ago and she said her shoulder was feeling better but what was worse is her her leg and she said she had walked twice and had this terrible pain from the outside of her hip halfway down her thigh and actually bruised but she was walking she never did anything but she said it was torn and she'd had physical therapy before for strengthening but she couldn't even walk from my clinic to her car without hurting so I said okay well let me just take a quick look at you so I had her walk down I said come on up walk down the hallway she walked down the hallway and immediately I could see that her left leg was turned out like that so I said okay any injuries other than this thing and she goes no no foot surgeries no knee pain no no no no just hip pain it's terrible so I had her lay on the table before I started stretching her shoulder and I checked her hip range of motion and I saw that that one leg that was turned out the motion was really impaired on the way in and she had no injury history so I asked her do you just sit on the sofa with your legs off to one side all the time because I don't understand why your hip range of motion is so limited and she said yes I sit like that all the time so I massaged her quad and outside of her thigh similar to I showed you in the knee video I stretched her hip into a little bit of internal rotation showed her how to do a stretch at home and told her to walk with her foot straight and then we worked on her shoulder for 45 minutes she left that day and she came back in like five minutes later to tell me she goes I just walked from here to my car and I had no hip pain for the first time I've had this for year and a half I can't believe how simple that is so I'm just mentioning her story so that you can hammer home to you how simple achieving normal movement even at your foot can impact your hip so let's look at video number eight to learn more hip pain to be pain free right now these are the things that you need to do whether you have hip pain because you started a new walking program and you have barcidus or you have hip pain from arthritis and it's a stiff hip this is how you get started so you're going to lay on your back and you're going to hug one knee up to your opposite chest don't feel bad if you can't go all the way you're going to stretch the night glutes muscles so you could pull it in here knee towards your opposite chest and breathe you want to make sure you do both sides and again if you're here that's okay if it's tight you want to give it time to lengthen another great great way to get hip range of motion when you have hip arthritis is to get hip rotation so you want to spread your legs as far as you can drop one knee in as far as it goes let the other one fall out you can use your hands to help your hip if you need to and then you go the other way drop one knee out the other one in and stretch breathe and then repeat so if you have very bad hip arthritis and you find it difficult to move your leg this is a great way to gain some range of motion you want to get on your hands and knees and sway your back a little pull your belly button in to protect your spine and rock back it's very important not to let your background because when you rock back this way it enables the hip joint ball to get back in the socket where it should be and not where it's riding causing you pain and wear and tear from hip arthritis so after you do these motions it's important to strengthen so you want to use your big glute so you want to lay on the surface and you can lay on your bed if you'd like but this is a nice table for it you're going to put your hips right here put your head down pull your belly button in and lift up just high enough to squeeze the glute you don't want to lift this high and get your back involved you only want to use your glute muscles you do 10 times on one side then you're going to repeat and do 10 times on the other side and because when we walk we alternate our legs and a lot of people are walking right now to get fit during this COVID shutdown we're going to alternate suck your stomach in alternate your legs if you can't go high that's okay you want to be pain-free so those are the things to do to get out of hip pain right now excellent so I hope everybody really tries to work on these exercises to get out of your foot pain out of your knee pain and improve your hip range of motion so that you can save your back it's so important now at this time that you have the tips and the tools to do in your home so that you can get yourself out of pain and enjoy life to the fullest right now during these difficult times so I have a little bonus for you and I bet you never thought that having poor posture could affect your hip or your knee or your foot however it can because if you're very rounded for 30 or 40 years you become more rounded and it alters the position of your spine which alters the position of your pelvis and your hip joint socket and can place more strain on the front of your knee so I'm going to give you a bonus posture stretch in video number nine okay so you just finished your surf session you just finished hunkering down over your laptop and everything in the front of you is tight and you don't want back pain and you don't want hip pain and you want to be able to go and enjoy your day so it's very important I do this every morning and I do it at work to stretch out the stuff in the front that got tight so you're going to stand in a doorway I like to have my arms and what I call it is the T position straight at shoulder height or slightly above you put one foot in front of the other and use your back foot you exhale and then you press forward your back foot and inhale and you stretch you'll feel it get tight right across here then you push back and you inhale now you can exhale from that position when it gets very tight inhale again lift your chest you can also look a little bit to the right breathe in now you stretch more of the left side then you come back look a little bit to the left breathe in and then relax and enjoy better posture the final piece of the puzzle that I gave that gentleman that hurt his left low back after his hip replacement from trying to lose 40 pounds is to suck it in I told him you need to suck it in before you move and engage your deepest abdominal muscles so let's review video number 10 for that let me introduce you to your secret weapon to end lower back pain and get a flatter stomach your transverse abdominis or ta as I call it is your deepest abdominal muscle it's been around since the beginning of time just waiting for you to call on it to help support your back but no one told you it was there if you have back pain you need your ta to form the anatomical girdle with your deep back muscles to support your spine you see the ta contracts just after your brain has the initial thought for motion your brain thinks I want to reach for that glass of milk your ta fires then your arm moves try it pull your belly button in towards your spine go ahead suck it in now that you know it's there use it before you move and enjoy a pain-free lower back excellent so thank you so much for tuning in please try these strategies so you can decrease your pain and get to pain-free and enjoy your life aloha everyone


Crops are susceptible to various types of pathogenic microorganisms, which can cause significant economic loss to farmers. To combat this problem, several laboratory-based and indirect detection methods have been developed to identify and monitor plant diseases. These include PCR, IF, FISH, ELISA, FCM, GCMS, thermography, fluorescence imaging and hyperspectral techniques. Additionally, biosensor technology has emerged as a promising alternative for early detection of plant diseases. This article was authored by E. Fung and Ramaraja K. Ramasamy.


everyone and welcome. This is Melissa Armo with the Stock Swishin' and Reviewing BA. This was a really nice option call and I called the 327.50 puts expiring this Friday yesterday. Actually before the open in this. And it depends again where you entered this, but this was, I mean this dropped yesterday to be honest with you, but I told people it's probably gonna go tomorrow. I was right. I did not anticipate it gapping down the way that it did today. Low today was three 2061 and actually it was a really nice gap down and it's, I mean, I still think this is lower, but I mean with it expiring Friday, again going into holiday, I definitely think this is lower and I think it was a good idea to take profits on this today, but the fact is that this could even be lower this week. But if it is, I'll look for another trade, but another example of weakness in the stock that is following through. So finally we're getting some follow-through. We're getting some follow-through in longs. We're getting some follow-through in shorts. It's nice to see. I feel like the market's getting more situated here into the end of the year from November into December. Actually let's look at this here tonight. No, this is pretty much flat here. So we're just flat here tonight. But anyways, long story short, this was a nice option straight. I did watch this this morning to day trade. It did not set up as a day trade. We did not day trade it. We did not short it and you wouldn't have gone long here. But the reality is that this is lower at some point I don't know when and with going into next week and it being a weird week with a holiday in the middle of the week, I think you've got to take fast trades. One day, two days, three days of the most. Excuse me. So I think that this was a good exit today. Call it yesterday, boom, out. And I mean, it was almost 200% return in investment which is a solid beautiful trade. And to have that into a week before a holiday at the end of the year is just phenomenal. But this is one of these ones I've been watching for some time. I've caught a lot of good trades in this. Last week was good trades in this. This has been just a great stock to make money and capture the moves in. Lot of volatility. It's a good stock to trade doing options but you got to capture it in the right move. You've always got to trade a stock in the right direction if you're gonna make money. I mean, it's just the only way to go. And you would never have gone long on this today. Okay, it was a short but you wouldn't have shorted this today. And it was long story short. This was a nice put, a nice short. And it was a good option trade. If you're interested in options you can take my golden gap course and learn how I pick the trades which is based on the 26 point rating system that I teach in the big class. Next class is not until 2020. In January, it's January 11th and 12th. If you want to take the gap options course it is this Thursday, December 19th. It's a half day course. You're not gonna learn how to make the picks though in that course is how to manage the trades, how to do options. And if you wanna sign up for the options newsletter that's a separate subscription. It's an annual subscription. You can sign up for the whole year. You get all the trades, the options trades are emailed to you. When you get the trade, you take it and you manage it yourself. So you could have got out of this yesterday. You could have got out of this today. You could still be in it. Again, this could drop lower tomorrow. That I think this could have some effect on the market but I will say that today, just look here how we close. I mean, the market is still strong, still strong. So I mean, it's just, you know, who knows where we go between now and the end of the week. But I mean, this market just is, it's just really holding up phenomenally quite frankly. Nothing is moving it down, not even BA. So good luck everyone and congratulations to everyone that did the put. And everyone that was on the option newsletter got the trade. Have a great day everyone. Email me on the list of the stockswush.com if you're interested in the Golden Gap course, the Gap Options course this week or the Gap Options newsletter which is $59.99 a year, $6 grand a year. Thanks everyone, have a great day.


Okay, it's one o'clock, so let's get started. I have an important announcement to make about what we're going to do in here on Thursday. So one of the things, most of this class is about learning how to use the Mathematica software to do calculations, data manipulations that are relevant to chemistry, problems, experiments, et cetera. But another one of the things that we like to teach you in this course is how to use online resources for finding chemical information. So for example, from the literature and also how to search the literature and databases for chemistry data. And so on Thursday, in order to give you a quick crash course on some of the many resources that are at your disposal through the UCI Science Library, the Science Librarian for Chemistry, Mitchell Brown, will be coming to give a presentation during Thursday's lecture. And there will be a homework assignment that will involve using the tools that he shows you how to use on Thursday. So you should all make sure to come to that. In addition, stay tuned to your email because one of the tools, very powerful tools that you're going to learn how to use is called SciFinder Scholar from the American Chemical Society. In order to have access to that, you have to register. And I'll be sending you registration information soon, probably as soon as this afternoon or evening. So as soon as you get that information, please go ahead and sign up to be registered for SciFinder Scholar so that you'll be able to use it to do your homework and also to follow along on Thursday. Okay, so that's one thing. Another thing is that you should have gotten an email from me that told you that I activated mid-term evaluations. And I know you're all very, very busy, but if you have any comments about how things are going so far in this course, good, bad, or otherwise, please let me know because it's still not too late to make some changes to make the course more effective. And then, of course, at the end of the quarter, you'll be asked to do the final evaluations which will be submitted to my department. And those will be ones that will be formal and taken very seriously. But in any case, if you have anything to say about the course so far, please let me know. You don't have to write long essays unless you want to. Okay. All right. So the next thing I'm going to do here is we'll go over this week's homework assignment. And notice that in addition to the homework assignment, there's two other files. These are the mid-terms. And that's because the first problems, the first two problems in the homework, I'm asking you to solve problems from the other sections mid-term. So we're in section A here, all right? And so since we're in section A, what I want you to do for problem number one is to do problem number two from the mid-term B. And then for problem number two, I want you to do problem number three for mid-term B. And the reason for this is that as I was walking around in both sections during the mid-term and looking at what people were doing, it became evident to me that probably a little more practice doing derivatives and integrals and interpreting equations and typing them into Mathematica properly would be helpful. So that's the point there. And then the rest of the assignment has to do with series and expansions. Okay. So problem three, I want you to enter that sum that you see there, that series, and evaluate it in Mathematica.  And then in problem four, we're going to consider the relation between the energy and momentum in relativistic mechanics, all right? So these formulas are different from those of classical mechanics when you take relativistic effects into account. And here each of these, the energy and the momentum is written in terms of the velocity. And the velocity here is in units where the speed of light is equal to 1. Okay. So I want you to do a few things here. This is basically to give you some experience with expansions and then seeing actually how they work and how useful they can be. So the first thing I want you to do is go ahead and just enter these in to Mathematica. And then I want you to evaluate the first one squared and then the second one squared plus the mass squared, okay? And I instruct you for the second one to go ahead and put the post fix full simplify on there because it will give you a form that will make it easier to compare to the expression for E. And if you do that right then you should see that in fact this one equals this one so this is an important relation in relativistic mechanics, all right? Now in part B what I want you to do is take both of these and expand them to fourth order in V. And around the point V equals zero so it's a Maclaurin expansion of each of those formulas and put them inside the normal wrapper thing so it will truncate the of the order of V to the fifth that comes out if you don't do that. And also assign each of these to a variable because you're going to use those series in the next part, all right? So the next part what you're going to do is you're going to use your series to evaluate the square of the energy and then the square of the momentum plus the mass squared, all right? And what you should find is that the two expansions give the same identity E squared equals P squared plus M squared to fourth order in the velocity. In other words the expansions reproduce the identity to the same order of accuracy that they're carried out, okay? So that's an interesting thing to see. Now in order to see that properly what I want you to do is enclose each of these evaluations inside this new thing that you haven't seen yet but you'll see how it works, try it with and without, it's kind of cool this one which is called expand, all right? So that will multiply everything out so that you'll have expressions that you can easily compare between those two and like I say you should find that they're identical up to the terms of order V to the fourth. Okay. And then in the last problem what I want you to do is a so-called harmonic analysis of an energy function that's meant to represent the energy as a function of bond length and a diatomic molecule. It's called the Morse function. It has a pretty realistic shape and it's given by this formula here where B is the position of the minimum in the energy where the preferred bond length is, R is the distance between the two atoms, D is the dissociation energy or the depth of the energy well and A here is just a parameter determining the shape of the potential, okay? So what I want to do is enter that in and then I want you to generate what's called a harmonic approximation to that function. Harmonic approximation means a second order Taylor expansion around the minimum which in this case is equal to B, all right? So you can type that in and then you're going to make the second order Taylor expansion which you should assign to a variable and then I want you to plot the actual function V of R on the same graph as your expansion so that you can see how the harmonic approximation works and what you should find is that the harmonic approximation which is out to terms in R squared should be a good approximation to the shape of the potential energy V of R when you're close to the minimum and those of you who are in Chem 131 will hear more, you've already heard about that I guess in discussing the harmonic oscillator approximation to the vibrations of molecules, all right? So probably you already saw something like this but now you get to do it yourself and I give you some details here about the values that you should put in as well as the range over which you should plot the function, all right? So that's this week's homework due Saturday as usual and are there any questions on that, all right? Okay, so last time we finished up our treatment of expansions by looking at the Taylor and McClurrin expansions of functions. Now this time what I'd like to do today is talk about a useful way of expressing periodic functions, okay? So these are series representations of periodic functions and they're called the Fourier series named after Joseph Fourier who's a mathematician who studied the properties of these series, okay? So I'm going to go to the board so I can make some definitions and then we'll play around with it and see how it works. All right, so the first thing is that we're going to consider that our function is going to be periodic on the interval x goes from minus pi to pi, all right? So this just means that one cycle is over the 2 pi range from minus pi to pi and then it just repeats itself infinitely in both directions, okay? And now the Fourier series representation of the function and this is just some arbitrary function. All we're saying is that it's periodic over that interval. We write this function in terms of sines and cosines because sines and cosines are functions that are periodic and therefore we ought to, if we add up enough of them, be able to reproduce the behavior of f of x and so that's what we're going to see how that works today and this will be our introduction to how we can represent periodic functions and then analyze their oscillatory behavior using a transformation that we'll learn about shortly. All right, so the way this works is that there's a constant term and then there's a sum which in principle can go to infinity and this is, the first sum is the cosines and then we have another sum which contains the sines, okay? So in practice you might represent it by a finite number of terms and we'll see how that works. Now, it turns out that these coefficients you can calculate as integrals of the function, all right? So the first one, a0 happens to be equal to and we'll verify this in a minute. The integral from minus pi to pi of the function and then the rest of the a sub n's equal to 1 over pi integral minus pi to pi of f of x times sine, sorry, cosine nx dx and then the b's are the same thing but with sines, all right? You can kind of see how this works by taking into consideration the symmetries of cosine and sine, even and odd symmetries. You can see that when you multiply this guy by sines, odd times even gives you odd, we're integrating over an even interval so all the terms in the integral of f of x times cosine nx go away and you only left with the cosines and same story for the sines, okay? But in any case, we'll verify these for a couple of terms. All right, so that's the introduction, all right? So what we're going to see is that there's a function in Mathematica that allows us to take an arbitrary function f of x and then generate the Fourier series to whatever number of terms that we want. So, yes, right, so if you look at the functions for the coefficients a, what they say is take the function which we're writing in this form and multiply by cos and then integrate, all right?  Sine times cosine is an odd function, cos squared is an even function. If I integrate an odd function over a symmetric interval minus pi to pi, I get zero. And a good example of that is if you take, right, if you take sine, all these areas cancel, right? So the one thing to keep in mind, it's just like with numbers, odd function times even function gives you odd, even times even gives you even, and odd times odd gives you even. So these terms in principle survive in this particular integral where we multiply by cos and then conversely when we multiply by sine, all these go away when we integrate and these are left. All right, so let's go ahead then and play around with this a little bit, see how it works. All right, so first thing, let's just define a simple function. So I'm going to define f of x underscore colon equals x, okay? And the assumption here is that that's ranging from minus pi to pi, so this is just a line with the slope of one and interceptive zero. Is that an odd or an even function or can you tell? If I plug in minus one, I get minus one. If I plug in plus one, I get plus one, so that means f of minus one is equal to minus f of one, so it's an odd function, okay? All right, so now what we're going to do, and this is, you know, it's just a straight line. So if you like when it repeats, it looks like a sawtooth, right? So if I draw that line here, okay, if I draw that line again and again, because remember, we're assuming it's periodic, you can see it's like a sawtooth function. It just keeps going on and on, okay? All right, so now let's go ahead and generate a Fourier series of that. And being a purely odd function, what do you suppose that we'll get in terms of the terms that we get? Will we get only signs? Will we get signs and cosines? Or will we get only cosines? If you had to guess, we represent an odd function by other odd functions, so it should be the signs, right? So let's try it. So there's our function, and then the command is, it's called Fourier trig with capital T series, all right? And we put, what we do is we put in the function we want to expand, and then we say what's our variable, X, and then we say how many terms we want. So the maximum N. So I'm just going to do three terms here. When you enter that, you see that you get three terms with only signs. One of them is proportional to sine of X, the second one, sine of 2X, and the third one, sine of 3X. And you notice in this case, we didn't get an A0, and we didn't get an A0 because F of X equals X is an odd function when we integrate it over the even interval, it gives us a 0. Okay? And we can verify those things. So let's do, for example, integrate F of X, and then we're going to say X goes from minus pi to pi. So according to the formulas for the coefficients on the board over there, this should be the A0 term, right? And we get 0. Now let's verify the formula for the B1 term. Well, actually the A1 term. So if we take F of X and multiply it by cosine of X, that should be, oops, cosine of X, that should be A1. Notice what we're doing. We're multiplying an odd function, F of X equals X, times an even function, so we get an odd function, and we should see that the integral is 0 and it is. All right? Now let's see if we can reproduce the first B coefficient, so there, we should multiply by sine. Now we have odd times odd, which gives us even, and we see that, in fact, oh, we have to divide by pi to get the actual coefficient, so we get 2, which you can see here is the first B coefficient. And we can do one more just to make it clear that we're consistent with the formulas. Now we can get the B2 coefficient by putting in 2 times X here, and what we find is that it's minus 1, which is what you can see here. Okay, so this is just a verification that the series we get is consistent with the definition of the Fourier series there. Okay, so now let's see how it actually works. How it is that we can take a line, a sawtooth here, and actually reproduce the shape of that thing by using sines and cosines, which are, you know, round functions. And the way I'm going to do this is similar to the way we checked how a Taylor series better and better approximates a function as we add more and more terms, we'll do the same thing here with the Fourier series, okay? So the way I'm going to do that is I'm going to make a table of Fourier series with increasing number of terms, all right? And I'm going to call that expansions, okay? So I'm going to say table and then Fourier trig series and then put in f of x and then x and now i and i is going to range up to some maximum number of terms, okay? And now we'll say i goes from 1 to 5, okay? So we're going to generate 5 Fourier series with maximum numbers of terms, 1, 2, 3, 4, 5, okay? So if you enter that, you get your list of the 5 series and the last one here, as you can see, goes out to sine 5x, okay? Now I'm going to define a plot, which I'm going to call function plot equals and then I'm just going to plot the line f of x and it's going to be from x going from minus pi to pi. So we're going to plot 1, period. And I'm going to make the line thick and black so we can distinguish it from our Fourier representations. So we'll say a plot style, arrow, gray level, bracket 0 and then thickness, bracket 0.005, okay? So that's just going to generate this plot, which you can see here, and stored it in function plot. And now we'll generate a plot of our Fourier series expansions and then we'll draw the two plots together using the show command, all right? So I'll call that set of expansions, expansion's plots equals plot expansions. And now just to make it look a little more interesting, rather than just going from minus pi to pi, we'll go from minus 3 pi to 3 pi. So we can actually see the reproduction of the periodicity. So we'll say x goes from minus 3 times pi to 3 times pi, all right? So I'll put a semi-colon there and then say show function plot. Whoops, I need a square bracket there, function plot. And which one? Oh, okay, let me correct that, sorry, thanks for pointing that out. Just re-enter, so we're consistent now and then we put in expansion plots, all right? And now let's put in, let's see, I'm going to put in here plot range arrow all so we see everything because that's only one cycle, okay? So there you can see how it works, all right? So here's our line and then you can see how the line here, the sawtooth, is more and more accurately reproduced as you add in more signs in this case, all right? So here's the first one. You can see the first one is not a very good representation of the function, that's just one sign, all right? But then you build up and, you know, you have coefficients that can build in more character of the signs that have more oscillations per period. And let's just add a couple more in here so you can see a little better, all right? So let's go ahead and make this go up to 10. I'll put a semicolon so I don't have to look at it, all right? And now go down here and re-enter this, all right? So you see that as you add in more and more, you tighten up the representation of the straightness and you also build up more and more of a point, okay? Notice that it's not perfect but it's interesting to see how you can take such a non-round function and represent it in terms of these signs and cosines, okay? All right, now the next thing that I want to show you, so I'm going to go back to the board now, all right? So here we have a very special case. The very special case is that, you know, the full period here is 2 pi, all right? It's very strict 2 pi, intervals minus pi to pi. Now in general, you can't insist that every periodic function that you would be interested in has 2 pi as its period, all right? So you can generalize, though, to another interval. This is the default interval assumed by Mathematica and I'll show you in a minute how it is that you can actually change that interval. And so the way this goes is we're going to make the interval arbitrary by saying now that the function is periodic between minus L and L, okay? Where L is now a variable that we can specify. And what that does is it causes a slight adjustment of the formulas. Now we have these arguments are N pi X over L. The intervals should go from minus L to L and the 1 over pi factors now go to 1 over L, okay? So a generalization for an arbitrary interval. And we'll do an example of this in just a minute. But now that we've changed the interval from minus L to L, it's no longer the default situation in Mathematica so we actually have to tell Mathematica that we've changed our interval. And unfortunately, the option that we add to the Fourier trig series command here is a little bit confusing, okay? So what I want to do is go from minus pi to pi the default. Now I want to go from minus L to L. Now the way Mathematica would write this is minus pi over some number that I'm going to call B to pi over B. B is what's called a Fourier parameter in Mathematica, okay? So suppose I wanted, for example, suppose I wanted L to be equal to 1. What should I set B equal to? Pi over B equals 1 so B equals, please help me. Pi divided by B equals 1 so B equals pi, yes, okay? Now the way we specify that in Mathematica then is we use an option, it's called Fourier parameters. There's actually two but I won't get into what the first one is just yet. And then we put the option 1 and then the value of B that we want. So if we want to go minus 1 to 1 we would say Fourier parameters arrow 1 pi, okay? So now we'll do an example where we change the interval from minus pi to pi to minus 1 to 1. Can you guys see if I don't have the lights on over here because I forgot to tell you what function we're going to do? Okay, now we're going to do what we just considered in the first example was the sawtooth wave. Now we're going to consider a function called the square wave, okay? And basically what that looks like is the following, okay? It's going to be, let me see how am I defining this one, yes, okay? It's going to be equal to 1. When X is between 0 and 1 and then it's going to be equal to minus 1 when X is between 0 and minus 1, okay? So you can see that if I repeat this guy over and over and then draw dotted lines this makes this thing that we call the square wave. Okay? All right, so that's what we're going to do. We're going to develop a Fourier series for the square wave defined over the interval from X equals 1 here to minus 1. Now in order to define this function we have to do something a little tricky. So let me go ahead and put it in and then I'll explain to you how it works. So now I'm going to define F of X underscore colon equals and we're going to see something new here which is if X and then I'm going to go over here and grab less than or equal to 0 minus 1, 1, okay? Enter. Now what does this mean? This is kind of a cool little way to define this square wave. What it says is the function F is defined as follows. If X is less than or equal to 0 it's minus 1. Otherwise it's 1, okay? And you could put any things in here you want and any condition you want. So this is a way to define some discontinuous function. You see how that works? If whatever this is is true it gets the first value or function and if it's not it gets that one, okay? So what this does is it says over here where X is less than 0 it's minus 1 and then otherwise meaning greater than 0 it's 1, okay? So that's our square wave and now we're going to do more or less the same thing that we did before. So I'll go ahead up here and grab this and then we'll grab function plot, put it there. I'm going to put a semicolon and then we'll grab this here and put it in and then I'm going to make a few modifications because first of all I said that now we're defining, we're going to define f of X between minus 1 and 1.  Now since I changed the interval from minus pi to pi I have to tell Mathematica what's my new interval and as we just discussed over on the board I'm going to put in the option Fourier parameters arrow 1, pi. Where? Well we've changed f but this command is essentially the same except because we changed the interval we need this, okay? I think I'll start out with a few less. Well we can go ahead and keep 10 why not? There was one other thing I think we need to change here. No I guess not. Okay well let's go ahead and let that rip and see what it looks like. Oops well I have a problem there. Oh okay, right let's see what we have here. Oh this Fourier parameters has to go inside the Fourier trig series command, thank you. Yeah that was my problem. So let's do that, let's get rid of that comma and we'll put it in here. All right now let's see if that'll work. Ah there we go. Okay so how does that look? Well one thing I can point out is maybe you notice this also on the previous one the number of curves that we get is not 10. How many is it? It's actually half because remember this is an odd function so our expansions are going to only have the sine terms, right? And so, wait a minute what am I saying here? Let's have a look at what we got here. If we undo this, all right, it looks like there's half as many curves because there's a repetition, right? Notice there's two of each. So if we wanted we could get rid of all the extra stuff by just saying go i from 1 to 10 and put in an increment of 2 so we'll get 1, 3, 5, 7 and then it looks a little nicer because we're not drawing curves on top of one another. So what do you think? How well is that square wave being represented by the Fourier series? As you add more and more terms you see you tighten up the wiggles around the square. One thing that's interesting and this is beyond the scope of the class is that you could add in an infinite number of terms here and you'll still have a finite amount of this what's called the overshoot. So this function is not, you cannot represent it perfectly in a Fourier series even with an infinite number of terms. There's always going to be a little overshoot which is kind of an interesting thing about that but you can see that you get better and better at reproducing the interior there as you add more and more terms and also that this straight part, discontinuous part gets better and better and you can see that, you can crank up the number a little bit more, let's go up to 20 and you see it's getting tighter and tighter in here. Okay, so there's your introduction to Fourier series. I think we'll go ahead and quit now but what we're going to do next time is we're going to generalize the Fourier series and then turn it into a continuous version which is known as the Fourier transform and then we're going to see that the Fourier transform is a very, very useful mathematical function that allows us to take, let's say for an example, a time signal that has some oscillations in it and to pull out those oscillations in terms of their frequencies. So in other words, we take a wiggly time signal and we can then decompose it into oscillatory components that have particular frequencies. It's a very, very important transformation that's used in signal processing, spectroscopy, there are many, many applications, all right? So we'll see how that works next time, tomorrow.