 So, that was the, that is the shock, right, they pick two points on it, one was A and the other was B and they pick these points in such a fashion, right, that I was able to take a control volume, in fact I took a control volume in such a fashion that they pass through these points, it is deliberately chosen, okay, with the proposal that I am going to let the limit A go to B or B go to A, this point is not only identified as A but in our finite difference scheme, in our finite difference scheme, right, this is a time level Q plus 1, this is a time level Q and in our finite difference scheme, this is, this point is P-1.5, that point is P-1.5 and this midpoint is P, is that fine, so it turns out that midpoint is P, which is a critical thing that I need, okay. And what was the equation that govern, that the governing equation that we had, what was the equation that we had, you remember UP plus 1.5, I will put the Q in the superscript and I am not sure that is what I have been doing so far, minus UP minus 1.5 into XP plus 1.5 minus XP minus 1.5 and what was this equal to? That was equal to F of, there was an integral but okay, let me write the full thing, so it is TQ to TQ plus 1, F of X plus XP minus 1.5, that is not the way I did it last time, so I will remove that, P minus 1.5 minus F of P plus 1.5, DT, is that right? And what we said was that we are going to consider the situation where to the left of the shock, you have UL, the state is UL and to the right of the shock the state is UR. I am using, see I am introducing extra notation, you know I have got AB and I have half, half, you know because these are UL and UR are something that you will see in standard text and papers and so on okay, but it is also important that so to the left of the shock I have the state UL and to the right of the shock I have state UR, so the question is what is the value at Q plus 1, UP Q plus 1, it is UL, you understand, so this becomes UP Q plus 1, what did I do? You are not on top of things okay, fine okay, so this gives me UL minus UR XP plus half minus XP minus half equals, this flux is a function of, the flux is a function of U and therefore if I look at P minus half on that phase F of P minus half equals F of UL okay, so this gives me F of UL minus F of UR times, I will write it just as delta T here just so that I can squeeze it in here times delta T which of course turns out which gives us if I divide through the shock speed US is XP plus half minus XP minus half divided by delta T which is nothing but TQ plus 1 minus TQ and that equals F of UL minus F of UR divided by UL minus UR, is that fine okay, everyone right, so this is whatever delta T that we have and that is whatever delta X that we have and this we get this because you get it, it is a speed US if you take the limit A going to B right, limit A going to B but this still holds, limit A going to B or this this still holds, the shock is a discontinuity this still holds and we are able to divide by UL minus UR because the states are different because there is a discontinuity okay, is that fine right, are there any questions okay, so I am going to do one other thing that I promised earlier right, I said I will do the stability analysis in a slightly different way and I promised to do it for heat equation, so I will just do it for heat equation and then we will get on with right with the rest of the class. So if I apply FTCS forward time central space to heat equation, heat equation that is one-dimensional heat equation dou U dou T equals kappa dou squared U dou X squared, I am going to skip ahead to something that I had written earlier, we have already done this, UPQ plus 1 equals UPQ plus lambda times UP plus 1Q minus 2UPQ I guess I should stick with either subscripts or superscripts plus UP minus 1Q okay. So if I combine terms remember what was lambda just to remind you lambda is kappa delta T by delta X squared okay, so you get, I can rewrite this as UPQ plus 1 just combining terms is lambda UP minus 1Q plus 1 minus 2 lambda UPQ plus lambda UP plus 1Q okay right, I had already suggested that you take these as A, B, C and work it out right, I do not know whether you have had a chance to look at it but in this particular case, you know that if lambda is less than half, this is positive. So if we are given lambda is less than half right, if we are given lambda is less than half, this is a positive quantity and what does it tell us? We can use triangle inequality, you are familiar with triangle inequality, you have the sum of these three and you have that and triangle inequality basically tells us that mod of UPQ plus 1 is less than or equal to, all of these are positive quantities lambda mod UP minus 1Q plus 1 minus 2 lambda mod UPQ plus lambda mod UP plus 1Q okay, triangle inequality gives us that and then we can use for example the maximum value, you can use the infinity norm which I have not really introduced here but basically you can take it that if I say UP minus 1Q that what I mean is max of U, you understand what I am saying? Max of U over P, is that fine okay, maximum value. So again I rewrite my triangle inequality UPQ plus 1 is less than or equal to lambda U, I just write it as U plus 1 minus 2 lambda, this is at U, I have gotten rid of the whole point was to get rid of that lambda, these are all at time Q, that is what I need, I have eliminated the P. So each one of these, at time Q plus 1, each one of these satisfies this, the maximum also satisfies that. So I can replace this by mod U at Q plus 1 infinity is less than or equal to mod U, so the different ways to do, you do not have to take that exponential substituted that is one, that is the von Neumann stability analysis, the different ways by which you could do this okay, different ways by which you could do this and in this case I am talking about the function over the whole region, I am talking about it over the whole domain, I am not talking about it at a point okay, that is what I sort of promised you to do earlier, I just wanted to make sure that, so there are lot, there is the analysis of these schemes right, there is a lot of scope for you, there are lots of things that you can learn from the analysis of the schemes, it is outside the scope of this course, but there are lots of things that you can learn in the analysis of the scheme, is that fine okay, are there any questions? Okay, so we are now formally what I will do is, we are out of the part of the course that I call simple problems, we have looked at Laplace's equation, we have looked at wave equation, we have looked at heat equation right, in the last class I indicated to you Laplace's equation falls into the category of problems called elliptic problems right, wave equation falls into the category of problems called hyperbolic problems, and of course heat equation falls into the category of problems classified as parabolic problems okay, so these are the right now, you may have seen it already in your partial differential equations course, but we have looked at simple problems, typical prototypes of problems that belong to these classes okay, now what we are going to do is, we are going to switch, we are going to, though we have been talking about propagation and in a sense diffusion, those are the two phenomenon that we have been looking at propagation and diffusion, it is still not fluid flow okay, so what I will right now do is, I will quickly derive the governing equations, I will quickly derive the governing equations, but my objective in this course is to restrict myself to one dimensional flows right, so if you have had gas dynamics already, you are familiar with the solutions, there are standard algebraic solutions to these equations that you have cleared around with at least in simple cases right, in fact cases that are more complicated than what I will study, what I will look at right, so the idea is to put it in a familiar setting right, the idea is to put it in a familiar setting, for those of you who are not familiar with gas dynamics, the parts that are necessary hopefully I will cover or otherwise you will have to go look up whatever it is that right, that does not sound familiar, but first the governing equations okay, so to derive the governing equations, I will set up my coordinate system, this material is actually available should be available on my website, the derivation of these governing equations in greater detail, there of course I also use tensor calculus to do the derivation, here I will stick to the standard Cartesian coordinate system, so before I derive the general ideas of conservation right, generalize conservation principle, we need to find out something, so I will look at some terms okay, so the question that I have is what is the field property, are you familiar with the term field property, I am not asking for something that is mathematically precise right now, but what is the field property, it is a function of position and time, any field property is a function of position and time, can you, can we make it a little more, what do I need to give you, if I have given you specified a field property, what do I need to give you in order to say that I have given you a field property okay, I will say I have put it in a sort of a, what do you call it funny way, so the field property is basically a region right, you are saying function of position, so there is a region, you have to give me a region right and maybe even a period, but let us just look at a region, because we need not look at an evolution in time, you need a region, that is the field as we call it and you need a property that is defined on that region, is that right okay, so if I say that I have a corn field, then I have specified a region which is corn, I mean which is field and I specified a property that is defined on that field which is corn right or it is paddy field or whatever it is, so the idea would be that if I go look at that region, any point in that region and I look at any point in that region, I am supposed to find the property, is that right, I am supposed to find the property, but in reality you know that that does not happen, in reality you go to the corn field close your eyes and stick your hand on your likely to hit dot right, so there is a property, in that case it is discrete, we want to treat it, you want it as a function of x and t, you want to treat it as a continuum right, so we will convert it in some fashion to a continuum, we will come up with a density right, so you can say, so how does this work, so if I give you a region, say a country the size of India or something of that sort and you are, you want to find out how much rice are we going to grow this year, so you could take a satellite photograph of the region right, with the appropriate filters, you figure out that red coloured spot there, those red coloured spots that correspond to paddy fields where rice is growing, the intensity of the red may tell you what is the paddy density right, from a satellite photograph it may be of the order of paddy density per square kilometer, we know that if you go to the paddy field as I said that if you stick your hand and you are going to hit water and mud right, but from seeing from there, it is a continuous distribution of red right, the saturation may be changes, so you can look at what is the density, you can infer what is the density and typically we would break up the country into small small squares, multiply the local density in that square right, into the area of that square, add up all the squares and you have a good estimate as to how much rice you expected the end of the year, fine okay, so we have basically looked at an idea of quadrature, we have, we have introduced, we have thought of, so as I said I do not want to, if it were a fluid mechanics class I would spend more time on it, but here we will rush along, so we have introduced the idea of a continuum, so though it is discrete we are looking at, we want a continuum, we want to define a density right, we want to define a density okay, so if you have some property, you need, you need a region on which that property is defined, so I would basically say that let me take a, in my region of interest let me take a arbitrary control volume, I will take an arbitrary control volume, the volume that it occupies is sigma okay, and this is placed in some kind of a flow field, my region of interest, my actual interest of course is, my actual interest of course is that I have this, I have this flow field, so it has a surface area, it is bounded by a surface, whose surface area is s, volume is sigma, volume is sigma, surface area is s, area, surface area is s, is that fine, at any given point on the interior of this volume, okay, this volume is chosen in an arbitrary fashion, I could have chosen any volume, I take a small elemental volume at the point, I need a position vector, at what point, at the point x, at the point x, I take a small elemental volume d sigma okay, so if the property that we are talking about, we will just pick some arbitrary property, if the property that we are talking about is, well since I pick Greek symbols, I will stick with Greek, is chi, right, this is the property that I am talking about, as I said it could be con, it could be pad, it could be anything, right, so it is the property that we are talking about is chi, the corresponding density chi prime is basically what are the units that it has, chi prime has units, if I say the units of chi prime are the units of chi divided by L cubed, right, divided by whatever units you are using to find, because this is 3D, if it were 2D it would be L squared, I get an aerial density and so on, fine, everyone, okay, given this, so now we can answer a simple question, what is the amount d chi of the property chi that is there in the volume d sigma, that is chi prime d sigma, so that is like taking 1 square out of that satellite picture and multiplying by the area, right, now we know this is a standard game that you can play now, so now you integrate 0 to chi, the total amount which just gives me the amount of that property that I have in that volume and this is integral over sigma chi prime d sigma, so it is integrated over the whole volume, fine, is that okay, everyone, I just want to make sure that this idea of conservation is fine, we will apply it to conservation of mass, this is my plan for today, right and then we will just, I will quickly write out momentum and energy equation, as I said I do not want to spend you know a whole lot of time on this, okay, so at this point, so this is like an analogy that I like to give for, this is like a bank account, okay, this is like a bank account, chi prime is like, let us say we go to the bank, we go to the bank, you go to the teller, right and you are, there is currency, you have an account there, each account they actually have a box, let us just say they physically have a box, so you give them money and they put that cash in, so they are sitting there and they are adding up all the 1 rupee notes, all the 10 rupee notes, all the 100 rupee notes, all the 1000 rupee notes, you understand, they are adding up all the, all of it and they are saying this is what you have in your bank, this integral is the total amount of money that you have in your bank account, the control volume is the little box in which they kept all your money, fine, in our mind we will imagine that there is little, so as far as you are concerned the bank is this control volume, right and this control volume is the account and you have just added up all the money in your account, right but we would like to know what is the rate at which this money is changing, we would like to know whether my bank balance is growing or decreasing, right, okay, so what you do is you take the time derivative of that, so the amount of money in my bank balance d chi dt is d by dt of the integral over sigma chi prime d sigma, okay, is that fine? Now you know there are lots of rules regarding how money gets in and gets out, if you go on to your bank and you fill out forms, right, deposit money, there is a set rules by which things can come in and come out, so the set ways by which money can show up in your account, so you have to look at what are all the possible ways by which it can happen, so one is that somehow inside you have some way of creation, okay, in the case of a bank account that could be in the form of interest that the bank gives you, it could be a source term, right or there could be a sync term, you go and you say I want demand draft for 100 rupees and 120 rupees are disappears because they take 20 rupees as a service charge, right but as far as you are concerned you may have withdrawn only 100 rupees, so there may be source terms, sync terms, there may be charges that they have that they charge you and so on, so there are possible sources and syncs within creation of money and at the boundary you can have inflow and outflow, so you want to know what is the net inflow and outflow, okay, is there any other way by which there can be changes? Well as far as we are concerned with what we know, if it were currency, I can talk about interests and so on but if there is no currency all I have is sky, some unknown quantity then there can be sources or syncs inside and there can be flow in and out, okay. So in order to find out what is the flow in and out is, that is an outward unit normal and that outward unit normal is on a little elemental area, it determines a little elemental area DS on the surface because now I realize that I have to do some integration through the surface, okay. So the rate at which the amount of chi prime that I have in the volume is changing, what is it? If the velocity vector here is V, V dot n DS gives me the rate at which something is either coming in or going out in this case because n is outward normal going out to the volume, okay. What that thing is, is chi prime is being carried out of this volume over the whole area S and because it is going out it is negative, it is going to create a decrease because we are looking at how much is going out, right, we are looking at how much is going out. Therefore you understand, so you can go to the teller and you can stand in front of that person, right, the guy or the lady at the teller and as long as there is a barrier, there is a little hole through which you can put your money, you can do anything you want with the currency side word, it does not matter. If it does not go through she is not going to take it or he is not going to take it, there has to be a normal component, right. So I can watch, I can stand here watch students walking past the corridor and so on but if the student does not have a normal component when he comes to the door he is not going to enter my class, you need the normal component, you need the normal component, you understand what I am saying, the tangential component does not really help me, tangential component gives you nothing, right because you are just going past, you are not getting in or getting out. So the v dot n, you have to look at the normal, it is not enough to look at v, you have to look at v dot n, is that fine plus any other sources, things, any other production terms, production terms and this has to come from, this has to come from the physics of whatever chi is, okay. So we will quickly apply it, now that we have, this is the idea of conservation, are there any questions, okay. So this is a generalized conservation principle, I will take chi to be mass, right, chi prime is mass density which we encounter so often we just call it density, right, mass density and we have a special symbol for a row. So we substitute there, now that we have this, we just have to turn the crank, every time we do this, so you have rho d sigma is the amount of the mass in that d sigma, integrated over sigma gives you the total amount of mass in the control volume, d by dt gives you the rate of change, so there is a little thing that goes with it, you have to repeat it in your mind, sometimes it is worth it. So what is it, v dot n ds is the rate at which something is going out, rho is going out over the whole area and because it is going out, whatever I have in the control volume is going to decrease, so I need a negative sign, is that fine and there are no sources and sinks, so we are not looking at either relativistic effects or you know, we are not looking at any of these kinds of, no sources and sinks as far as we are concerned. That is what we have, is that fine, so this is the integral form, most general form that we have right now, of course I have taken a rigid sigma, you can derive an even more general form with the control volume itself deforming and so on, but we are not interested in that. We have this, what next, what is the next step that we do, from here I can take the time derivative and I want to get to a differential equation, we have been solving differential equation so far, this integral form is useful, you will see it in a different class, but in this class we have been looking at differential equation, so I want to get it into a form of a differential equation. So this gives me the integral over sigma dou rho dou t d sigma, I have taken the time derivative in all the necessary properties, continuity related properties, we have presumed that we have, okay, it becomes dou rho dou t because rho is a function of x, y, z, t whereas here you have integrated it out and therefore it is d rho d t, you understand d by d t, I have taken it in, it becomes a partial derivative, okay. What can I do to this, I should convert it to volume integral using theorem of Gauss, so that becomes minus integral over sigma divergence of rho v d sigma, is that fine, I use theorem of Gauss and of course combine these two and you get integral over sigma dou rho dou t plus divergence of rho v d sigma equals 0, right and now comes the critical assumption we made earlier, we chose the control volume arbitrarily, right. So this is valid for any control volume sigma, so I want an integrand here, whatever the control volume you pick, I have to guarantee that the integral is 0, that is possible only if the integrand itself is 0, so this tells us, this gives us conservation of mass, I will write it there, I will write it here, dou rho dou t plus dou divergence rho v equals 0 and I am going to now constrain myself to one dimension, I am going to constrain myself to one dimensional flow and say dou rho dou t, I will have only the x coordinate plus dou rho u dou x equals 0, is that fine. This is my one dimensional conservation of mass, I will just write Cm there, conservation of mass, you will hear a bit called mass balance, right, conservation of mass, where I write your names that you will have, okay, are there any questions, so you have seen all of this before, let us do conservation of linear momentum, conservation of linear momentum, the same equation, for the same equation now I am going to substitute, what is momentum, it is m times v, right, so momentum density will be rho v, again just blindly substituted into that equation, rho v d sigma, right, is what we have in that little elemental volume, whole volume, time rate of change is v dot n ds, rho v is being carried at that rate, integral over the whole surface, negative sign because it is an outward norm, plus. Now we come to the point that I was talking about, you have to consider the physics of what you are looking at, so what are the ways by which you can change the momentum in that control volume, you have to apply forces, right, forces come in various flavors, the forces that we are used to come in various flavors, you have the action across the distance where you can reach into the volume, right, you are talking in terms of something like electromagnetic forces or gravitational force, you can reach into the volume and apply a force, right, that is basically one way, and the other that you have is surface forces, right, forces that are applied presumably at our length scale, that is what I said, I do not want to get into the idea of continuum and so on, but at the length scales that we are talking about, at the continuum level, surface forces, right, the fact that I am pushing happens, the mechanism actually happens to the contact, okay, I cannot keep my hand here and expect to apply on this bone a compressive load, but if I come in contact, yes, then I can apply that force, am I making sense, that is a surface force. So the surface forces or traction forces, these forces are there, they are called traction forces, when we walk around, when I am walking around between my, the sole, right, of my sandals and the podium, there is a force, a traction force, so if I am standing erect, it is likely that the traction force is perpendicular to the surface, but when I start to walk or I am leaning forward, the traction force is actually at an angle, okay, so we have to take all of these things into account, so at this point there could be a traction force, the figure is getting messy, but it does not matter, there could be a traction force T, through the volume at any given point, there could be a body force F, are there any line forces, we have finished volume, we have finished surface, anything that corresponds to a line force, not that you are familiar with, I do not know, maybe surface tension or something of that sort, but we are going to ignore those, we are going to ignore, even at this point we are going to ignore those, okay, so Fd sigma, that means that I have considered F to be a force per unit volume, Fd sigma is the volume on that element, integrated over sigma gives me the net body force, volume force or body force, okay, on that volume. Similarly Tds is the net force on that element and that force can be a, there can be a shear component, it need not be normal and the integral over S gives me the total force that is acting on the surface, is that fine, okay, for the sake of this course I am going to ignore body forces, no body forces, we are going to ignore them, right, in fluid mechanics anyway you have already studied when you need to take gravity into account and when you can ignore it. And again for the sake of this course I am going to ignore viscous effects, which means that I am not going to go through that full game of Cauchy theorem and coming out with a stress tensor, defining a stress tensor and so on, right, instead of which you have already done, instead of which I am going to just write Ts, what is the direction of T in this notation, I can hear something, what is T, if it is only, if you have only pressure if there is no viscosity, minus p times n, is that right, minus p times n, okay, right, okay. So we are back here, maybe I will leave that just for your reference. So what happens here, so I get the integral over sigma dou rho v dou T d sigma, I have taken the time derivative inside, equals integral over S or over sigma if you want, divergence of rho v v d sigma, minus sign, plus integral over sigma, gradient of p d sigma, right, p and dS, again you can go through the same process and convert it to a volumetric, is that fine, rho minus sign, thank you, thank you, okay. So this tells me that dou rho v dou T plus divergence rho v v plus grad p, grad p equals 0, I have done the same thing, I have merged them all together, the integrand has to be 0 because it is an arbitrary control volume. In the one dimensional case which I will write here is dou rho u dou T plus, I will combine the two terms, dou by dou x of rho u squared plus p equals 0, and some of this from your gas dynamics should start looking familiar, is that fine, any questions, we do not, we go to the last one which is total energy, look at energy equation, so we have looked at conservation of mass, we looked at linear momentum, right, conservation of linear momentum, now we look at conservation of energy, right. So we go through the same process, if I have total energy, I will write just the density here, chi prime, similar fashion, it depends on how you want to write it, I will write it as rho e T because I have rho v and rho, right, I mean it makes me, I like the pattern, right, so you could have just written it as e T and gotten away with it, we need to do a little worrying here, what is e T, I have just introduced e T, e T is e plus one half rho, say v dot v, okay, that does not help us, so what is little e then, little e depending on what we are doing is c v T, right, and every time we do something we get, yes so we need closure, how do we get closure, typically you take a constitutive model or an equation of state, so we need an equation of state here which is p equals rho R T and now we have something, we have T in terms of the other two quantities we already know, fine, so we have settled that, right, otherwise this could go on forever, right, so we have settled that, is that fine, okay. So what do we have, let us write the thing again, rho e T d sigma integrated over rho sigma, the time rate of change of that is v dot n ds is carrying out rho e T integrated over s with a negative sign, what are all the ways by which we can change the energy in that system, in that control volume, sorry, you have two possible processes that we know, right, there are only two possible processes, heat and work, right, heat and work, so you could either have work of course can be done by these forces, we already know we have volume forces, so we have enumerated the forces, we can deal with that, we keep it simple because we have already ignored certain forces, right, and heat can come in, what are the mechanisms of heat, can have radiation, radiation conduction, that is energy carried by convection, right, so that is taken care of, okay, so that is a different game, right, now we are talking, whether it is conduction or radiation, there can be source terms, you can have exothermic reactions of some kind, endothermic reactions of some kind, there can be source term, there can be other sources, so we will ignore chemical reaction, you understand, we will ignore chemical nuclear, all those reactions, all those things that are, no sources and sinks, no sources and sinks, we will ignore radiation, right, fine, while we are at it because we already ignored viscosity, we will ignore conduction also, and we have done heat equation, what mode you want, no, no, seriously, we will ignore conduction, so what is left, work done by body forces, and we have only one, no body forces, body forces is gone, anything else, we have pressure, right, work done by pressure, so you can figure out how that is going to, plus or minus, you can figure out how that is going to, I will let you figure that out, I am going to come back here, I am going to jump now, right, I will let you go because I have thrown away, I have been a nice guy and thrown away a lot of things, so you can simplify that and figure it out, I am going to just write the final form here, dou rho E t dou t plus dou by dou x of rho E t plus p times u equals 0, that should give you a clue, you see the work done by pressure, that is power, the rate at which work is being done, p times u, these are rates because it is the rate at which the total energy in that volume is changing, okay, right, so you can work that out, it is just p times u times n whatever it is, it is not, right, you should not be, right, okay, you want me to work it out, you can work it out, okay, that is fine. So we will know, so this is linear momentum, this is energy, okay, we have the three equations and then we have a whole bunch of auxiliary equations, right, we have the whole bunch of auxiliary equations, E t is E plus 1 half rho u squared, this is for the one-dimensional case, E equals C v times t, C v times t, what is C v, specific heat at constant volume, okay and then t equals rho r t with which we have closure, okay, right, we have spent so many weeks looking at linear wave equation, so you should be legitimately upset that this does not look like linear wave equation, right, so we will try to make it look like linear wave equation, we will take a little effort in trying to make it look like linear wave equation. The one first obvious thing to do is combine them into a vector equation, right, at least the chalk dust will look very similar, we hope, okay, so let me come here, we will combine all of that into one equation, we do not need any of the stuff now, we are done with it all, so the one-dimensional gas dynamical equations can be written as dou q dou t, so dou E dou x equals 0 where q is a vector, I should really indicate these as vectors or matrices but I am not going to, right, even the 0 is actually 3 zeros, right, I am going to drop them, it is very clear what it is that I mean, so this will be rho u rho E t and E is rho u rho u squared plus p rho E t plus p times u, okay, close but not good enough, close but not good enough and this looks like a generalized one-dimensional wave equation, right, it looks something like that, right, I mean this resembles what we wrote as dou u dou t, this u is not the same as that u plus dou f dou x equals 0 and there was a trick that we played here, we could do dou f dou u, right, in the last class I have written this very light, okay, I will use capital U instead of little u because I have already, so we looked at this last class, we looked at this, we are looking at this equation, a generalized, right, one-dimensional wave equation kind of a thing and what did we do? We could write this, we could take the Jacobian, flux Jacobian, we could write it as dou f dou u, we could write this as dou u dou t plus dou f dou u dou u dou x using chain rule, okay and this was the propagation speed, right, at least it looks like what we have it, we are trying to get it to that form, then we can, so I write this as, I write our one-dimensional equation dou q dou t plus dou e dou q dou q dou x equals 0 and I will call dou e dou q, I will give it a name A and like we said this is called the flux Jacobian, very often people will refer to it as the flux Jacobian, right, e is the flux term, you look at this, e comes from the, e is the flux term, right, e is the flux term. So this equation then is written as dou q dou t plus A dou q dou x equals 0, that is good news and it is bad news, it is good news because it looks like dou u dou t plus A dou u dou x equals 0, so I have not wasted a lot of your time working on that equation, it is bad news because this is a system of equations, right, so we will have to do something, right, so before I close for the day I want to point out, this is again in the standard form, right, that mathematicians call the divergence free form, right, it is first derivative, some of these first derivatives dou q dou t plus dou e dou x equals 0, so it is called the conservative form. From gas dynamics also, from gas dynamics, we have our own little reasons for why we call it a conservative form, what happens to these quantities across a shock, these quantities are conserved across a shock, right. So we look at, we have our own little game that we play, we say it is a conservative form because right rho u is rho u rho u square and in fact the derivatives will work because they are actually continuous, they do not jump, these quantities do not jump, okay, fine. So we have our own reasons for calling them, calling it the conservative form, I say this because you will get, you can get confused, sometimes you may get asked why is it called a conservative form and depending on who is doing the asking, the answer may have to appropriately change, is that fine. This of course therefore is called the non-conservative form, it is written in terms of what we call the conservative variables, these variables because this equation when written in this one, so these are called very often referred to as the conservative variables, right, it is written in terms of conservative variables q, but the form is called a non-conservative form, is that fine, non-conservative form and q is called conservative variables, simply because you can write the equation in terms of q in this conservative form, is that fine, okay. So in the next class we will make some effort to see if we cannot simplify this, we will find out what is the nature of A and how to find A and I can tell you now it is not going to be satisfactory, so then we will make an effort to try to make the equations look more and more like our linear one-dimensional wave equation, is that fine, so that everything that we have done there we will see what carries over from the one-dimensional equation, what have we done that carries over, is that okay, thank you.