Hi you guys! It's Lindsay here. Welcome back to my channel, Inside the Hem. As you can see, I've got a huge pile of fabric in front of me, which can mean only one thing. This is my plans video for October. It's all the things that I plan to work on for the next month. So without further ado, I'm gonna jump right in. The first thing is the one pattern one week projects. One pattern one week is this initiative or this like, I don't know, just fun event on social media where you wear the same pattern all week long for one whole week. So that can either be one pattern, halved into different variations. It can be different ways that you style a pattern, like you could take something like a black skirt and style it many different ways. However, you want to do it. You just take one pattern and you wear that same pattern, regardless of how many different versions of that you make into garments, and you wear that all week long. So I have chosen to make seven different garments, which seems crazy, but I did choose a fairly simple top to make those seven garments. And I'm doubly challenging myself to only use fabrics in my stash. I cannot under any circumstances go out and buy fabrics for this challenge. So it's not as hard as it seems. I've got a rather large stash, which is the point of making myself use it up. But you know, it does limit me a little bit. So I'm going to be using M7660. I'm not going to make all seven of them this month, but I have already made one and a half of them. So I'm wearing one now. This is basically like a shortened version of F, give or take. It's like a combination of F and E. Anyways, and then this is version A with those sleeves, with those really fun sleeves. I'm having some fit issues. The sleeve was too long. And so I took it all apart. And I also took in this part of it because I felt like it was really big. And then that posed some problems with this little bubble part. So I'm not 100% sure how this one's going to turn out. I mean, it's going to be wearable, but it might not be like my favorite top ever, if you know what I'm saying. And then some other versions that I have planned that I want to get done this month is to make, um, let's see, a dress. So you can take any bodice and without too much pattern hacking, make it into a fit and flare dress. You just have to like taper in the bodice and add a skirt. Or you could just add a skirt that's the same width of the shirt already and then add an elasticized waist. That's also a really easy hack to do. And that looks really great in sweater nits. So I might take this sweater knit and make an elasticized waist dress with, I don't know, one of the necklines, whichever I'm feeling. I do like this one. I feel like it's very flattering and really it was easy to do. And then for this one, I might actually put the work in to make it a fitted fit and flare with like a, you know, a fitted waistband. So those are the two that I want to do for this month, um, fabrics. This came from Joanne's clearance section like last year. And I don't remember what I was going to make with it why, which is why it's still in the stash. And this one came from fabric mart, I think fabric mart or fashion fabrics. I don't get them confused all the time. I can't remember, but isn't it cute with this like gold bows? I just could not resist that. So once I do that, I'll have one, two, three, four of them. And then the week that it, um, that you post your seven garments or seven, wait, seven ways you are using the pattern, I think is the very last week of November. So I'll still have those few weeks of November to knock out the rest of them. Maybe I should make three in October. Hmm. Okay. We'll see. Either way, that's the plan to make, um, different tops and dresses. I even have an idea to take the, like one with the funnel neck and add a zipper and make it like a, like a workout jacket. And you know, do the extended thing with the thumbs. I don't know. We'll see how much time I have. Otherwise I just might end up wearing the same top or dress twice in one week. Okay, so that's that. Next is so my style. I am still trucking along with so my style. Um, if you saw my makes video, then you know, I finished the darling ranges dress. Um, which gets me up to, I think the only ones I'm missing now are the briar tee, the valley blouse and the Yona coat. Well, and then October's is the Hampshire trouser. But I looked at that earlier and her sizing, her extra, extra large is still like three inches too small in the waist and the hip for me. Um, according to her sizing, I would be a four XL in her pants, which I mean, if that's, if that's what it is, that's fine. The numbers and sizes don't matter to me, but I don't know. Something seems off about that. And I just haven't had a lot of success with pants patterns in general. Um, but especially indie companies. So I might skip the Hampton trousers. Would that be bad? I just don't want to spend the money on fabric or even if I decide to just muslin them, like I don't even want to spend the time on something that I have like 20% confidence that it's going to come out right. You know what I mean? I don't know. So we'll see what I decide about that. But the briar tee, I know last month I had that blue and silver stripe and a lot of you commented that y'all all liked it. But the more I looked at it, the more I was just like, I don't think I'm going to wear this. That might go in the donation bin or I don't know. Maybe you'll just, I don't know. Maybe I'll make a briar tee, but I didn't want it to be my like official make for so my style. Instead, I found this in the stash. It's this really pretty dusty pink color, which is like the color of the season. So I thought it would just be more trendy and more wearable. So I'm going to make the briar tee out of this one this month. I promise. Also the valley glows. I haven't made that one yet because I was struggling to find fabric. And although I really only wanted to sew from the stash for so my style, I have straight away from it for a couple of things. Well, actually it's about half and half. So I don't know. But I ended up finding this really pretty, I don't know what it is, something from Joanne. It has this like diamond pattern on it. So I'm going to use that and then underline it with just this like crinkly gauze. So it'll be a white, you know, valley blouse, but that's okay. The pattern is like kind of boho anyways. So I figured just don't fight it. Just go for it. So that's what the bodice will look like layered together. So I think that that's really cute. So I got those things specifically for the valley louse. So that'll get made this month as well. My friend Amber and I, she's so, I sewed this on Instagram, if you have Instagram. She is my real life friend. She lives here in Charleston with me. She and I are going, well not with me, but she also lives here in Charleston with her husband in like a separate place. She and I are going to the ballet this month. So we went shopping at Joanne together a couple of weeks ago. And for whatever reason, we were in the fancy, shiny, like event, ball gown, whatever section. And she and I were like, you know, it would be really fun to make something really fancy, something we wouldn't normally just wear, but something for a special occasion. So we said, hey, let's make something for the ballet. So she got a really, really pretty. It's like metallic gold and black. Is it jacquard? I think it's jacquard. And then I picked up this metallic brocade. And I spent a lot of time over the last few days muslinning, like I muslin one pattern, two different bodices from there didn't like it for this fabric. Then I muslinned this pattern, two different bodices to decide which one I wanted and ended up going with this one. So I have already made the shell of it. And the shell is really, really cute. But as explained earlier in my makes video, what turned something from a work in progress to a year old work in progress and prevents it from being an actual make is that I never have lining fabric. I don't understand why every time I'm at the store, I'm like, Oh, yeah, I've got lining for that at home. I always think I have some and I get here and I either don't have enough or I don't have the right color or whatever. So here we are again. I have ordered some it should be here this week. So I will be able to finish this for the ballet does have a deadline, you know, so that'll help me get it done. But it is really pretty and I'm excited to wear it to the ballet. But also, it would be a really great holiday. If I get invited to any holiday parties, like for work or if my boyfriend and I go out somewhere, I think it'll be really perfect for lots of different things. The weather here in Charleston last year, I feel like it barely dips below 60. So with like a jacket over this, I should be able to wear it like through the holidays, fingers crossed, that's the weather I like anyways. Okay, so there's that that is going to get finished this month. And then again, if you saw the makes video, then you know that getting through some of these work in progress projects is like paramount. It's my biggest goal that I have for myself right now. So I'm committing to one a week. And we'll see how that goes. I got two done in September. So we'll see what I can do in October. The first one is this. Well, it's inside out. I don't even know if it's in a state where I can flip it right side out. Okay. Basically, I was feeling I don't know ironic or something. And I decided it would be really cool to have a cold shoulder sweatshirt, which I know seems like those are complete opposites when you want to be warm if you're wearing a sweatshirt or when you want to be cool if you're wearing cold shoulder top. But I don't know. I guess I just thought that juxtaposition of those two things would be cool. So this is where I've landed with it. It was a total hack job pattern hack job.  So all I've got left to do is to get this neck band on and then him everything and she will be done. So, um, or am I going to do bands? I don't know. I got to make those decisions. So I'm really close on this probably just a couple more hours worth of work and this one will be done. But the sweatshirt place is so cute. It came from Style Maker. It has like, I don't know, it's like Hawaiian tropical flowers on it. So maybe that's what made me think of cold shoulder when I saw it because it was like sweatshirt, but had like tropical vibe. I don't know. I just think it'd be really wearable in Charleston. I think it'll be really cute. So I want to finish that. And then I pulled these two out because they're in my work in progress bin, but I don't know why. So I think it must be a fitting thing. I mean, this looks completely done to me. Um, and I just have no idea why it's in the bin. This is, I think, the seam work, a story out possibly that I turned into like a ballet wrap, but I don't know why it's in the work in progress bin. So I'm going to figure that out and it'll either be a make or it's going to go in the garbage one or the other. I don't, I don't know what's wrong with it. So we'll have to wait and see on that, but the fabric is really nice. I don't even know what the fabric is or where I got it, but it feels great. I hope it's not stretched out being on this hanger for so long anyways. So there's that. I must not have very much of it because I put a seam in the center back of the, this is an old work in progress. And then this one is this super cute coat. The fabric came from, oh lord heart maybe? Hearts fabrics? And like I think I'm done. I think all I have to do is some of the finishing work on the inside. I think what happened was is the pattern called for shoulder pads. And so it's like, okay, I'll get some of those and see what it looks like. And then I did and I didn't like them. So I just need, I'm literally just need to attach this facing to the sleeve and she will be done, but it's so cute. So this is like literally 30 minutes of work and this one will be out of the fabric stash as well. And then for the fourth one, I am going to choose one of those, one of many, many patterns that I have that is all cut out, but not actually sewn. Like how does that happen? Do you guys do that or am I alone on this island? I don't know if, because in the old sewing space, you know, I had to set everything up and then put it all away. Like as needed, you know, if I needed to cook, then I had to move everything out of the way. So I think that's why that was happening is because I was like, oh, yeah, I need to cook something. So I've got to pack all this stuff up. And then I just got like, you know, distracted from the project or like another more exciting thing came up. I don't know, but I have like lots of projects that I've cut out and I have never sewn. So I'm going to choose one of those. I've got so many, but there's a lot of really cute things in there. So I'm going to do that. So it sounds like a lot. But if you think about the works in progress being like either most of the way done, halfway done, or like at least the cutting done, you know, you can knock out several projects in a couple of days. So that's the plan on that. My biggest concern is the so my style projects. September's was a coat, the Yona coat. And I've been going back and forth back and forth on whether or not I was going to make it because so my styles all about adding things to your wardrobe that you need. And we just don't wear coats that often here, you know, it like doesn't get that cold. So I don't need a closet full of coats. I really only need a couple and that gets me through. It's the same thing that's held me up on the Hampshire trouser. It's like I don't really wear pants a lot. And I haven't had a lot of success in making them, you know, but if it doesn't really fit into my wardrobe anyways, then doesn't let defeat the purpose of so my style. So I might try and adapt so my style for my lifestyle and the climate we have here, meaning I might make like a second Saunio cardigan, or maybe I'll find another like lightweight coat, like the trench from deer and dough, the Lucerne trench that seems totally wearable here. And then for the trousers, like maybe instead of making pants, I'll make a skirt or I don't know, something else so that I'll have 12 garments at the end, but there'll be 12 garments that I'll actually really want to wear and fit into my wardrobe. So we'll see. This is how plans go, right? Like you think about them and then you work on some and you don't own the others. So we'll see what ends up happening. But let me know what you're planning in October. If you have an event you're sewing for, or if you've got a big project you've been working on, or if you are making the Hampshire trouser, let me know how it's going. I'd like to know, especially from those that might not fit into the size chart, what is, what's the verdict on pants? Um, anyways, but that is it for today. Thank you so much for watching. Sunny, the superstar of these videos, and I say bye-bye.


Whenever I find that I have a hole in the toe of my sock, super annoying, right? Maybe it's halfway through the day. I might try to rearrange it, but we all know the toes are gonna break through, right? Hate that, it's uncomfortable, but all I have to do, get a new pair of socks full. There's a little loophole here. You may have figured out where this is going, but I have a perfectly good sock on my prosthetic foot. So we just do a little switcheroo. I'm not gonna feel it on this side. And just like that, I'm good to go.


Good morning everybody. I am Deepak Phatak from the Department of Computer Science, IIT, Bombay. And I am very happy to welcome you all to participate in this second subject of the ISD workshops on database management system. We have started these workshops in order to empower our technical teachers so that they can in turn empower the students whom they teach. I would like to use this opportunity to share with you some background on the conceptualization of these workshops and how we are proceeding with the execution. Long time ago we realized that the conventional ISD or QIP workshops which are conducted for 30 or 35 teachers at a time, while they serve an extremely useful purpose, they are unable to cater to the large number of engineering teachers that today we have in the country. Further, we also noticed that while these two week courses are useful, since there is no subsequent engagement envisaged as part of this workshop activity, teachers who benefit from these workshops go back and are not necessarily able to use that knowledge gained in their regular teaching. It is in this context that we conceptualized our model. The physical workshop that you are attending for two weeks is only the beginning.  And that will be an open source portal that portal will contain not only the recorded audio video lectures of this course, all slides, all the material for laboratories and assignments, but it could eventually have a whole lot more contribution from all of you and from many others in the country. Hopefully this will become the database knowledge portal for all the learners in this country in coming years. That is the long term hope. We are thankful to M.H.R.D. who have approved the inclusion of this model in the national mission on education through ICT. And I'm also glad to report that this is the first time the actual registration for the workshop has crossed thousand teachers which was our ambition right from the beginning. A few things more about the model, ordinarily an IST or QIP workshop is held on a specific subject and it typically embodies the views of the expert faculty on the subject matter. Since the teachers coming for such workshop have to teach the courses in their conventional way in their respective colleges, we decided that when we run our model we will try and use the syllabi that prevail in most universities including the examination pattern that is seen in most of the colleges. Therefore we have decided to cater to the common syllabus that is used in the Indian universities for teaching. We believe that the course material designed in this fashion will permit participating teachers to go back and use this material directly in their own teaching efforts in their respective colleges. Of course we would like to make a difference because in the IIT system as you know and in several other autonomous institutions the syllabus is not necessarily fixed. There is a written syllabus which acts as a guideline and the expert faculty member is free to further explore the new developments that are taking place in the field. More importantly the examination system that obtains in the IIT like institution does not have a conventional straight jacketed makeup. We never have questions like solving six out of ten problems. We never necessarily say that there will be at least one question from each portion and so on. Of course you would be unable to change the examination pattern in your colleges immediately after you go back, but during this workshop we would like to give you a glimpse of how the questions are set, how the difficult questions are useful in challenging the minds of the participants so that hopefully most of you will be encouraged that when you conduct your examination at least as part of the in semester evaluation you will be encouraged to use harder problems as sample problems, more difficult lab assignments and tutorial assignments which will challenge your students. In this workshop we hope to give you a glimpse of all of this coming to the subject matter of this workshop database management system. When we did the survey of the prevalent syllabus we found that in most places the book used on database management system was the one written by Kurt Selbersah and Sudarshan. We are very fortunate that one of the authors of that book himself, Professor S. Sudarshan, who is my colleague in the department of computer science for more than two decades now, well one and a half decades has kindly agreed to be the instructor in charge of this course. I am therefore very pleased to see that the known database guru in the country is personally available to deliver lectures and interact with you. I would request you to use this opportunity to the hill, interact with him, ask questions. The course coordinators at respective remote centers were assembled in IIT for a week and they have personally interacted with Professor Sudarshan imbibing the philosophy of this workshop, imbibing the lab sessions and the tutorial sessions that have been planned. They will in turn guide all of you at your respective remote centers in conducting your labs and tutorials in the afternoon. So please use this opportunity to make as much as is possible from this workshop so that the take away will permit you to be greatly empowered when you teach these courses later. In conclusion, I would like to add one more thing. Since the objective of this workshop is a long term engagement, I hope you will remember that the workshop certificates will not be distributed at the end of two weeks when the workshop closes. Instead, participants at respective remote centers will be broken into teams and each team will be required to make a specific contribution in terms of solving an assignment within two weeks of completion of this workshop. It is only on the receipt of their assignment which will be gauged by the respective coordinators at the remote centers that the certificates will be issued. The objective is not to again do anything forcefully but to request you that this kind of engagement model where you don't forget what has happened in the course but remember it extensively over the next two weeks by working on the assignments will hopefully prompt you to further remain engaged with us in our efforts to increase the quality of our teaching in database management systems. With these words, I would like to introduce my colleague, Professor Sudarshan and the database guru of the country. So, welcome Sudarshan. Thank you. Thank you so much for agreeing to engage these teachers. Let me tell you, now I speak on behalf of these thousand teachers. If I am one of them, I represent a band of about 150,000 engineering teachers in the country. We work at small colleges. We do not have the kind of opportunities that teachers at places like IIT has. We learn on our own mostly and therefore, we look forward to this opportunity to learn from you. All yours. Thank you so much. Thank you very much, Professor Fadak. It is a bit of a stretch to call me the database guru. There are many more database gurus here. I am merely one of many. So, in any case, welcome to this workshop. We have 10 days of work ahead of us and hopefully all of you will enjoy the workshop as much as I do in teaching it. So, let me put up a few slides on the goals of this workshop and on how we are going to structure the workshop before I get into actual technical details. I hope you are able to see the slides on Avue now and you should be able to see the cover slide. So, here is how we will be running the course. We have lectures in the morning for 3 hours as you have probably found out by now. Followed by lab sessions in the afternoon where all of you have to get hands on in doing stuff on computers in the lab. The lab component of this course is, if anything, even more important than the theory component, because the theory component is something which all of you can pick up by reading a textbook or by reading the slides. Though I do hope that when I cover it here, we will include little more insight than you would pick up just by reading a book or by reading slides. But in the lab component, we give you exposure to actual software which you will be in turn using to teach your students. A feedback which I have heard from several people is that the lab component of many courses is inadequate and many faculty have expressed the need for better support in terms of labs. And I hope that this course will play a significant role in that and we will have assignments every day which are either lab or tutorial. Thanks to the efforts of Professor Fartek and his team, we also have a new gadget which is the clicker which we are going to use in the course over the next 2 weeks. Now, I hope all of you have clickers. I believe due to overwhelming response to this course, some of you will be sharing clickers which is fine. Either one of you, you can take turns or whatever. So, we are not going to be using the clickers to evaluate your performance directly as a result although it will be indicative. But what we are going to use it is to get a broader feedback on how people are finding the course, how they are understanding it, are there parts which they have followed or they have not and we can tailor the course appropriately. So, what I would like you to do is at this point if everyone is ready, please test your clickers at this point. Just click on any one number. So, we will find out which is the most popular number among 1 to 4. So, please use the clickers now and pick any random number from 1 to 4 to make sure the clickers are working fine. We will have more meaningful questions coming up in a couple of minutes. So, while we wait for the clickers to come online, let me continue on to the course structure. So, the course is for 10 working days from today through the 23rd. Let me give you a quick outline of what we will be covering in this course. Basic curricula as professor Fartak said covers what most of the courses across the country cover which also happens in this case to be what we cover in the database concepts textbook. And we will essentially be going sequentially through the chapters of this book with a few minor changes in the chapters that we cover. First of all, we will be giving a lot of importance to the relational model and SQL and we are going to devote several lectures to this topic. For the simple reason that pretty much everyone who goes out of this course into the industry will be using SQL. A very large fraction will be using SQL on a day to day basis. So, that is something which is very important to understand how to write SQL queries. SQL is also a rather confusing language for someone who has only seen imperative languages such as C++ or C or Java earlier. And therefore, it is important to take time to explain the concepts and how SQL is different and give enough examples so that students are able to learn how to program effectively using SQL. So, that is going to be one of the first focuses of this. So, right after an introduction to SQL, we will move on to how to design relational database schemas. Now, certain books including ours earlier on used to cover the schema design first followed by SQL. Now, we realize while teaching our courses that this causes a problem. We are not able to conduct lab exercises in parallel with the course for a while because we have not covered material which is required for the lab. As a result, we switched to having SQL first followed by database design. It turned out that this was also a wise move for another reason which is that when students first see relations, they really want to understand what you can do with the relational model, what kind of queries you can write. Only after that do they have a level of maturity which is required to do a schema design. So, it serves a double purpose to cover SQL first followed by database design. So, as part of database design, we will be covering the ER model followed by normalization. Now, both ER model and normalization have been around for many years now. However, in the ER model, the notation has not been fully standardized. There are in fact three broad schools of notation in the ER model. One of which is the standard Chen's ER notation which came first. The other is the notation which is widely used in many industries or was widely used called IDE F1X. And then third which is more recent is to use UML for modeling rather than ER modeling. In the industry, a mixture of all of these are used, but UML has come now to dominate modeling activities. And in this edition of the book, we decided to go with the UML modeling. However, UML has a lot of other things associated with it which are not relevant to a database course. And therefore, we chose a subset of UML and then we added a few more features which are not there in UML, but have been there in traditional ER notation. And as a result, we have something which is almost completely a subset of UML with a few extra features which we will use as the basis for our ER modeling. We will also show you how to use certain tools to do this modeling as part of the lab. After that, we will have a little bit of theory and more of lab on how to build web applications using databases as the back end. Now, this has become the standard mode for deployment of most web applications. Almost all of them have a database behind because they all need to store data. And it is very important for students to learn how to build these. And as a result, we cover them in the database course although the web is not specific to database systems. And finally, we are going to cover database internals. Again, 10, 15 years ago, most courses did not cover internals and externals in one course. There were certain places which covered mostly internals and there were many more places which covered mostly database external, external meaning SQL and schema design and so on. These days, many courses across the country have combined these two aspects and they cover both the SQL and building applications that is externals as well as the internals to some extent to the extent that it is important for programmers who use SQL to understand what is going on behind the scenes in the database. And it is very important that students know what is happening to some extent at least even though to understand fully what is going on requires a lot more time and energy. So, in this course, we will be giving an overview of database internals and hopefully, we will cover all the material which a practicing programmer will find useful. In this course, we are actually covering a little bit more than most of you will probably be able to cover in a course in your college or university. But I decided to go ahead with it because I believe that this is a topic which some of the instructors may not have covered in detail earlier and you may benefit from knowing a little bit more than what you will be teaching your students. And we will cover a few advanced topics in the last day of the course. So, that is the summary of what we will cover. I will also mention what we will not be covering. These include a few topics which used to be a standard part of database course, but we believe that they are not as important as they used to be anymore for a basic course, although the topics are certainly still important. So, two of these topics are the relational calculus, the domain relational calculus and the tuple relational calculus and object based databases which include object relational and object oriented databases. These are appropriate today for advanced courses, but probably not for basic courses. However, there is a small twist in the game these days where a technology called object relational mapping which has some connections to object oriented databases and some connections to object relational databases is used in the industry. So, if time permits we will give a short introduction to that although we will not cover the other aspects. There are many more topics which would be nice to cover in a first course, which have traditionally been in a second course in most places and therefore, we will not be covering them. These include parallel and distributed databases and more detailed coverage of object oriented and object relational databases. So, maybe in some future course those can be discussed. So, that was a quick summary of what we will be doing over the next few days. So, now let us move on to the actual technical content. So, let us start the technical part with quick overview of what database systems are meant to do and why they are important. I have quite a few slides here in this chapter, but I am not going to cover all of them in detail although it is pretty useful to go and read them offline. So, all of you have been teaching databases. I would assume maybe a few of you are here to learn about databases before you start teaching them, but all of you I am sure have used database systems whether you knew it or not. Pretty much everything you do on the web today stores data somewhere and I am sure all of you have used many many applications on the web. You have also used applications in your university for various tasks in all likelihood such as registering students, admitting students first of all, collecting their fees, maybe entering their grades on a computer and so on and so forth. All of these use databases as a back end and their applications built on top of databases. These of course date back quite a long time with some of the early applications including airline reservations and in India one of the early uses of computer with applications with a database as the back end was the Indian Railway Reservation System which has been so useful to us across the country. There are many many more and I will not bore you with all of them. So in the context of a university database backed application as I said does many things starting from admitting students to running courses to saying what courses are running and which students are registered for which courses, what are the marks or the grades that they receive and so on. In fact we will be using this as a running example for our database queries and we will be introducing you to a particular schema which models a university database. This is going to model only a small part of a university because a real database can be very large. Typical large scale database today has thousands and thousands of relations and it is obviously not feasible to teach a course with a schema containing so many relations. So we will use a small subset of it to introduce SQL in relation model and how to write queries. I should also mention that in early days database applications were built directly on file systems but there were many drawbacks which people realized as time went on. In the original era of mainframe computers when they were first introduced there was no such thing as a database. Data resided in files and people had to deal with file formats and so on. They had to actually write out the files which were on tape in those days and then read data back from tapes. Now this was a very low level of abstraction meaning that programmers had to deal with every little aspect of how data was stored how it was to be retrieved and so forth. And this seemed like a good thing for programmers because it gave them job security or so it seemed but it turned out to be a bad idea for enterprises because doing everything was very expensive and certainly small enterprises could not afford any of these things. So a database system really gives a much higher level view of data and provides many features which are important both for ease of access to data and to ensure data is robust in the face of all kinds of failures. So as an example of some of the problems which database system which is properly designed can help to avoid are database redundancy and inconsistency. And we will see later what these topics mean but inconsistency should be clear to you. If there are two parts of the system which give two different addresses for you then whoever has to send you a letter is going to be confused which one to use. So that is an example of redundancy and inconsistency which can be caused by redundancy. I already mentioned the difficulty in accessing data if you have to write a complicated program to do even simple tasks. A third problem is data isolation which is really multiple different file formats which get created if you store data in files and different people will use different things and now building an application which accesses two different parts of the database is complicated by the fact that they use different formats. A single internal representation in a relational database system avoids these problems. Integrity problems there are several kinds including inconsistency between copies that is one kind. Another more common kind is what is familiar to many of you is foreign key constraints. If I store a name of a degree program that had better be a valid program. I do not want to have a student who is registered for the degree x, y, z which is not valid. They should be registered for a meaningful degree. There should be other constraints like this such as the department name should be valid and the role number should be a valid role number and so on. Now at the systems level and this aspect is what we will be covering in the second half of the course on database internals are issues such as atomicity of updates. If you have a transaction doing multiple updates what if there is a failure in the middle? Is it going to leave the database in inconsistent state? This was a question which database people answered long ago and it turned out people in other areas ignored for a long time. In particular file systems people ignored this for a while till they realized what problems there were because of failures that happened in the middle of some update. And today this idea of atomicity is used in many many things even outside of database systems. A second issue at the internal level is concurrent access by multiple users. What if two users go and update the same data at the same time and cause a mess? How to deal with it? So we will be seeing that later in the course. A third aspect is security. Who can be allowed to do what to which pieces of data? Again we will be covering that later. So database systems of course have to provide solutions to all of these. Any database system has to abstract away from the lowest level of representation which are basically bits and bytes sitting on a disk or in memory somewhere which is a level which most programmers will not be able to deal with and give a much higher level view of the data. So the highest level is the view level. In this slide they are shown upside down with the physical level being at the top but it is actually the lowest level. The logical level is what is stored in terms of the schema design and the view level is what can be made visible to specific applications or programmers as then and we will see this later in the context of SQL. So this figure shows the same things the right way up. So as you can see there has to be one physical representation, one logical representation but there can be many view representations depending on who needs the data. We have a notion of a schema which is what is the information stored not in terms of what are the actual role numbers or actual names but at a higher level which is what are the relations or the entities or relationships which we represent, what are their attributes, what are the types of their attributes and so on. So this is the schema. Then there is a notion of an instance which is the actual data sitting in there which is who are the specific instructors, who are the specific students, what are their names and so on that is the instance of the data. And there is a notion of physical data independence which says that the lowest level of the schema, the physical level should be decoupled as far as possible from the logical level so that we can make changes at the lowest level without affecting higher level. So this is called physical data independence. Why is this important? It lets us for example add indices to improve performance and to change the file representation from something to something else if it improves performance and so on. We will see a little bit of this later. There is a concept called a data model which is a high level abstraction or collection of tools for describing data including the relationship between data, the semantics, the constraints and so on. So there are multiple data models which have been proposed and for the bulk of this course we are going to focus on the relational model although we will talk a little bit about, in fact we will talk a fair deal about the entity relationship model which is widely used for modeling data initially and we will talk a little bit about a few of the other things including the object oriented or object relational data model and a little bit about semi-structured data models such as XML. There are older data models also which are used in some very old legacy applications but we would not be covering them here. The relational model which all of you can intuitively see what it is even if you have not seen it formally before stores data in the form of tables such as the instructor table shown on this slide. As you can see there is a bunch of instructors one per row and each instructor has a number of attributes. What are these attributes? There is an ID which is a unique identifier. There is a name, there is a department name and there is a salary. Now of course this is a toy example. In reality any organization would have to store a lot more information about instructors or for that matter students or anything else. But to keep our example simple we are just modeling this much. A little bit of notation if you are not familiar with it. The individual rows of this relation are referred to either as rows or as tuples. The term tuple comes from the theory folk who view this as a n tuple representing the data. I will be using these two terms interchangeably. So row and tuple mean the same thing. Similarly this table has multiple columns. Each column is referred to as an attribute of the relation. So again I will be using the term column and attribute interchangeably. And you will find in the systems world people tend to call it rows and columns. In the theory world people tend to call it attributes and tuples but they mean the same thing. Now here is a very very tiny sample database consisting of two relations. The instructor relation and the department relation. You can see intuitively that each instructor has a department name and the department relation also has a department name attribute. And it should be clear that the department name in the instructor relation refers to a particular department which ought to be present in the department relation. If it is not then there is a problem. So we will see later how to ensure that such a row will be present for every department name which is there in instructor. Now a little bit about the languages used to access databases. As I have told you already SQL is the most widely used language. Although in the early days there were many other languages which were proposed but most of them have fallen by the wayside because not enough people use them. But the concepts behind a language are common and even today new languages keep getting generated. In particular variants of SQL are many but there are languages which are SQL-ish but not quite SQL which have been proposed for XML or other semi-structured data models and so forth. And some of these concepts are common across all the languages. So there is a data manipulation language and then there is a data definition language which we will see in the next slide. So data manipulation language is a language which is used for accessing and updating data in the database. Now there are two classes of data manipulation languages. There are the procedural languages such as C, Java or C++ which could be used in theory at least to access databases. Although in practice today the most widely used one is SQL. And SQL is a more declarative language where you do not specify exactly how to carry out computation but you say what you want and it is the job of the system to figure out how to do it. It turns out this idea of declarativeness has a lot of applications across many areas of computer science. If you specify exactly what is to be done the system is limited by how cleverly or how foolishly you said how it is to be done. If on the other hand you can say what you want declaratively. If you have a clever implementation inside which can figure out the best way of doing things that can be a lot faster than what even a good programmer would have written by way of detailed instructions. So this paradigm has found enormous success in the relational database area but has begun to creep up in many other areas as well. A data definition language is the language which we use to specify the schema. Again SQL has a data definition language component which is the most widely used. So we have a small example on this slide where we have a SQL statement to create a table called instructor with the four attributes which we saw earlier ID name, department name and salary. In addition for each of these four attributes a type has been specified as you can see. The first thing is a character 5 which is a fixed length character string. The second one is a wire care 20 which is a variable length character string of up to 20 characters. The third is again a wire care 20 and the last one a salary is numeric 8, 2 which means 8 positions total of which 2 are after the decimal. That is how you specify the schema of a relation. There is also a lot more which goes into the data definition language we will see that in detail later and these are all stored in a part of the database called the data dictionary. And these include both the types you have seen as well as integrity constraints such as primary key and foreign key constraints which we will see later on as well as authorization information. Now the SQL data manipulation language is what we are going to use extensively. Here is a small sample of it which does the following. It says find the name of the instructor with a given ID 22, 22, 2. Now how do you do this? To understand how to write the query we always have to go back to the schema and see which relations contain the information that we need to access. In this case our life is simple because the instructor relation as we saw already has an ID as well as a name attribute. So all the information we need is contained in this one relation. In general as we will see later we may have to connect information from multiple relations to answer a query. In this case it is all in one relation. So the SQL query is simple. It basically looks like select name from instructor where instructor dot ID equal to 22, 22, 22. So that is going to look for rows in the instructor table, find which all rows have this particular ID attribute and print the name information for all those rows. Now if we declared ID as a primary key for this table instructor as we ought to have done and what will happen is there can be only one row with a particular ID value. So we will find the unique name of that instructor. Now there is another example right after this which connects information from two relations. So what is this doing? It is finding instructors in departments whose budget is greater than a specified amount. So first of all we have to look at a department relation to find which all departments have a budget which is larger than 95000. Then we have to connect this information up with information from the instructor relation and what we are doing here is called a join of information. We will see this in more detail and from that we extract the information which we want. So if you are not familiar with this we will cover it in detail shortly. SQL is a language used to talk to the database but application programs are typically written in some other language such as C++, Java, the .NET family of languages or PHP and Python, Perl and so on. There are many many languages used to build application programs. Now all of these need to talk to databases and the way they do it is to construct SQL queries and ship them to a database which executes the queries and sends the results back which the application program consumes. So we are going to see one of the interfaces called JDBC is an API. ODBC is an earlier one for other family of languages JDBC is for Java and there are other similar ones. But the one we will see illustrates the basic features which all of them have. The second part of the course after SQL will be on database design and again there are two levels of database design. The logical level which is going to be our focus. There is also physical level which is best done only after understanding what is going on inside a database. So we are not going to cover that initially but we will discuss it as we cover database. As a small example of database design gone wrong to motivate why we need to be careful about it. Here is a relation which has combined information from instructors and departments. Now we showed you these two separately which is what a good designer would have done. But let us say we had a designer who did not quite understand how to do design. And they said here is the instructor, here is the name, here is the salary, here is the department name. These are the same as what we had. But in addition they said well we really want to know which building their department is and what is the budget of their department. So they put all of these as attributes in one relation and here you have a relation with all of this. Now if you look at this relation carefully if you see the rows in here there is a computer science department down here. I do not know if you can see the cursor it is the fourth row and the computer science department also appears in the seventh row. If you will notice the department building and budget have to be unique in our schema. In reality some departments have multiple buildings. In our simplified world the department has one building and one budget. So if you have two instructors in that department the building name and the budget get repeated. So this is an example of redundancy which should be avoided. So it is intuitive that this is a mistake. So you can treat design as an art and let people figure out how to do it by being artistic or clever. But that does not give the best results. What you need is a theory for understanding what designs are good, what are bad and a set of practices which help you come up with a good design. That is what we are going to cover in the database design segment. Now here is a quiz question. Let me ask you to think about this question anyway. So the question says what is the problem with this particular relation? Is it missing information, repeated information or everything is fine or perhaps the instructor's salaries are too low. Which I am sure all of us would love to have higher salaries but that obviously is not the answer. It is one of the other three and if you have been awake at all you will know that the answer is P. So as I mentioned in the topic section there are two broad approaches which I use in conjunction for database design. One is normalization theory and the other is the entity relationship model. We will see this in more detail later including the diagrammatic notation I am going to skip it here. And as I mentioned earlier there are other data models for lack of time I am going to skip this and you are welcome to read these slides later. Now here is the schematic picture of the internal level of a database system and if you can read this the letters are rather small even I have difficulty reading it. But if you can read it you will see that the top part is a query processor which includes something which understands or parses the SQL data definition language. Whatever other language they may use and then figures out how to execute any particular query which comes up with a query plan and gives it to a query evaluation engine which then executes the query. What does it execute the query on? It executes a query on the actual data which is there in the database. Now the database may be stored on disk or may be these days on flash storage but to operate on data you have to bring it into memory. So there is another component called the storage manager which abstracts away the details of the underlying data storage and gives you a somewhat higher level view which the query evaluation will use. We are going to see this in detail later in the course and if you submit a query it actually goes through multiple steps including conversion to a simpler relational algebra notation or some variant thereof followed by optimizer which figures out how to run that query followed by actual execution of that query. We will again see this later. The second part of our internals is on transactions and what is a transaction? It is basically a collection of operations that form a single logical step. So if you go to a bank and deposit money or withdraw money that is a transaction as far as you are concerned. As far as the database is concerned a transaction consists of operations inside of the database. To you receiving cash is part of the transaction to the database it does not know that you are receiving cash or whatever it is just a set of updates on the database. So what if there is a failure in between? These are issues which need to be tackled. The transaction management component ensures that the database remains in a consistent state even if there are failures. There is also a concurrency control manager which controls the interaction between concurrent transactions. So let me quickly wrap up the rest of this chapter. There are several different database architectures including centralized client server, parallel and distributed things and we will see them briefly later on. And there are many different kinds of uses of database starting from those who have no idea they are using a database because they are only talking to an application program. Moving on to application developers and on to data analysts who basically look at the data and try to help the business make decisions on what to do like what products to manufacture which are selling well. Should we offer a discount? Should we run an ad campaign with ad campaign work? All these decisions can be studied based on data which has been collected. So it is very important these days for every organization to have good data analysts. And finally database administrators who deal with the database itself and make sure it is operating properly. There are a few more slides on history which I am going to skip for the moment except to note that electronic computers were born in the early 50s and very soon after they were born they were used for data management application. There is a pressing need for it. In the 50s we had magnetic tapes and subsequently a lot of database systems have been driven by the underlying technology. When people moved from tapes to hard disks that completely changed the design of database systems. They were basically redesigned and the relational model could come about to a large extent because of this new technology. And over each decade there have been significant trends. Now anything which was a trend in the 1990s when the web was born is probably something which looks very old to many of your students although to us it still seems new. For students it is something that was there when they were young kids so it is old stuff for them. So what is happening today in the context of database systems? Well in the last 10 years or so XML and other semi-structured data models have become more important. That was also work on automating database administration to reduce the dependence on DBA. But most recently in the last 5 years or so there has been an explosion of growth of really really big database system on a scale which was unimaginable sometime back. And these are the database systems that power the web applications which all of us use and take for granted. We take for granted that you can have an application with hundreds of millions of users. This is a scale that was unimaginable some years ago. How do they do it? Well unfortunately we do not really have time in this course to cover that but there is a lot of information on the web on how people are building very highly parallel database systems to handle this. It turns out that if you want to have really high levels of parallelism it is easier to do it if you sacrifice certain features which a bank would not at all be happy sacrificing. Such as certain kinds of atomicity. So those trade-offs are being studied in the research community and industry today. But again it is beyond the scope of this course. So that is the end of chapter 1.


Our journey of studying Cauchy problem for general nonlinear equations has started in the last lecture that is lecture 2.10. Today we continue. The outline for this lecture is first we recall once again the hypothesis and notations involved in general nonlinear equations and in the last lecture we started our search for a characteristic direction we were not at successful. Today the search resumes and we will find a characteristic direction for the general nonlinear equation and then resulting characteristic ODEs are incomplete we are going to see that and therefore we need to extend the system of characteristic ODEs to a new system which is system of equations for characteristic strips. So, let us recall the notations and various hypothesis which are involved in the Cauchy problem for general nonlinear equations. F is a function which is going to define as the general nonlinear equation the arguments of F have been written as x, y, z, p, q. So, defined on for a phi tuples so lying in omega phi which is an open subset of R phi. Such an F defines the most general form of a first order nonlinear general nonlinear PDE by f of x, y, u, u, x, y equal to 0. Though we know that this the this class GE contains in it linear equations, semi-linear equations and quasi-linear equations but we refer to this kind of form of general equation as fully nonlinear equation or general nonlinear equation. So, hypothesis on the function F which defines the PDE or that F must be a C1 function on its domain of definition which is equivalent to saying that the function along with all first order partial derivatives of all the arguments which are x, y, z, p, q are continuous functions on omega phi. And we need to assume that Fp and Fq do not vanish simultaneously at any point in omega phi. Omega 2 and omega 3 denotes the projections of omega phi to x, y plane and x, y, z space respectively. The Cauchy data is prescribed by gamma x equal to Fs, y equal to Gs, z equal to Hs for S belong to an interval I and these functions are C1 functions on the interval I. And we assume the projection of gamma to x, y plane which is denoted by gamma 2 is that gamma 2 is a regular curve which means F primes and G prime do not vanish simultaneously at any point on the gamma 2. Recall that Cauchy problem means we need to find a solution of the equation which satisfies this U of Fs Gs equal to Hs. In other words Fs Gs Hs belong to the surface z equal to Ux, y. Now let us recap from the last lecture the search for a characteristic direction. So, we wanted to extend the ideas from Cauchy linear equations to general equations. The idea in Cauchy linear equations was construct integral surface using characteristic curves. To think of characteristic curves we need to have a characteristic direction. So, how do you obtain a characteristic direction for GE that is a question. Recall for Cauchy linear equations the equation gave us a characteristic direction. The equation QL is AUX plus BUY equal to C. What we observed is if you take any point on integral surface given by z equal to Ux, y. The normal direction at P is Ux, Uy minus 1 and the equation tells that ABC is a direction in the tangent plane at P. This observation led us to define the notions of characteristic system of ODE and characteristic curves. Once again there is no automatic choice of such a characteristic direction for general equations. So, then we went on to consider this family of possible tangent planes to possible integral surfaces through a point for general nonlinear equation. Take a point P naught in omega 3. We get a one parameter family of possible tangent planes given by T1 and T2. What is T1? T1 is simply equation of a plane passing through X naught, Y naught, Z naught. But by restricting FP to satisfy T2 will mean that this is going to be a tangent plane to a possible tangent possible integral surface. It should satisfy these conditions T1, T2. Note this you just the only information coming from the equation. We are not pretending that we know the integral surface that is not needed. Now, we observed that the T2 for quasi linear equation is this and as one of them is nonzero we know that ARB has to be nonzero by assumption in the quasi linear equations. We assume B nonzero then we can solve Q as a function of P. In fact, it is a linear function of P that is what makes things much simpler and T1 for QL becomes this after substituting for this. So, this is a family of possible tangent planes. We have explicitly got one equation. So, this represents the Tp represents one parameter family of possible tangent planes indexed by a parameter. For general equations we have to have both the equations. If you are able to eliminate P or Q from here that is express Q as a function of P go back and substitute here then once again you have only one equation like this. In the case of quasi linear it was very easy to express Q in terms of P. But in case of general non-zero equations it is not clear. So, this is still a one parameter family of possible tangent planes at the point P naught. Now, envelope of one parameter family of tangent planes for QL we computed it turned out to be this. It turned out to be this. What is this? This is nothing but the characteristic direction at P0. This is a line passing through the point x0, y0, z0 and the direction ABC is a characteristic direction. So, therefore the plan for getting a characteristic direction for GE is like this. Obtain an envelope of one parameter family of possible tangent planes for GE which is this. So, get the envelope of this and choose a direction in the tangent plane for the envelope. This plan will be successful because we have a lemma that we proved in the last lecture which is here. If you take a parameter one parameter family of surfaces given by z equal to g xy lambda and you find its envelope which is this z equal to g xy g xy and c lambda denotes the intersection of the envelope and the family then look at the second conclusion. The envelope and s lambda touch each other that means wherever they intersect which is along c lambda the intersection is precisely c lambda at every point on c lambda they share the tangent plane. Therefore, if you can find a direction in the tangent plane for the envelope we are done that will also be a tangential direction for s lambda that will also be yeah s lambda. So, let us resume our search of characteristic direction. So, this is t1 and t2 that represent one parameter family of possible tangent planes at the point p0. Let us find the envelope that is what we need to do. To do that we need to assume that q can be expressed as a function of p in a differentiable way so that this equation is satisfied. In other words we are solving f of x0 y0 z0 pq equal to 0 and q is expressed in terms of p. Question is is it possible? It is always possible to find such a function locally. Implicit function theorem tells you come with a particular solution and nonzero or invertible derivative I guarantee the existence of such a function. This is typically what implicit function theorem says. Now we will go to the implicit function theorem with a particular solution with that means you first find out a p0 q0 real numbers such that f of x0 y0 z0 p0 q0 is 0. That means we have the particular solution. Now we need to see what is this nonzero derivative and fq at this point x0 y0 z0 p0 q0 is nonzero. Please note when we are applying implicit function theorem x0 y0 z0 is fixed. So it is only as a function of p and q that we are trying to solve this equation 1. So fq is nonzero if you assume then implicit function theorem guarantees existence of such a function but locally that means in an interval containing p0 you can express q as a function of p. That is the implicit function theorem. Therefore the one parameter family of planes becomes simply one equation now because I have solved q in terms of p and I put it here. Now only thing to do is differentiate this equation with respect to p and we get this equation these two equations. You differentiate this equation you get 2a, differentiate 2tp equation you get 2b. From 2a and 2b you can eliminate q prime p and we get this x minus x0 y minus y0 is proportional to fp fq okay something we got. Now what is this p0 pqp p0 actually x0 y0 z0 f is a function of 5 variables right we need a 5 tuple here for want of space here I have just made it a small notation p0 it stands for this. Using this and the equation for tangent planes which is tp here we get the curve of intersection and that is this okay x minus x0 is proportional to fp y minus y0 is proportional to fq therefore z minus z0 is proportional to pfp plus qfq that is what we have written here. So this is a line right cp is a family of lines of course indexed by p the same thing was actually one single line it never depended on p for a quasi linear equation all of them passed to the point x0 y0 z0 and having a direction fp fq p of p plus qfq the 3 tuple which is given in 5 is non-zero because we have assumed that fp square plus fq square is not equal to 0 throughout the domain of f that is a hypothesis on it. An envelope of the family of planes tp by definition is a union of cp that means union of these lines so envelope is a cone having vertices at p0 so visualize the way a family of lines passing through a point p0 for example you have this point p0 right so imagine lines in 3d okay like that so that is how looks like a cone this is called monge cone thus we have a monge cone field defined in omega 3 so omega 3 is in R 3 some set okay at any point you have a cone sitting there at another point you have another cone like that so this so every point you can attach a cone. So this is called cone field similar in spirit to a vector field to any point if you attach one vector associate one vector it is called vector field if you associate cone it is called cone field that is to every point of omega 3 a monge cone is associated for quasi linear equations the monge cone degenerates into a straight line because we observed that the envelope is actually a straight line it is independent of p so therefore just one single straight line so the monge cone field reduces to a characteristic vector field in the case of quasi linear equations. Now each of the lines Cp corresponding to each fixed p is a generator of the cone generator of the cone means it is a line in the cone so you have a for example you have a cone like that you have a line here so at a point on the cone if you take tangent plane okay that line is going to be there on the tangent plane therefore what we wanted was to look at the envelope and take one direction in the tangent plane of the envelope now here it is very nice tangent plane of this cone at these points the one of the direction is clearly the line Cp so we can take that Cp to be characteristic direction so a characteristic direction for GE is found for every p in omega 3 and every pq we do not we no longer require q of p here pq after all q is a function of p q is some value right it satisfies this equation still we have found a characteristic direction at p what is that fp fq p of p plus q of q now we have to look at the characteristic odys we are going to say they are incomplete we will see why so inspired by the success in finding an integral surface using characteristic curves we hope to repeat the same for GE also in ql we were successful so we hope same thing happens for GE as well so we are in search of curves in omega 3 which have tangential direction equal to the characteristic direction that we just found which is this fp fq p of p plus q of q at each of its points p p such curves we are looking for now pq of course depend on the point x y z right they satisfy this equation so it depends on x y z let us write a curve having a characteristic direction so let j be an interval 0 belongs to j gamma p0 be a curve given by xt by dz t as t varies in j such that it passes through the point p0 at t equal to 0 and it has a tangential direction which is characteristic direction at each of its points this is what we would like to call a characteristic curve therefore the following system of characteristic odys hold along gamma p0 so dx by dt dy by dt dz by dt are fp fq and p of p plus q of q and at t equal to 0 we want to be at the point x0 y0 z0 so fine now why is it incomplete I do not know what is p and q that is a problem okay let us denote the above system by Kara word e exactly like we used for quasi linear equations the solutions of this its images images will be characteristic curves only thing is that this is not it does not determine characteristic curves that is the problem because there are pq's here it is not solvable as unknowns pq appearing in it depend on xt by dz t via this equation f of xt by dz t pq equal to 0 p and q are also depending on t thus we also need to determine p and q as functions of t since xt by dz t themselves are unknowns there is no hope of solving for p and q from this equation and note this is not the case for quasi linear equations because characteristic odys never involved p and q and this is a second difficulty in extending the ideas from the quasi linear case what is the first one finding a characteristic direction that was the first difficulty we have overcome and second difficulty we will overcome by supplementing two more equations for 1 for p and 1 for q so how to find equations for pt and qt we need to supplement Kara word e system with equations for pq along the curve xt by dz t and what should be the property pt qt minus 1 should be the normal direction to a possible integral surface they are not arbitrary functions so we have to determine five quantities xt by dz t and pt qt and not just the three quantities because we are unable to determine the three we would have been very happy if we would have got xt by dz t we could have proceeded but the equation for xt by dz t involved p and q therefore we need to find p and q also so geometrically speaking what we are saying is we have to determine not only a curve but also a tangent plane to a possible integral surface that contains this curve so in other words we are trying to find a curve like that so suppose this is the point xt by dz t so we want to find something like that pt qt of course minus 1 I write but then it is okay minus 1 there is nothing to find so we need to find this so let J be an interval in R and gamma be a curve given by x equal to xt y equal to y dz equal to dz t be a curve having tangential direction as char dir that is characteristic direction at each of its points where pt qt satisfy this equation f of xt by dz t pt qt equal to 0 the phi tuple xt by dz t pt qt is called a characteristic strip maybe you may put like that because we are saying tuple but even without that it is fine all these five together is called phi tuple that is called a characteristic strip and the point xt by dz t is called support of the characteristic strip because there is a point at which you are putting a plane with normal pt qt minus 1 so let us illustrate that pictorially so here there are two points I have considered index by t equal to t0 here and t equal to t1 so x t0 y t0 z t0 is a point this is a tangent this is a plane with the normal pq minus 1 this is another point where the normal is pq minus 1 so it can be strip can be thought of as a point along with an infinitesimal plane element passing through that point xt by dz t with normal direction pt qt minus 1 now how do you get equations for pt qt so here we assume some more nice properties about the solution u see u is supposed to be a solution to first order pde we have not yet found but we are assuming that it is c2 is it okay that is a question it is okay because we are going to use this only to derive certain equations and for deriving equation you assume what you want does not matter but after getting the equations then you should show that solution exists there you should not suppose that c2 etc okay we will comment on this in the next lecture also so this is only to derive the equations for pt qt that we are assuming u is c2 so how do we derive the equation for pt what are at our hands this equation f of x y u u x u y equal to 0 differentiate that with respect to x so first is fx here I am using the zeta to stand for the x y z pq okay so fx and then with respect to z in z you have u x y therefore you need to differentiate fz and u with respect to x then with here also p fp and then whichever is here with respect to x which is u x x and here it is fq and u x y or u y x equal to 0 at every point in omega 5 where we use this notation zeta equal to x y z pq no actually this is zeta is not x y z pq it should be x y we have to substitute x y z pq equal to x y u x y u x x y u y x y that is what it is because we are going to differentiate here this u is a function of x y u x is a function of x y u y is also a function of x y so that is why we get this by chain rule and x t y t z t is a point lying on the integral surface z equal to x y we require the following thing to hold namely p of t I want it to be like u x so p of t is equal to u x of x t y t we are demanding this okay because pt qt minus 1 should be such that f of x t y t z t pt qt should be equal to 0 right so p is supposed to play the role of u x along the curve x t y t so that is my pt we want this so on differentiating the above equation with respect to t and using chain rule we get p prime t is equal to t appears in both the variables so differentiate this with respect to x u x x at the point x t y t into derivative of x with respect to t that is x prime t and differentiate this with respect to y that is u y x at the point x t y t into derivative of y t which is y prime t since characteristic od hold for x t y t z t denoting zeta t equal to this phi tuple x t y t z t pt qt we get p prime t is equal to u x x x prime is f p y prime is f q so we have got this equation for p prime and from the equation 7 that is the one we got after differentiating f of x y yeah this one this equation and here we almost got an equation for p prime the only problem is there is u x x and u y x we do not want that right equation of p should involve only f it can involve activities that you know problem but not u x x and u x so that needs to be removed now so we solve for that which we do not want in terms of what we know f x p is what we want to find f z which we know therefore p prime t equal to this and now becomes p prime t equal to minus f x plus p f z this is the equation for p prime similarly we do for q what do we do we look at the same equation ge differentiate that with respect to y we get something then we propose value for qt qt is supposed to be u y of x t y t differentiate this with respect to t it will involve this u x y and u y y eliminate this using the previous equation I think it equation 11 and then you get an equation for q prime x is replaced by y that is the only change p is replaced by q so this is the equation for q prime so we got the equation for q prime also so this is the system where the first three we have obtained or we have proposed using characteristic direction we realize that p q also depend on x t y t z t and we said we have to find the equation for p t qt and we have got we have appended Kara ODE with these two equations now it is called Kara strip ODE equations for the characteristic strip now there should be no problem because x y z p q here also x y z p q anything else f we know f therefore we know f p f q so no problem of course this is a non-linear system of ordinary differential equations how to solve this we have to solve this to get the characteristic strip and that we will do in the next class so let us summarize what happened so far we saw that the quasi linear equations has a characteristic direction right away given by the equation q l g e gives only possible tangent planes and using the idea of envelopes we found characteristic directions for g e and we observed that the system Kara ODE is incomplete so the system is supplemented with two more equations and we got characteristic strip equations ordinary differential equations for characteristic strip so in the next lecture we are going to take up the Kara strip ODE system which is a system of 5 equations non-linear equations and we need to solve them and what is the strategy in quasi linear equations we take the datum curve take a point on the datum curve and pass a characteristic curve through that so now if you take datum curve what is known is only x y z values on the datum curve namely f s g s h s which are given to us we will use them as initial conditions in the characteristic system of ODE in the quasi linear case but in the general non-linear equations case we have two more equations which is p t and q t equations for them therefore we should know on the datum curve what should be the values of this p t q t on the datum curve so that is what is called initial strip so initial data or datum curve is given that must be extended to a strip and that is called initial strip using initial strip and using the initial conditions coming from the initial strip we will solve characteristic strip ODE's and get characteristic strip we will determine characteristic strip and from there will come characteristic curves and from there we try to take their union and get the integral surface so these steps will be implemented in the forthcoming lectures thank you


Hello and welcome to the session. In this session we will learn about loss of logarithms. Now let A be a positive number which is not equal to 1 that is A is greater than 0 and A is not equal to 1 and let x and y be any positive numbers that is x is greater than 0 and y is greater than 0 and n be any real number that is n belong to the set of real numbers. Then the first law of logarithm is log x1 to the base a is equal to log x to the base a log y to the base a. Now let us start with its proof. Now let log x to the base a is equal to m. Now let us give it equation number 1 log y to the base a is equal to m. Let this be equation number 2. Now we learn the definition of logarithm that is if three numbers are so related that a is equal to x is equal to m then in the log book form we can write log m to the base a is equal to x and here a x and n are all positive numbers. So from the first equation by definition it is equal to a raise to power n and from the second equation we have y is equal to a raise to power m. The supplies x into y is equal to a raise to power m into a raise to power m which is equal to now applying the product log of x1n this will be a raise to power m plus n. The supplies xy is equal to a raise to power m plus n. Now again by definition of logarithm we have log xy is equal to m plus n. Now m is equal to log x to the base a. So this implies log xy to the base a is equal to log x to the base a plus n which is log y to the base a. So we have proved the first law of logarithm assumption of this law that is log xyz and so on. Only the base a is equal to log x to the base a plus log y to the base a plus log x to the base a. So we have proved the first law of logarithm that is log of a product to any base is equal to sums of logs of the factors to the same base. Logarithm that is log over y is equal to log x to the base a minus log y to the base a. Now let us start with its truth m is equal to log y to the base a. By definition we have x is equal to a raise to power n and y is equal to a raise in case of the experience this will be equal to a raise to power m minus x over y is equal to a raise to power m minus n. Now again by the definition of logarithm minus n that is log y to the base a logarithm which is log of a fraction to the base a is equal to log of the numerator of that fraction to the base a minus log of the denominator of the fraction that is of this fraction to the base a. log x to the base a that is log x to into x 3 and so on log x to the base a plus log x to the base a plus log the whole minus log y to the base a and so on. The third law of logarithm and that is log x to the base a is equal to m let this be equation number so by definition of logarithm we have this is equal to a raise to power m. Now raising both sides to the power n this is equal to a raise to power m by a raise to power n. This fourth law of logarithm is equal to a raise to power now again by definition of logarithm we can write this equation as log is equal to m into n. And therefore this implies log x raise to power n to the base a is equal to that is m into log x to the base a here we have proved the third law of logarithm. And that is log is equal to 1 over n. Now let us start with its root log raise a raise to power n is equal to p so by definition this implies p is equal to x by using this definition of logarithm. Further this implies a raise to power n into p is equal to x which further implies. Now here again logarithm this will be is equal to n into p is equal to 1 over n into log x to the base a. We have taken p is equal to log x to the base a log x to the base a raise to power n is equal to 1 by n we have learnt about the laws of logarithms. And this completes our session hope you and have enjoyed the session.