Greetings from the National Archives flagship building in Washington, D.C., which sits on the ancestor lands of the Nacotja Tank peoples. I'm David Ferriero, archivist of the United States, and it's my pleasure to welcome you to today's conversation with Laura Edwards about her new book, Only the Clothes on Her Back, which shows how studying the clothing of ordinary people can reshape our understanding of law and the economy in 19th century America. In the authoring conversation is Adam Rothman, professor of history at Georgetown University. Before we begin, I'd like to tell you about two programs coming up soon on our YouTube channel. On Friday, March 18th, at 1 p.m. we'll present a panel discussion on the topic working for suffrage, how class and race shaped the U.S. suffrage movement, with historians Paige Harrington, Kathleen Cahill, and Allison Parker. On Wednesday, March 23rd, at 1 p.m., Linda Hirschman will discuss the alliance among Frederick Douglass, William Lloyd Garrison, and Maria Weston Chapman, which she details in her new book, The Color of Abolition. One of the digitization projects supported by the National Archives through the National Historical Publications and Records Commission is an effort to collect and publish a database of fugitive slave advertisements. A great many of these notices published in the newspapers of the day are quite specific about the clothing possessed by the men and women running toward freedom. For example, a 1779 advertisement in a Baltimore newspaper declared that a woman named Nanny took with her one Indian calico gown, one new black and white Lindsay Petticoat, one old striped ditto, one white linen handkerchief, one single ditto, one pair of high-heeled shoes, and a felt hat. In writing only the clothes on her back, Laura Edwards examined the materials at archives, libraries, and historical societies in 15 states, including federal court records at the National Archives in New York. The resulting book shows the importance of textiles as a unique form of property that people without rights could own and exchange, even enslaved persons could claim their clothing and accessories as their own. In her Washington Post review, Marjolene Carrs wrote, Edwards has written an engaging, if complex book that reminds us that there is more to legal history than formal rights. Recorded history is not limited to the written word, the things people wore and handled and used tell their own stories about the people themselves and about the times in which they lived. Laura Edwards is a class of 1921 Bicentennial Professor of American Law and Liberty at Princeton University. A legal historian whose research focuses on the 19th century United States, she is the author of four books, including A Legal History of the Civil War and Reconstruction and The People and Their Peace. Adam Rothman is Professor of History at Georgetown University and is the Principal Curator of the Georgetown Slavery Archive. His books include Facing Georgetown's History, Beyond Freedom's Reach and Slave Country. Now let's hear from Laura Edwards and Adam Rothman. Thank you for joining us today. Hi everyone. It's really a pleasure to be with you today at this National Archives event. I am so excited to be speaking with Professor Laura Edwards. I'm a huge Laura Edwards fan. She is an amazing historian of the 19th century United States. I teach a class on the 19th century United States graduate class at Georgetown. The last time I taught the class we assigned Laura's previous book, one of her previous books of People and Their Peace, just a master piece of legal history. So I was so excited to get an opportunity to read Laura's new book, Only the Clothes on Her Back, and then to have this chance to talk with you about it. It's pretty amazing.  And to, you know, just also to publish a book like this two years into COVID, that in itself is a tremendous achievement. So a mausoleum on the book. Thank you, Adam. And thank you so much for being here. And thanks to the National Archives for hosting this. It's really fun. Yeah. So let's get into it. You're known as a legal historian. Nobody who does meticulous archival research and courthouses on a range of topics. But as far as I remember, clothing and textiles hasn't really been in your repertoire until this book and the articles around it. So I'm always curious to know how my fellow historians arrive at the topics they arrive at. You know, I'm curious about the rabbit holes you went down to get to this book. So how did you, how did you come to write this book? It's a good question. And you know, it actually started in the book that you mentioned, the People and Their Peace. And I had all of these cases, well, several cases that really puzzled me because they involved people and clothing and textiles. And one of them is actually the Elizabeth Billings case, which starts at the first paragraph of the book. And Elizabeth Billings is a white servant who is working in the upcountry of South Carolina, and she flees with her mistress's clothing. And among the clothing items that she flees with are high heeled flowered shoes.  Ultimately, Elizabeth Billings is found. And the reason she's found is that the clothes don't seem to match her. She also, unfortunately, escaped when she was exceedingly drunk on gin. And so she passed out on the side of the road, but they found her in the clothes. And it's not so much the gin or anything else. It's like, wait a minute, those clothes don't match. And then the clothes were discovered to belong to her mistress. And then it was all over.  It's like Elizabeth Billings, high heeled flowered shoes, South Carolina clay. There's something interesting in this case. And then there was another case involving an enslaved person in the upcountry, South Carolina. And her name was Suki. And she was accused of stealing some banknotes and several bolts of cloth. And what was interesting about that case is, I mean, she's enslaved. She has all this property. You would think that that would be like evidence of theft in and of itself because she can't own property. But they have all of this testimony based around whether this property is hers. And I'm like, well, why do you need to do that? And the banknotes, you know, they have all this testimony about that. But the cloth was like nobody questioned that at all. And that was presumed to belong to her. And it was only one cloth of a similar kind and quality was found to be missing that people then suspected her of stealing it. And I find that really strange. And then there was another case where there was this is North Carolina and a man files a case for having his dress stolen. And I'm like, well, that's interesting because this case would be about way more, way more of interest than just a stolen dress. But why is it his dress? And then you that was the outside of the court record.  And it's very clear that the dress belongs to his wife. And all of a sudden it hit me that there are two kinds of ownership here. There's the one of property rights that somehow it's the husband's property. But the dress actually belongs in another legal way to his wife. And that really puzzled me, too. So I ended up trying to work through some of this stuff in a chapter in that book. And I was sitting at the desk in the Newberry Library in Chicago with the light coming in in late December, late November. And I was sitting there writing and writing and writing and pretty soon the chapter was eighty pages long, and that's not a chapter anymore. So I sort of put some of that aside and decided that I needed to look at it more carefully to try to figure out who these people were and what was going on. And then that let me down the rabbit hole of other kinds of cases. And all of a sudden, textiles appeared everywhere. And so I would kind of go from archive to archive and look for anything I could find on textiles, which ended up being a lot. And I discovered that these cases weren't actually unusual, that there were legal principles in place that were widely observed that attached clothing to people. So it's presumed that the clothes that you wear belong to you. And then people use that principle to basically claim that any kind of textile that passed through their hands or that they made was theirs, or at least they tried to do that. And then that opened up this whole economic world where, oh, if people without property rights can make legal claims to these kinds of property, which are really valuable in the early 19th century, then that property is all of a sudden more valuable because they can claim it and they start participating, using it as currency is the basis for capital credit. And so you see this whole economic world open up to. So it was kind of Elizabeth Billings and Suki and the man who claimed to dress. And I imagined him in his dress behind his pile, and that didn't work. And that kind of got me on board with this. Yeah. Yeah. Wow, I mean, there's so much in what you just said that we have to follow up on every dimension of it. So just thinking about the accumulation of these cases, you start to notice that all of these cases have to do with textiles and especially about how ordinary people, including people without who we think of as without rights, use the law to stake claims to these to these textiles. And so, yeah, I want to get into all of that, but I'm still, you know, I wonder somebody else might come across these cases about textiles and and put them aside or not really know what to make of them or not even necessarily be interested in them. I have to I don't know if this is I don't know if I should bring this up, but historians always read the acknowledgements first. And I did notice in your acknowledgements is the only book I've ever read that acknowledged and thanked a sewing class. But I have a suspect that you actually have a kind of interest in textiles, cloth and clothing that is not just academic. No, indeed, indeed. And this is like one of those quiet sides of your life as an academic that you don't always own up to. But yes, I do. Some acknowledgements. It is. It is. That was a big leap for me, though. It's like I'm like fessing. I'm coming out of the clothing closet here. Oh, that's good. Yeah, there's so many metaphors with this. So, yes, I when I my mother sewed all my clothes when I was growing up and I would sit across from her when I was very small and I would sew, which would mean that I would get the scraps and then try to cut little armholes in the scraps so that they would fit around my stuffed rabbit who was my friend. And she her clothes did not look as good as the ones my mother did, which was frustrating. But I grew up doing needlework of all kinds and sewing. And then later on in life, I got dropped it. And then about 15 years ago, I took it back up. And there was a fabric store down the street from where I lived in North Carolina in Carborough. And it was very convenient. And so you walk in and they have sewing classes. And so I signed up for one. It's very hard to get in and signed up for one. And then I became sort of admitted into the Tuesday sewing class with a core group of people who actually been there forever. And so I've known them for, you know, decades now over a decade. And so, yes, since I'm trained to look at needlework and I understand my mother, when we would go clothes shopping, when she wasn't always sewing my clothes anymore, we would she would go into, you know, department stores and she turned the garments inside out and explained to me how they were made and what made good quality and what was bad quality. And we bought good quality stuff. And I was told, you know, 10 years old, we're paying more for this. But that's because they did the hem in the right way. So it's something that I've always noticed. So I think that's part of the reason why I notice this stuff. And I think it's also too part of it fits with my what my historical interests are, which is I write a lot about, as you know, Adam, ordinary people, people without rights, people that most other historians, many historians might walk by. And so I am interested in kind of seeing the fullness of their lives and what struck me is like what struck me about Elizabeth Billings and her high heel flowered silk shoes. It's like, wow, here's a woman who you would think is so exploited and oppressed that she kind of turns into a two dimensional figure. Her life is about misery, and yet she loves flowered silk shoes. And somehow that means that makes her into something else to me. And the way that all of these people in this book are not just it's not just a story of what the law does to them to strip away pieces of their life, but then what they do to bring some beauty and some color back into it and how they make meaning in their life. And it is what textiles are about as well.  And so clothes aren't just consumer items. They're also an expression of an individual. They're also efforts of people to bring some beauty into their life. It's about things that are more sort of moving history to the arts and humanities in that sense as well. And I just enjoy seeing that aspect of these people in this book as well. Yeah, it comes through. The book is richly populated with people. There are a lot of people in this book, but it's also richly populated with textiles of all sorts. And the care and attention that you bring to the description of the material culture that people are embedded in is quite extraordinary. And I think it's something that helps what could be what could be in somebody else's hands, a dry legal history, just really rich and often emotional, like really emotional. One, you mentioned just now that clothing or textiles is not just a consumer item. And you you talk in the book a little bit about the the emotions that people invest in textiles and the relationships that are formed through textiles. You have a really striking example of this, I think, with the London Foundling Hospital. Is that right? Yeah, talk a little bit about those those emotional investments that people have in in textiles above and beyond their function. Yeah. And we have a picture of that. And this is the London Foundling Hospital, just the tokens that are left by mothers to reclaim the children that they're leaving there. And actually, the London Foundling Hospital is still in operation today. It's quorum and they gave us permission to use this particular image. So this is London Foundling Hospital in the in the 18th century.  But this is a culture where, you know, writing is in literacy is not as widespread. And it's also the case that if mothers come back to claim their children three years later, the three year old does not look like the newborn child. And there needs to be some way to connect the child to the mother. And so mothers would leave tokens and the tokens were often cut from the children's clothes or from the mother's clothes. And this token, I think, is particularly heartbreaking. It is a heart, right? So she's cutting this red wool, which is actually particularly fashionable red wool. She also has another piece behind it and then a ribbon behind it as well, the blue ribbon. And she pins it all together and they pin it into a book then. And that's how she can know which child is hers. And so mothers are very careful in selecting the tokens. It wasn't just anything. They were clipping fabric from their clothes and their children's clothes to reconnect them in that relationship, which actually just, I don't know, that that touches your heartstrings.  And clothing here is the way then that they're establishing a legal claim to their child, which I think also has a metaphor and the imagery of that and the metaphor quality of that extends beyond this particular instance of mothers claiming children to the ways that people understood their clothing more generally. So, for instance, there are these examples in the court records where people, a neighbor would see somebody they didn't know going down the street with their friend's coat and that person would then chase the culprit down, attack them, right? Grab the coat and not only take the coat back, but then haul this person off to court. And the way that they do this is really they're very invested in this. And it's not just because it's a stolen piece of property. It's because these what was stolen was a piece of their friend. That that connection is really very profound. And the way that people saw and described each other was often through their clothes. You see this in some of the court records. Well, they'd be very careful about no, he was wearing a certain kind of cap or certain kind of hat, his coat, it was worn or it was too short. They're very specific about describing the clothing and sometimes less so about the actual physical features of somebody. But certainly the clothing was all of a piece of that. And then the really interesting kind of irony contradiction here is that it's because those things are so closely identified with people that then people are able to alienate that property. So it is that close identification, which I think goes beyond a consumer item. It really is a way of it's not even just self-fashioning.  But it's because of that that then people can alienate that property. It's like it is so clearly mine that I can then sell it and make legal claims to it. And I can claim the value of that. So it's that personal connection that also allows them to turn this into currency to turn it into, you know, like people save clothes, it's capital, it's their savings. They transform the value of their labor into this particular form of property precisely because they can make those legal claims to it. But I also find it's interesting. It's hard to remind around that in some ways. Yeah, let me let me back up to something you just said. And I had that kind of record sketch moment. So textiles as currency. Yeah, can you explain this is one of the more fascinating and they're all fascinating chapters, but this was really one of the more fascinating chapters. The way people use textiles as a form of currency in everyday life. Could you explain that for us a little bit? We're talking really about the 1820s, 1830s, a moment when currency is a very live issue in American political economy. Yeah, absolutely. So this is coming in at this moment. It's it's first in the the founding moments in the late 18th century, where the United States is kind of struggling to have its own currency and people are still denominating things in pounds and shillings. And then in the 1820s, 30s, where we have the big bank crisis with Jackson that people may be familiar with. Then you also have all sorts of state banks issuing currency denominated in dollars, but not just state banks, but even, you know, local areas. There's this proliferation of currency, but you don't know of the value of it. So we have a I think we have an image here again. This is a caricature of a treasury. Yeah, there's a bug on it, eating it. There's all sorts of negative references in here. You can go on and on and on and on. Actually, this is from the Library of Congress's collections. And you can go find it in their in their online resources. And it gives you a whole explanation of how to read this. But it is a parody of the banknote in the sense of there is no value to this banknote.  And people call these banknotes rags, getting back to sort of removing over into textiles. And they call them rags, bad textiles, in a sense, worthless textiles, because you can never tell how much they're worth. So there's one court case I have where somebody has stolen banknotes and they will list out the face value and then the actual value. So one of it is like a banknote from some bank in Southern Canada. They don't know what bank it is. And they say it's listed at, you know, I forget the exact amounts. Now it's like ten dollars, but it's worth seventy three cents. Or another bank that was listed at five dollars, but it's worth nothing. So when you get banknotes, you're just faced with, how do you know how much they're worth? You don't know. But with textiles, you actually do know. And this is one of the key things here. And so people are very familiar with these particular goods. They're widely available. Even poor people can afford textiles at this point. And they can afford an array of textiles. And they're at least very familiar with understanding the value of those. So when people are walking down the street, they're looking at textiles and they're thinking, oh, that's sure. That's nicely fit, good linen, that's worth so much. This is not such a good linen, it's worth less. And textiles, so they can value them. Everybody sort of has consensus about what they're looking at. And they come in an array of denominations. So you have like a handkerchief, you have a shirt, you have a length of cloth, you have a shorter length of cloth. So they're really useful that way. If it was just like if you were dealing with tools, for instance, like axes or hammers, I mean you bring in an axe and you want, you know, a loaf of bread is not going to work. But you can make the textiles work. I can hand you a shirt and you can give me two handkerchiefs in return. And this is not barter, even, because they're valuing all these things in what we call the standard unit of account, right? So the unit of account is what you account for value in. And for a while it's dollars and, or it's pounds and shillings. And then they flip sort of in the teens into dollars and cents. So you value these things in dollars and cents. So what you're doing is you're taking a handkerchief. Handkerchiefs are usually worth the 1830s about a dollar if they're new in cotton and good cotton. And so you put it up here, you abstract that into a value of a dollar. And then you trade it for something else worth a dollar. So textiles are not barter as some people have thought. It really is currency for people who don't trust currency, but also people who can't have currency. So getting back to Suki, the enslaved woman who stole some currency. People were willing to admit that it might be hers, but she really had to prove that because an enslaved person with currency is that's a dubious thing. She really shouldn't have that. And she has no basis of making a legal claim to that. And it's the same thing with married women. Actually, there's all of this proliferation of literature is great about women always are passing counterfeit bills. It's not like every woman out there is passing counterfeit bills, but it is that the idea of women with banknotes is suspicious because married women generally are not supposed to be trading in their own names, but textiles are completely different. So Suki is questioned when she has banknotes. But when she has textiles, it's like, yeah, whatever, you know, people walking down the street, black or white, rich or poor, and they have a shirt and it's like they're supposed to have a shirt, right? But a banknote, not so much. It can slave people with a watch, but sometimes painted brown so that it wouldn't be it would be seen as cheap and not expensive. It were silver and gold. So this is a secure form of property. It works really well because it comes in different denominations. And then it's widely accepted. You can like spend this stuff really easily. And it's easy to get rid of because there's such demand for it. And this to me was like, oh, my God, I thought this was barter. It's not barter. It's currency. It's just taking a different form. And then when you read up on this, this goes way back centuries where textiles were used as currency in the global textile trade for a very long time. So it's not like these people are making this stuff up. They're actually using established practices that have been, you know, sort of standard across the economic spectrum and even with wealthy merchants. They're still using those and they're persisting into century in the United States because it is so difficult for these people to use the other mediums of exchange that are available. Yeah, I want to return to the issue of how enslaved people fit into this history because for me and I'm a historian of slavery. So I found all of that exceptionally fascinating and significant. But I want to just focus on something else for a moment, which is the many connections that your book has to the new history of capitalism in some way and in some ways what people call the new history of capitalism is actually very much focused on business history, the new business practices that emerge across the 19th century. I was really struck by something you said earlier that your mother made your clothing when you were young and you have a point in the book where you argue that hand-crafted textiles persist through what we think of as the age of industrialization, particularly in cotton textile manufacturing. And I wonder just if you would I want to get into the legal history in a minute, but what do you think all of these ways that ordinary people make use of textiles in this age of tremendous capitalist transformation? What does that what does that tell us that we didn't know already or had overlooked in this new history of capitalism? You know, I kind of struggled with this because the evidence that I was reading was a lot of hand labor. So, you know, like they there's women who I talk about in the book who are doing weaving and spinning in the 1850s. Now, why are you weaving and spinning in the 50s? When I would assume that, you know, machine made cloth and machine slug thread would have replaced that. It was really mysterious to me. So, you know, I was also at one point asked, it's like, well, why are you doing this? We already know this history. And we know about the low mills and women and textiles. I'm like, oh, but that's not actually what I'm seeing in my sources. I'm seeing a lot of women and it's like people who are doing hand labor still in their homes. So this led me down another kind of rabbit hole where I was realizing, oh, I started looking at different the ways that different kinds of cloth are made and the different fibers. You have silk, you have wool, you have linen, you have cotton. And cotton is mechanized first. Wool and linen and silk later. In fact, linen is very late to be mechanized, which also explains part of the reason why cotton is taking over is because linen has been and still is done by hand. You it's very difficult to mechanize because the fibers themselves are different. Again, sort of the material qualities of this are really important. But then I also realized that in fact, this idea that the mechanization is the key moment sort of misses the longer global history of textile production, where people have been scaling up for the market for a very long time, in fact, centuries. And the way that you produce mass produce is actually through scaling up the traditional means of hand labor. So you have more people producing more goods, essentially. You use more and more labor to produce more and more. And you segment that labor out. And so you have kind of like the peace workers that you have think about as peace workers in New York City, the women working sewing shirts and whatnot in the 19th century. You have various sort of iterations of that going on for centuries before. And so mechanization is just one piece of this larger history. And the larger history is actually about hand labor, which involves women's labor. And then I had an aha moment when I realized I was looking at two pieces of cloth and one of the archivist was trying to explain to me that well, one was produced for the market and the other one was not. And I'm like, well, where were they made? And she said, well, all stuff is made in households at this point.  And I was like, oh, my God, I was thinking household production was for the household. And in fact, all the stuff that's produced for the market is produced in households.  But, you know, when you join household production. But what distinguishes goods made for the market is skill. And I'm like, oh, so a skilled worker can produce for the market. And basically, Homespun is badly done cloth. And that was interesting to me because it struck me that we've been missing that one piece of the puzzle, which is skill. And women and men both have skills to do textiles. It's not gender specific in any way, except with some weaving, which takes somebody who's a little bit stronger and bigger and men are more inclined that way. But the skill part is really important. And the skill that people can put into this makes them a good saleable. And that skill is really important throughout the 19th century. And in fact, alongside all of the domestic production that is mechanized, you find all sorts of people still doing hand labor because essentially, too, the more fabric you produce, the more hand labor you need to turn it into clothing, too.  And Seth Rockman is working on this where, you know, you have wool production that is happening primarily within households. But people are scaling that up as well. And women are doing a lot of that production. Linen is always and still produced by hand primarily throughout this period. And even as we're moving from linen to cotton, there's still a need for linen, which is actually more durable and works well for a range of things. And then not everything can be mechanized. So, you know, heavier fabrics, really fine wools even after you have mechanization are done by hand. So, in fact, it was this it's again, it's this contradiction. The more mechanization you have in some ways, the more hand labor you need. And the more hand labor you have, the more opportunities there are for women to sort of enter into that economy, use their skills and then market, keep the value of those skills too. So it was that skill part that I think we're missing. And I think we miss it sometimes because we assume that things that are produced in the household are for domestic use and that women can't own them because their wives or daughters and the value of that would belong to the husband. But textiles are different. So women can produce, use their skills to produce textiles and then keep the value of what they produce. But that's the last point is the key, keeping the value of what they produce. And your argument is basically there is a there is a space in the law, especially at the local level that recognizes ordinary people, women, slaves, kind of attachments to textiles and recognizes on some level that there's stuff that belongs to them and can't be taken from them. Yeah. Could you could. So could you explain the how how the law works to recognize those attachments? Because that's really at the center of the book. Right. And I was confused to make it back to about this. It's like, how can these people keep this stuff? They don't have property rights. And essentially, you know, what you have are ways of owning property through property rights. And I'm going to be my one hand over here. You have this law over here, but then you have customary practices, which are also recognized as legal principles, but which are established through practice. And that token would be an example of that that we showed earlier, right? That this is established through practice, that these pieces of garments that are belonging to the infant and the mother can then establish a legal connection. And that is done through practice. It's not written in a treatise. There's not out there in a statute. It is simply what happens. And then the law recognizes that because it is it is practice. It's custom. It's what we expect. So the law is going to uphold that. So when you have, though, like, I'm going to give you an example from the book. There's a woman named Sarah Allingham, and she claims that her sheet is stolen. And actually, this is really complicated because she loaned her sheet and she now wants the value of this back. She seizes the sheet to get the value of the loan back. And then she claims she marches down to the magistrate and says, you know, you do something about this. So the magistrate here is faced with an angry married woman waving a sheet around saying he needs to do something about this, that somebody stole it from her and she explains the situation.  She's an Irish, but the other women are Irish in New York City. Happy St. Patrick's Day. Right. Today's St. Patrick's Day. But that's what they say in Patrick's Day. So I have to. There you go. Absolutely. So he takes this seriously. This is a cheap sheet, too. But he doesn't he can't actually this is a loan. This is a debt case. If she were a white man who had rights, then he would put this into private law, what we call civil matters today, which would be prosecution for a debt. But he can't do that with her because she's a married woman. She has no property rights. She can't contract. So she can't enter into a contract for property of credit and debt. And she has no property rights to that sheet. He still has the angry woman waving the sheet around and he knows the sheet belongs to her. And he also knows that these sheets are loaned oftentimes and there's whole array of economic practices attached to them. So what is he did? He ends up putting it to another legal form, which is theft and theft lies in the area of what we call public law. And it's distinguished, you know, we're called the state versus, you know, Adam Rothman state versus lawyer Edwards. That's a public case so that the matter rises to the level of public significance, violates something about the public order and affects us all. So you don't you aren't charging. You're not prosecuting based on property rights there. Whereas if it were a civil case, it would be Laura Edwards versus Adam Rothman because it involves two individuals and we're protecting the rights of those individuals. So the magistrate puts it into the area of public law where he can put things back where they belong. He can maintain the order of the community where we all recognize that textiles can belong to people without property rights without actually recognizing the woman's property rights. And this explains that case that I first ran across in North Carolina where the man's dress is stolen. That's the legal form that you have to use on the outside. You know, you're filling the chicken in the boxes, filling out the actual legal form. A husband can file charges because they have rights. And then once the charges are filed, it becomes a case against a person on behalf of the state. So you open up the records and then you can talk about who that sheet or that dress or whatever other textile, who that belongs to. And then you have a whole series of, you know, kinds of evidence that are really routinized weirdly up and down the Eastern Seaboard in a very, you know, kind of standardized way where people will bring in evidence that they're connected to that piece of property. Either they have made it or people have seen them buy it or people have seen them with it. Sometimes they will bring in the item itself, which I found interesting because it's like, OK, you're just accused of stealing a dress and you're going to bring the dress in as evidence that you own it. How does that work? Right. I mean, compelling evidence to me now in the 21st century. But it is then because it's like, look, it's my dress. I'm showing you it's my dress. I am making a public statement about my connection to this item. I mean, if I had stolen it, I wouldn't be bringing it in to show you. And then they bring in people who've witnessed the sale, who have witnessed them with it, who can provide some testimony about the person's relationship to that material item. And so then the magistrate will follow the evidence and try to make an assessment about who that who belongs to that that good, right? So it is understanding, but it collapses at the whole range of transactions into this one form theft. So it's not really theft, though. Some of these are loans that are not repaid. Some of it is disputed ownership for a sale that went sideways. There's a whole range of economic transactions that you would be able to see in this other side of the law, private law, civil law matters that get sort of obscured in this one legal form theft. I think at one point in the book, you say that these in these cases, the legal concepts like theft are masking. They're kind of masquerades where the real issue about the control of this property is concealed or or mislabeled in a way, but you you're able to see through that masquerade and tell us what's really going on, which is the one of the wonderful things about the book is you it's kind of you manage to make kind of a history of small claims court utterly fascinating opening up this this window into how ordinary people think about textiles and and other goods that that they think belong to them, which is really quite extraordinary. There's a cost you point out that there's a huge cost to this because on the one hand, this is a way for ordinary people without rights to make claims to things for themselves. And so it's a kind of alternative legal system where they have a presence. On the other hand, these things get criminalized in ways that are actually very hurtful for parties. So I thought that was an interesting tension that you draw out in the argument. Now, it's really sad to watch this because it's people, they don't care about the legal form. It's like they want their sheet back or whatever. And so, you know, the magistrates, the local court officials are like, God, what do we do? They're creatively using the legal system that they have to accommodate all of this. But the problem is that, you know, if you have a debt case in the civil side in the 19th century, it used to be that you would be in prison for debt, right? So debt, do you end up in the same place, which is prison? And but those start those kinds of punishments start going away for people who are operating and can't operate with rights and private law and civil matters, right? So imprisonment for debt goes away. If you're sued for debt, then it's like, oh, well, oops, boo boo. And then you can, you know, after a brief period, you can kind of resuscitate your your economic interests and start over again. You're not a criminal that doesn't follow you forever. You don't endure physical punishment and, you know, the fines aren't what they are overwhelming like they are in the public theft side. But if you're resolving the same kinds of economic disputes in the form of theft, somebody has to be a criminal. So it makes those kinds of economic transactions look different, which is, you know, the remnants of what we have now and the sources, they look utterly different, even if they're both debt cases. Once you put one in the form of theft, it's criminal, right? And we think, oh, it's really theft. And it is sort of in the sense that somebody doesn't repay a debt as stealing property, but it's not the same thing we don't think as a debt case in the other areas of law. And to say then that somehow the you stigmatizing all these people's property ownership by making everything potentially look like it could be theft or criminal in some way when people own property. So, you know, by the 1850s, if you see somebody who is a poor woman, a poor working woman and she has, you know, a big stash of cloth, you might be thinking, yeah, that seems kind of dodgy to me because we're so accustomed to now adjudicating those cases in criminal matters. Right. It tends to undermine people's claims to the whole. It tends to undermine their claims to this kind of property. Yeah. But one of the most striking forms of property in the 19th century contested forms of property were people themselves. And I think it's one way one way this connects is you talk in the book about advertisements for fugitives, fugitive slaves. And in, you know, in some sense, fugitives were stealing themselves, trying to reach freedom, but were criminalized in doing so. So we could talk about that. But what I want to really talk about is when you read through these runaway advertisements for 19th century newspapers, they are full of clothing.  There are many advertisements where there are just lavish descriptions of the clothing the fugitives are wearing or maybe took with them. In some ways, the descriptions of the clothing are much more precise and ornate, elaborate than the descriptions of the people themselves. Yeah. My students are always kind of astonished at the extensive language, the extensive descriptions of clothing. And of course, they mostly don't understand what's being described like the different kinds of clothing. And by the way, for the audience out there, if you if you want to just browse through these runaway advertisements and they're really quite extraordinary reading, you should go to freedom on the move, freedomonthemove.org. I think an incredible project that's collecting, transcribing runaway slave advertisements. So my question for you is why are these advertisements so much about clothing? It's a good question. And, you know, part of it is that, like I said before, people are seeing textiles. They understand what they're seeing in ways that we don't anymore. I mean, I hardly remember what anybody I just to I just saw was wearing, but people have a memory for this. I mean, they're seeing the world in a very different way. And so they're seeing people through their clothes. They're also doing this because it is a way that enslaved people use to kind of pass into freedom. So, you know, clothing is also a second skin.  And so by dressing as a free person, sometimes you can be a free person. And oftentimes that part is really important in making that transition from slavery to freedom. And this is also it gets to some of the larger sort of points and arcs of the book, which is like the importance of rights here as well. And, you know, we have this, there are many moving parts in this book. And one of them is understanding all of these different jurisdictions within the United States. So, you know, you have local jurisdictions, you have state jurisdictions, you have the federal jurisdiction.  It changes depending on where you are, right? In one state you can be presumption is you're enslaved if you're a person of African descent. In another state, the presumption is you're free. And so it's actually really very confusing when you're on the ground trying to figure out who is enslaved and free. And clothing can blur all of that by making enslaved people look free, too. So the really detailed description of that is, you know, the really detailed descriptions of clothing are built in around trying to figure that out, I think. And I think it also gets to the importance of ultimately rights and rights as the way and the primary way of owning property, too, because claims to clothing are very much rooted in particular places. So an enslaved person who has a full array of clothes. There's one person, James, in the book he's enslaved. He has striped velvet pantalons, which I find to be, again, one of those things is like an enslaved person with these like amazingly fashionable pants is an amazing, it's fascinating to me. But he can own those because of where he lives.  In fact, they're stolen at one point in their return to him because people know that they're his pants. But if he moved someplace else and he has those pants, those might not be his because people aren't there to establish that connection to those material goods, right? Whereas property rights flow over jurisdictions in the United States and are really important that way. So you can take your status with you. You have a right to your body. You can also claim an array of textiles over different states and across different localities if you have rights to those things. But it is more difficult if you're claiming using the traditional principles of textiles, which is your connection and your relationship to those goods. And that doesn't travel nearly as well. And I think that's also sort of tied up in those runaway slave advertisements, too, is that sort of clutch of legal principles. One about rights to your body, rights to property. And the other about claims to goods that are based in custom and your relationship to those items. So over time, yeah, that is problematic. Yeah, so your argument, basically, is that or one of your arguments is that over time, moving into the late 19th century, those kinds of customary the recognition of those customary attachments of ordinary people to clothing and other textiles is undermined. Yeah. Yeah. And that this rights based regime comes to the fore and that really disadvantages ordinary people, doesn't it? Well, it does.  But the problem there is that rights aren't extended out to everybody in equal measure. And rights are really important in the United States because, like I said, they move across these jurisdictions. It's the way that you can keep your status and move around. And it is more secure. Creditors like this better. Businesses like this better because they know who they can deal with. And it doesn't matter where that person is. They know they have property rights and it's all the same, right? It's uniform across these jurisdictions and they don't have to walk in and say, OK, who actually owns this stuff? Can you establish that? OK, let's bring in your friends. Let's bring in the goods. I mean, that's really complicated to do, right? So again, in this emerging economy, that really matters. So, yes, but over time, as the legal system starts then elevating the rules of property rights, legal procedure in ways that elevate that way of owning property over the other ways of owning property, though, it's really problematic for people who can only own property through those customary practices. And those are enslaved people, after and married women. Actually, that extends to all women and oftentimes people without who are poor, whose claims to property seem somewhat problematic because they don't have resources and cannot not to be able to claim that. So the extension of rights is that even and even after the Civil War and reconstruction, where in theory rights are extended to African American men, you know, in practice, it's hard for them to claim those rights. So if you go to a rights-based economy and legal system, but not everybody has equal access to those, it becomes really problematic for a lot of people who are left with only the clothes on their back. And it's the title, but those are not legally meaningful things anymore. They simply become more like consumer goods. We have a couple of questions from the audience. And I think this one is connected to what you were just talking about. Can you discuss the declining value of textiles in the decades prior to the Civil War? Was this change a deliberate result of a concerted effort to disenfranchise certain groups? So actually the economic decline has more to do with, you know, the the increased pace of mechanization and the decline in prices of cotton goods, which then also lead to the decline expectations of what other fibers of their kinds of cloth should cost to. So it leads to an incredible decline generally. But, you know, part of what I want people to think about in this book is that value is not just about what these goods fetch at the market. But value here is also about the legal qualities that attach to these items that make it possible for people to claim them, which means that those goods have more value than other property for certain people. So that part's really important, too. And part of what happens in this period is the legal qualities value declines, too. So it's kind of a double whammy. And in that way, you know, what you're talking about economic value, if you take out the economic and say the value of these goods, the legal value of these goods is also about elevating a rights-based framework that, in fact, excludes large portions of the population. And so that I think has sort of different implications here. And that becomes a problem. And it's at that moment where we're excluding all these people from participation in the economy by elevating this rights-based model that you get abolition, move for racial equality, rights-based claims by women. When rights become more important, you also get those claims of the necessity of extending rights to more people as well. Here's another question. Does this topic give a whole new perspective on the production of uniforms for the Revolutionary War after reading her 2,200 shirts for Washington's Army and the Civil War? You do actually have some things to say about uniforms. Yeah, yeah. Yeah, uniforms in the Civil War are really, I think, in particular interesting because one of the major claims and goals for African Americans during the Civil War is to fight in Union Army, but not just to be admitted into the Union Army, but to be uniformed soldiers in the Army. And the uniform itself is about clothing you in the status and power of the United States. And in fact, they're a whole, it's still the same sort of thing to this day, right? If you're a uniformed officer, a different whole series of laws apply to you about what you can do, where you can go and how you're treated, then if you're at a uniform. And so, you know, African American soldiers when clothed in the uniform of the United States have US protection. And then after the war, they're given rights. And I realize I'm looking sort of in a metaphorical way here, but they're given rights. They're clothed in rights. And that is actually a phrase you see often in law, but they have to give their uniforms in, right? So clothed in rights, they're a little bit more vulnerable, actually, than in that actual clothing of the power and authority of the US government. So it's interesting to think actually of what uniforms meant about the clothing itself, that outer skin and what that authority meant and why that authority was meaningful. And, you know, the fact that clothing can communicate that authority also is not something that is unique to the Civil War, but it goes back, you know, a very long time in other contexts, too. We just have a few more minutes left. I have one final question for you, which gets it gets us to the end of the book. Now, history books are not mysteries, usually. So it's not like I'm giving away, you know, solving solving the crime here. But your conclusion focuses on Mary Todd Lincoln. Yeah, why Mary Todd Lincoln? So Mary Todd Lincoln, after her husband is assassinated, she's left because of the will and contest over the will and settling the will with very little money. And so she needs something to live off of. So she decides to sell her clothes. This is not unusual. In fact, people have been doing this. Women in particular have been doing this for, you know, centuries. You buy clothes, they're not just consumer goods, they're capital. You sell them, right? It's the thing that you invest money in, and then it is something that you could spend. So the fact that she's doing this is really not unusual, but she's completely lampooned and rid of, filled in the press at this point. And it is interesting, because the whole narrative about this ignores the fact that women have been doing this. And it's totally acceptable.  Women pawn their clothes, they sell them. But she is seen as somehow prostituting herself. And it is this really interesting moment where what used to be currency and capital where people would see what she's wearing is. This is her personal duress, but it could also be something that you could alienate and use as currency and it had other economic uses. It's now seen as being something that is just her personal property. It's a second skin, the legal meanings of this. And then the economic meanings are beginning to flow away. And so it's seen as somehow she's selling herself rather than actually selling a piece of property that she could own and then use as currency and sell to raise capital, which is really fascinating and also says something about those declining value legally and economically of these goods. And what's really, really sad about this story is that Elizabeth Keckley, the picture of her as well, was Mary Todd Lincoln's modest and Elizabeth Keckley was an enslaved woman who bought her freedom through her needle. And she was a very talented designer and she actually, you know, designed and made clothes for the Washington wives of the Washington elite, including Mary Todd Lincoln. But as Mary Todd Lincoln's clothes are ridiculed, so essentially and indirectly, but implicitly, are Elizabeth Keckley's skills and what she was able to do with her needle and the beauty and value of what she made for Mary Todd Lincoln. And after all of this was over, Elizabeth Keckley could never work as a designer, essentially again. And here she is in a beautiful dress, which is also beautifully fitted. And so we have this moment where all of that value is going away. And instead, you have women seen as simply buying things that are fashion items that are disposable and that it's unseemly then to try to get any other kind of value out of them, which is very sad. Well, I think that's a sad but apt note to end our conversation. The book is only the clothes on her back by Professor Laura Edwards, and I encourage you all to read it. And thank you so much, Adam. This is fun. And thank you, everybody, for coming.


You're no longer connected to information or people. Is anybody there? Are they okay? You simply don't know. Now imagine you're in the middle of an armed conflict. Safety may be a distant memory and violence is not the only thing that can harm you. People are talking of war. Fighting is erupting in your city. There is no power. What if phone lines go dead and the internet connection shuts down? With no means to charge your mobile phone or go online, you find yourself cut off from the rest of the world. News doesn't reach you. You're left in the dark wondering and worrying. What's going on? Are my family and friends safe? Is anybody coming to help? Being able to connect physically and digitally can give you back hope and strength in a crisis like this. But this need to connect can also be turned against you. During armed conflict, every form of communication brings uncertainty. You can't be sure if it's safe to talk in the street. Local media can be controlled by armed groups. What if your conversation is overheard? Where do your messages end up? Who can you trust? What news can you believe? Can you distinguish fact from fiction? What used to be a network of connection can trap you in a web of fear and manipulation. Enabling safe and trusted communication makes a real difference. Your community can reconnect and reorganize, ask for help, make informed decisions and demand accountability. This dialogue turns frightened people in need into active responders. The humanitarian community has made a lot of progress in the last decade to help people communicating crises. But armed conflicts are very different from natural disasters where things may improve over time. Conflicts pose quite specific challenges. It's crucial that we take that into account in order to establish better, smarter and safer ways to communicate with people in need. Amidst violence, communication is a lifeline. No matter how difficult the circumstances, meeting people's need for information and dialogue is never a choice. It's our collective responsibility.


Okay, welcome to the audio and video instructions for Worksheet D, T-bills and mortgages. At this point, you should have good familiarity with the way Excel is working and making models based on the templates I've been providing. So this worksheet is slightly more open-ended in the sense that I didn't provide all the face value, interest rate, gear value here, that you need to write that, you need to type that in. Do it in a way that makes it look nice, show that you really have mastered creating models in Excel. So the first part, you need to create a model that calculates the price of a zero coupon bond. If you know the yield to maturity, the face value and the time to maturity, that's the first model. I'll create it right here on this page. And the second model, create it right below, is slightly different. You'll have to rearrange the formula a little bit to create a model that calculates the yield of a zero coupon bond. If you know the price, the face value and the time to maturity, that should be fairly straightforward if you've been following along so far.  This is a little bit of a more challenging sheet, there's a little more guidance here. You need to figure out a formula for calculating the loan that a person can afford, if the total amount of the loan that a person can afford, if they can afford a $1,000 monthly payment assuming that the interest rate is 5%. So your model should calculate the total loan amount if the monthly payment is $1,000 and the interest rate is 5%. This is a 30-year fixed, for a 30-year fixed rate mortgage. You're going to need to use the appendix to chapter 4 to figure this out. And you're going to need to look carefully at my hints up here because I want you to make this worksheet work given the standards of the mortgage industry. And that's where this hint 2 comes in. Point 2 says mortgage interest rates are quoted in a funny way. When the mortgage industry says that your yearly interest rate is 5%, what they really mean is that your monthly interest rate is 5 divided by 12, or perhaps a little bit more clear. If they say that your yearly interest rate is 12%, that means that your monthly interest rate is 1%. So in order to make this model work, you need to make sure that in your formula that you enter the interest rates in a monthly time period and that you calculate the formula for the total number of months being 360 months, not 12 years. In other words, your model should not calculate 12 years at 5% or whatever per year. It should calculate 360 months at 5% divided by 12 months. If you do it that way, then when you go online and you type in say mortgage calculator in Google and you enter this information, what you'll get on your mortgage calculator will be in Google on whatever website you feel like should be the way that this model works out. So the cool thing you should be able to do is create a mortgage calculator instead of having to go online. You should be able to create one right here that you can use. In order to make sure you've done it right, because I know that this is going to be challenging, you're going to have to play around with it, I have provided solutions for you. This isn't the formula, that's the trick, you need to figure that out. But your formula, if you enter these parameters should yield this result. If you get that, you'll know you did the formula correctly. In other words, at an interest rate of 5% and a monthly payment of $1,000, that person can afford a $186,000 loan.  So think about it, when you get a mortgage, there's sort of two ways you can think about it. You have a budget, say $1,000, and you ask yourself, well how much house can I afford with $1,000? That's what this first calculation gives you.  That's what this payment, that's what this model gives you. Obviously the two are very related. For the purposes of this class, there's some different economic meaning that we associate with that. That is going to be drawn out down here in this section of the assignment. What I'd like you to do down here is using the models you've developed up here, create some data tables. Up until this point in the class, you've always created a two by two data table. In other words, you're varying something in the row and in the column. This data table is only asking you to vary the interest rate. It's saying, hey, listen, leave the monthly payment fixed. In other words, what this data table is going to show you is assuming that you have $1,000 monthly payment. How much loan can you afford if the interest rate is 7% and what happens as the interest rate on mortgages falls? What you should notice is as interest rates fall, the amount of loan you can afford increases. The point of this is to show you that falling interest rates put upward pressure on home prices because they enable borrowers to spend more money on a fixed income. The next one is asking, well, what happens to your monthly payment if interest rates change? And this is particularly relevant for types of mortgages that have adjustable interest rates. Now, you've done this calculation for a 30-year fixed rate mortgage. But just to give you an idea, there is such a thing as an adjustable rate mortgage. That means that your mortgage payment will change depending on what the prevailing interest rate in the economy is, and it will adjust every month to that interest rate. So the message of this model is, look, if interest rates are low when you take out this mortgage and then they rise, your payments are going to start going up. And that's going to be a problem for you if you can't afford the higher payments if you can only afford the lower payments. And we're going to talk much more about why that mattered for the financial crisis. But this model should just give you some intuitive sense about the relationship between interest rates and monthly payments and interest rates and the amount of house that you can afford. Again, though, the challenge with this spreadsheet is figuring out the right formula in here. And let me tell you, it's not simple. It's not an easy formula. But if you focus on that appendix, you should be able to work it out. Again, I provide the answers to the right.


So, in our last lecture we looked at the way of using Taylor series approximations to discretize OD boundary value problem. So we are able to convert a differential equation and the two boundary conditions into a set of algebraic equations. So the problem got transformed from the original infinite dimensional space to a problem on a finite dimensional spaces which was set of linear or non-linear algebraic equations which have to be solved. So today let us look at some examples and let us see whether we can extend this concept to solving partial differential equations. So I wrote a generic boundary value problem, discretization of, so we had this original problem was or I would say y is equal to T of x, this original problem was, so you have this differential equation which holds over entire domain 0 to 1, this is the differential equation and then you have two boundary conditions, one is that, so you have boundary condition 1 which is f1 and you have second boundary condition which is f2, we have two boundary conditions. Then what the way we proceeded next was we discretize the domain, we discretize the domain, we mark these grid points such that using equidistant grid points we discretize the domain, we mark the grid points and then we had this notation that a dependent variable u, ui is equal to u of, ui was the value that this dependent variable takes at z equal to zi that we denoted as ui, note that this u here is the approximate solution, the real solution of this would be some u star z which would be a continuous function, twice differentiable continuous function and we are not able to in general find out this solution in many situations where the operator is not linear, we are not able to find this solution, so we want to discretize and solve this problem numerically and not analytically. So we have created this grid points, we have marked the dependent variable value at grid points as ui and then we converted this boundary value problem, the differential equation first we converted into a set of algebraic equations, so that was done by psi, so we approximated the second derivative using Taylor series, we approximated the first derivative using Taylor series, so at all the internal grid points, see here these are internal grid points z2, z3, z4 up to zn these are all internal grid points and then z1 and zn plus 1 are two boundary points, z1 and zn plus 1 are two boundary points whereas, so at all the internal grid points we enforce this equation, please note that enforcing this equation does not mean we are solving the exact problem, we are solving an approximate problem and for the approximate solution we enforce this equal to 0, there is an alternate way of solving this problem is to create two hypothetical grid points, one on the either side of the end points and then discretize, either way is fine if delta z is small, so for the time being let us write the equation in which we use one sided derivative approximations, so this what I get here is the approximation, this approximation is nothing but in the notation that we used earlier it is nothing but y cap is equal to or y tilde is equal to t cap x tilde, so we started with some problem we discretized and we got this discretize problem, so we started with a differential equation and two boundary conditions, we got set of algebraic equations, so the transform problem is completely different from the original problem, we hope that if you choose delta z is small then the Taylor series approximation is a good approximation and then this transformation will give you a solution which is close to the true solution, this will not this way of transforming will not recover the original solution, it will only recover approximate solution, if you reduce delta z then you will get better and better solution but the price that you have to pay is that smaller the delta z more number of equations and then you will realize that these equations are coupled for ith equation requires u value of i plus 1 and i minus 1, so these equations are coupled, so if you let us say take 100 internal points delta z is very very small if I take 100 internal points then I will have 100 equations here and then 2 additional equations, 102 equations in 102 unknowns to be solved simultaneously, how do you solve this Newton Raphson method or if these turn out to be linear equations solve them using you know ax equal to b simple Gaussian elimination, so you are transforming the problem and then solving it let us look at some examples today, so this is the background and we catch up from here, so my first example is going to be a tubular reactor example, so my first example is going to be T R AM or tubular reactor with axial mixing, I am not going to derive the equations I am going to just state the equations and then we will see how to discretize that is the key thing here, so this is a simple reaction where a goes to b, a very very simple reaction where a goes to b and the ODE BVP that is given to you is 1 by Peclet number into d 2 c by d z square by d c by d z minus this is Danckworth number into c square is equal to 0, so this should be obeyed between 0 z 1, okay this is my differential equation and then I have two boundary conditions, so my boundary conditions here, so my first boundary condition at z is equal to 0 the derivative at z is equal to 0 is Peclet number into c concentration at 0 minus 1 and my second boundary condition at, so this is my second boundary condition I am just going to apply the method of finite difference, this method that we are developing is called as finite difference method because we have developed approximation of first and second order derivatives using Taylor series approximation we have developed finite difference approximations okay, so now my first task what is this operator psi here, this operator psi here is nothing but this differential equation okay, so this second order derivative I am going to replace by its approximation at any ith point okay, same thing is true about this derivative I am going to replace approximation of this derivative at any ith point and so I am going to force the residual the term that you get is called as residual the residual is forced to 0 at all the internal grid points at the two boundary points I am going to use the two boundary conditions, so this equation will now be replaced this equation will now be replaced by 1 by Peclet number, so this will be ci plus 1 minus 2 ci plus ci minus 1, now what are these ci, c1, c2, c3, c1, c2, c3 are concentrations at the grid points, so at this point the concentration is 1 at this point is 2, c3 at in general at this point is ck, ck plus 1 and so on cn and cn plus 1, so these are the dependent variable values at the grid points okay, at any ith location at any ith location I am discretizing the differential equation, so the first derivative at ith location the first term that is second derivative is replaced by this then minus ci plus 1 minus ci minus 1 divided by delta z divided by 2 delta z then what is the next term, da ci square is equal to 0, this algebraic equation this converted algebraic equation should hold at all the internal grid points i is equal to 2, i is equal to 3 up to i is equal to n okay, so suppose I create let us say total grid points is 100, so 98 internal grid points 2 boundary points okay then I will get 98 equations like this okay each equation is related to the neighboring values of each equation is related to the neighboring values of c okay, how do I discretize this, what is my first boundary condition, my first boundary condition is this will be c1 this will be c2 minus c1 by delta z is equal to Peclet number into c1 minus 1, so this is one more equation this is one more equation I have just I have just converted this into a difference equation finite difference method alright I have converted the first boundary condition into a difference equation now I will convert the second boundary condition into a difference equation, so this will be cn plus 1 minus cn by delta z is equal to 0, how many equations I have now not 3 n plus 1 equations how many unknowns I have n plus 1 unknowns, if I solve this n plus 1 equations in n plus 1 unknowns I will reconstruct a solution an approximate solution of the boundary value problem okay, difficult to solve analytically because of this ci square and in general I have taken very very simple reaction okay it could be 2a going to so 2a going to be something like that, so I have taken a very simple reaction this could be a more complex equation here okay, so this n plus 1 equation in n plus 1 unknowns I have to solve them simultaneously to now I want to point out something else here in addition to talking about discretization I want to talk about some nice structure that appears okay I am going to rearrange this set of equations and then write them in a specific form to give you inside about something that will be talking about little later I am going to talk about a sparse system, now this particular equation is a very nice equation if you look at if you look at it there are n plus 1 variables but only 3 variables appear in one equation only 3 variables appear okay and when you have large number of equations you can actually make use of this fact that only 3 variables are appearing in each equation these kind of equations will give rise to what are called as sparse systems okay so we will side by side as a side note I will also introduce this idea of a sparse system and then later on we will be looking at special algorithms to deal with the sparse systems but right now it just to give you an idea which is so this first equation and this together I am going to rewrite in a matrix form which will give me a sparse equation okay so if I if I collect all the terms of you know ci, ci plus 1 and ci minus 1 together then I can write this equation as alpha ci plus 1 plus sorry minus beta ci plus gamma ci minus 1 is equal to da ci square okay where alpha is 1 minus 1 by delta g square okay I get equation like this n is equal to 2 3 up to n okay I will just group the terms together I will just group the terms together okay and then we have two more equations we have two more equations coming from the boundary conditions I am going to combine this and I am going to write this as a matrix equation okay so if I include the two additional boundary conditions okay what are the boundary conditions one boundary condition will give me right one boundary condition will give me this equation the other boundary condition will give me ci n plus 1 is equal minus cn is equal to 0 so these I have n plus 1 equations in n plus 1 unknowns and I am going to write them in a specific form I am going to write them in this particular form so this particular equation will be set of n non-linear equations in n unknowns you might be wondering initially when I started talking about Newton Raphson method in multiple multivariable domain where do you get this where do you get these equations classic example okay one boundary value problem is giving me if I take 100 grid points I will get 100 equations okay so large number of equations this equal to see this is like this is like matrix A this is the matrix A operating on vector x which is f of x you have to solve ax equal to f of x what is x here x here is this vector c1 c2 cn plus 1 in abstract form this equation this equation one matrix equation is nothing but matrix A operating on x gives me f of x I have to solve this I have to solve this problem or in other words I have to solve ax minus fx equal to 0 ax minus fx equal to 0 you can solve this by Newton Raphson you can solve this by simple iterative methods for example I could I could solve this by a simple method which is iterative method of this form ax k plus 1 is equal to f of x k I start with a guess x not I start with a guess x not so I start with a guess of concentration profile what is x don't forget that this is concentration profile this is where your input as a chemical engineer will come into picture you have to give a sensible concentration profile if you suppose give here negative numbers it is not going to work so giving initial guess somebody had asked is it very important it is very very important how do you give initial guess okay so this is my concentration profile discretize concentration profile and I can give a guess if I put this guess x not here I can generate the new guess by solving ax equal to fx not then x1 I can substitute get x2 x2 I can substitute get x3 and so on and then I can see whether you know difference between this and this is going to 0 or not by checking norm of this and see whether algorithm converges okay this is one way of solving right now that is not important what is important is that if I solve it like this this a I have to solve ax equal to b see because once I put x0 here okay this f of x is a known value okay I have to solve for x1 by solving the linear system ax1 is equal to okay then I have to solve another linear system ax2 is equal to fx1 and so on so I have to solve large number of linear systems linear algebraic equations now if I come back here if I come back here okay and then if I look at this matrix there is something special about this matrix this is a sum of three diagonal but the upper one is not strictly diagonal so this is a this is a matrix this is called as a tri diagonal matrix it's a banded matrix there are a lot of zeros how many elements this matrix has let's say if I take if I take 100 grid points how many elements this will have 100 plus 100 if I take 1000 grid points this will have 1000 plus 1000 and as we said more than number of grid points better is the approximation so I am forced to take smaller delta z if I take smaller delta z more number of equations matrix will have large number of elements okay if I develop some special methods that can deal with this banded matrices then this iterative calculations will become very easy okay so later on we are going to look at this methods for sparse systems this is this is a sparse so we will develop special methods for dealing with sparse linear systems okay which will reduce the computation because suppose you need 200 iterations 200 times if you have to invert a 1000 cross 1000 matrix not a nice thing to do okay instead if you develop a method which takes into account that there are lots of zeros okay you will be able to do calculations very very fast that's where all these sparse matrix method will come into that is a side note we will need this very often and when we do the discretization using finite difference you will see that almost every time you hit into this sparse matrices sparse matrix okay some form of the other okay so let's move on to some other example but it is not necessary that this finite difference method can be used to solve only boundary value problems it can be used to construct approximate solutions of partial differential equations as well okay I will take the same tran problem and show that how you can convert this problem a partial differential equation right now I took the steady state problem I will look at the problem which is time varying now in this case we got algebraic equations in the partial differential equation we might end up with something else okay so what is it that will end up with so my second example is a PD say same unsteady state same tubular reactor with axial mixing same tram okay with except that I am going to consider the unsteady state condition not the steady state condition okay so here so let's say this is a reaction in which 2A goes to B and then we have this dou C by dou t that is rate of change of concentration inside the reactor okay this is 1 by Peclet number now unlike the previous case where I was looking at the steady state behavior I am looking at a transient behavior I am looking at a unsteady state behavior so dou C by dou t is not 0 okay the concentration is a function of time and space not just space okay earlier we looked at the boundary value problem that came that arises when you look at a steady state of behavior of this particular system okay I still have the same two boundary conditions but I will also have an initial condition now okay there are two boundary conditions at z equal to 0 z equal to 1 and is also initial condition in time okay so at t equal to 0 so at t equal to 0 I have some concentration profile I have some initial concentration profile okay inside this is given by fz okay this is my initial condition there is some concentration profile inside and then I have two boundary conditions sorry this is not 1 this is t so this boundary condition is at z is equal to 1 this boundary condition is at z is equal to 0 so the solution now solution now is actually function of two independent variables one is time one is space another is time at z equal to 0 for all time this boundary condition holds at z is equal to 1 at z is equal to 1 at all time this boundary condition holds okay I want to now discretize this particular problem using finite difference method okay using finite difference method what I am going to do is I am going to discretize in space and leave the time untouched for the time being how will I solve the resulting problem something that we will see later but we will convert into a standard form which can be solved using a standard tool okay so in this case this partial differential equation I am going to convert into a set of ordinary differential equation okay I am going to discretize in space not discretize in time okay so what I will get is a set of differential algebraic equations what I will get is a set of differential algebraic equations so how will I discretize this okay the same trick that we did earlier in space we are going to you know we are going to denote this grid points z1 z2 z3 and then you have this c1 okay again I have discretized I have discretized my domain except now the dependent variable c that is concentration at the grid points is not only function of space is also function of time okay I will discretize only in space so c1 so the discretization in space appears through these indices 1 2 3 4 5 up to n plus 1 each one of them is a continuous each one of them is a continuous function of time so I am going to only discretize in space okay so now what will be the residuals I write I enforce the partial differential equation only at the grid points actually the original partial differential equation holds at every point inside the domain at every time we are not able to do that because we will get large number of equations okay if you start infinite number of equations if you start forcing it at every point we have to discretize and okay so now if I discretize this I will get dci by dt see the differential equation is forced at the ith grid point okay what will be this 1 by Peclet number into ci plus 1 t minus 2 ci function of time well minus ci plus 1 t minus ci minus 1 t by 2 delta z what I got here what I got here is set of ordinary differential equations I started with the partial differential equations I started with one partial differential equations one partial differential equation got converted into large number of ordinary differential equations okay so in this case the original operator t is partial differential equation the transformed operator is set of algebraic and differential equations why I am saying algebraic because I have two more conditions here okay one of them well in this particular case you will get two more differential equations what are the two additional differential equations the two additional differential equations that you get here are well this is a derivative in space so you will get algebraic equations here not the differential equations so what will be here c 2 t minus c 1 t by delta z I am taking forward difference approximation here and is equal to Peclet number into c 1 t minus 1 so this is one algebraic equation okay what is the second algebraic equation so what I get here is a differential algebraic system okay I got differential equations in time and I got two algebraic equations actually in this is a simple problem in this particular problem I can eliminate two variables and I can convert this problem into n minus one ordinary differential equations I can treat it like that or alternatively I can treat those ordinary differential equations and this two algebraic equations together into a single DAE differential algebraic system at towards the end of the course will be looking at how to solve differential algebraic systems so there are two ways to go from here one is to use these two constraint eliminate c 1 and eliminate c n plus 1 you get equations only in terms of only in terms of c 2 to c n n minus one differential equations in n minus one unknowns and then well you cannot stop here you also have to discretize initial condition this is a differential equation it will need initial condition so you cannot just stop at here discretizing the boundary conditions what about the initial condition what will be the initial condition so initial condition will be c 1 t c 1 0 is equal to f z 1 c 2 0 is equal to f z 2 and c n plus 1 0 is equal to f so these are the initial conditions these two algebraic constraints coming out of the boundary conditions and then n minus one ordinary differential equations have to be solved simultaneously okay have to be solved simultaneously so we have transformed the problem which was a partial differential equation which is difficult to solve analytically because you have c a square term appearing okay analytical solution could be difficult in this particular case in general if you have a more complex reaction rate equation it will be very difficult to solve this problem analytically you have to solve it numerically okay so you have to convert this problem into set of ordinary differential equation initial value problem okay or differential algebraic equation with initial value specified okay so these equations have to be solved simultaneously the transform problem is different from the original problem so remember this let us look at some other partial differential equations okay is the idea clear what is happening here here we transform a partial differential equation into a set of ordinary differential equations earlier a boundary value problem was transformed into set of algebraic equations which were non-linear which have to be solved simultaneously remember this set of ordinary differential equations is not a linear ordinary differential equations this has to be solved numerically using some method like Euler method or whatever we will be developing these methods later on right now I am just worried about problem transformation okay I am only worried about problem transformation yeah no I have n plus 1 variables and I have n plus 1 equations out of this n minus 1 equations are differential to our algebraic okay so I can eliminate two variables from this ordinary differential equations using the two algebraic constraints and convert into n minus 1 ODE initial value problem that is possible in this particular case it is possible if your boundary conditions had some non-linearity it is not possible to convert into very easily into ordinary differential equation then it will be a DAE system it has to be solved as a DAE system okay my third example is going to be a partial differential equation in two dimensions okay so in two spatial dimensions so this is a model of a furnace okay temperature distribution in the furnace the three walls of the furnace are insulated and at a constant temperature idealization okay and then there is a you know convective heat transfer from one of the faces I mean if you ask me how do I model this particular room temperature distribution in this room I would use equation something like this well why why is this like a furnace equation well you can see that this phase is not insulated okay we could say that from this phase there is convective heat transfer to outside okay these three walls let's assume as an idealization are insulated perfectly at the constant temperature where is the heat being generated well each one of you is like a bulb of 40 watts so there are 40 students into 40 watts so there is so much heat being generated inside this room okay so the temperature inside this room is function of two variables x and y okay the way the heat is distributed the heat sources are distributed all of you are sitting along different places okay this is like a so what is this model so this is the famous Laplace equation okay let's initially consider the steady state problem all of you are perfectly generating the same amount of heat okay at all time there is perfect steady state okay now there will be four boundary conditions here at four different boundaries right so we said that the three boundary conditions are so this equation should hold between 0 x 1 I have normalized the length okay between 0 and 1 so this partial differential equation should hold at every point at every point inside this room okay and then I have four conditions my four conditions are x is equal to 0 my t is equal to t star insulated boundary okay then x is equal to 1 t is equal to t star okay when I say t here which means t along the boundary okay I'm not I'm taking shortcuts and not writing the full then at y equal to 0 I have t is equal to so I should write t x 0 is equal to t star so accordingly I should write even these equations okay t at all x so three boundaries have a constant temperature and the fourth boundary so I have convective heat transfer at the fourth boundary at y equal to 1 I have k so I have four boundary conditions here I have four boundary conditions here okay now how am I going to discretize this I'm going to use Taylor series approximation in one dimension at a time okay in one dimension at a time so I'm going to I'm going to discretize this domain okay by constructing grid points or my domain is something like this I'm going to construct grid points here so this is my one two three four and so on this is my okay so my finally I'm going to get nx plus one grid points along x direction and let's say n y plus one grid points along y direction one to n y plus one one to nx plus one so so if you draw these lines here parallel to the so in general in general I'm concerned about forcing the differential equation at some i z point okay I'm constructing a grid here I'm discretizing see earlier we had only one special dimension we discretize in only one special dimension now I'm going to discretize in two special dimensions x and y okay and then these partial derivatives are going to be approximated in two special dimensions so my partial differential equation where should this where should this partial differential equation hold everywhere actually it should hold at all the points okay all the points even for one dimension we created hundred grid points suppose you create hundred like this and hundred like this how many internal grid points will be there hundred cross hundred okay we'll have hundred cross hundred internal grid points and you want to force the partial differential equation at every grid point at every grid point right so in general at i z point I can write this equation as well we are going to use this notation ti j is equal to t x i y j so temperature this is to simplify the discretization process I'm going to use t i j is temperature at x i y j okay so this is ith vertical line and this is zh parallel line okay at this point so I'm going to write this equation and so what will be what will be the the first one will be ti plus 1 j minus 2 ti j plus ti minus 1 j by delta x square plus ti j plus 1 see I'm taking partial derivatives in x direction partial derivative in y direction 2 ti j plus 2 okay how many such equations I'll get i going from 2 3 2 nx j going from 2 3 2 ny how many algebraic equations I get from a one partial differential equation I'll get nx minus 1 into ny minus 1 right okay where are the additional equations coming from boundary conditions so I should discretize the boundary conditions and then take all these equations together okay how many equations and how many unknowns nx plus 1 cross nx plus 1 100 cross 100 if you take 100 grid points very modest requirement just imagine this room 100 grid points not not too many if you want a better solution probably you should go 1000 cross 1000 but the number of variables will be very very large I'm not I'm not getting into the physics of the problem given this problem we want to solve it let's so now we will require additional equations right at the boundaries so what are the boundary equations no no let's not let's not get into the physics you do that in other course transferred course ask that question I'm just worried right now about problem discretization okay so now say this this will give additional equations so I will get t star at x is equal to 0 x is equal to 0 I will get all these points right along y so I will get t 1 j is equal to t star j going from 1 to ny plus 1 then the second boundary condition will be t nx plus 1 j is equal to t star j going from 1 to ny plus 1 okay then you discretize the third boundary condition okay and then this fourth boundary condition you discretize using forward difference okay sorry backward difference we discretize this using backward difference okay at all the grid points at all the grid points along x okay yeah mixed derivative you don't get second order derivative of the boundary so why don't we develop an approximation a difference equation can we develop a difference equation so you take first derivative in x then you take derivative in y so you will get large number of equations you'll get large number of equations which are coupled in general and they have to be this will also give rise to a sparse system you can see that any equation here how many equations how many variables appear only neighboring four variables appear if you take t ij ijth point there are only five variables appearing how many variables we have we have large number of variables nx plus 1 cross ny plus 1 but in one equation there are only five variables you can expect that if you try to solve this numerically you will get sparse system okay you get a sparse system and then of course we will be looking at solving sparse linear systems or sparse non-linear systems separately but the is the idea clear how do you do discretization okay so in this case the original problem is a partial differential equation discretize problem is a set of non-linear or linear algebraic equations which have to be solved simultaneously which have to be solved simultaneously so the transform problem is different the solution that you get from the transform problem is an approximate solution okay not the original solution so in the next class we'll see one more example one more way of discretizing this and then we'll start with some other methods of discretizing