 Welcome, from now onwards we will start analyzing the equations first order partial differential equations for their solutions. Today's lecture is devoted to geometry of Quasi-Linear equation. So in this first we set up the basic hypothesis and notations which will be assumed throughout the analysis of the Quasi-Linear equations. And then we discuss geometry of Quasi-Linear equations the geometrical objects which are associated to Quasi-Linear equations. So now prerequisites is that from now onwards to follow this course definitely for the first order PDs what is very much required is an exposure to ordinary differential equation theory mainly existence uniqueness theorems for initial value problems namely Picard's existence and uniqueness theorem that is good enough. On the other hand one more we require that is dependence of the solutions on the initial data that theorem will also be required but not for this lecture but in future lectures please refer to some good books on ordinary differential equations where you can find this material. Then from multivariable calculers the most important theorem I would say is chain rule and one should be very much expert with it and change of variables it looks quite easy while computing certain integrals which you would have done in plus 2 but actually it is not that easy in the sense you have to be very very careful. If you are not careful it can lead to lot of confusion and then how a partial differential equation changes under a change of variables to a new equation how to derive that that is also very important. So quasi-linear equations hypothesis and notations omega 3 is an open and connected subset of R 3 as I told you omega d will be an open subset in Rd but for the discussion of quasi-linear equations we have to take d equal to 3 because we are going to consider quasi-linear equations in 2 independent variables. So omega 3 is an open and connected subset of R 3 why do we need connected it is because we are going to take some curves in omega 3 curve is a quite connected quantity right I mean there is no gaps in that. Therefore that curve should be in omega 3 otherwise also does not matter that is why it is not that important at all here it is enough it is an open set there is nothing wrong in assuming it is also connected there is no loss of generality. Then the projection of omega 3 to the x y plane is denoted by omega 2 that is consistent omega 2 is a subset of R 2. So how do we define that those ordered pairs x y in R 2 such that x y z belongs to omega 3 for some z in R. The functions ABC which will be playing a role in defining quasi-linear first order PDE they are assumed to be C 1 functions on omega 3 that is continuously differentiable functions on omega 3 which is equivalent to assuming that all partial derivatives of ABC with respect to the 3 variables exist and continuous on omega 3 and we need this non-degeneracy assumption which says that at any point x y z in omega 3 at least one of the two functions a and b should be non-zero a way to express that is a square plus b square is not equal to 0 at every point x y z. Another way to write the same thing would be this a of x y z comma b of x y z is not equal to 0 0. This is another way of writing the same thing. Why do we need this assumption? Because imagine both AB are 0 then this equation which we are going to consider the quasi-linear equation there is no differential equation because at that point maybe a is 0 b is 0. So, it is just C x y equal to 0 some kind of degeneracy. So, we do not want that that is the reason why we assume a and b simultaneously cannot be 0 at any point in omega 3. So, quasi-linear PDE we often use this notation q l regularly we use q l by q l we mean this a u x plus b u y equal to c abc or c 1 functions in omega 3 and a square plus b square is not equal to 0 at every point in omega 3. So, by just q l it stands for all these properties. Now, let us discuss the geometry of quasi-linear equations that means this equation says something about the geometry of what? Of a solution surface. If you have a solution of this equation let us say u of x y defined for x y in some domain D then z equal to u x y will be a surface and this equation says what kind of restrictions are imposed on the surface. Geometrically what this PDE means for a solution surface that is what we are going to discuss. So, integral surface the definition of an integral surface let d be an open subset of omega 2 what is omega 2 it is a projection of omega 3 to x y plane and let u be a solution of the equation q l u is defined on this domain D is a solution of the equation. What does it mean the equation a of x comma y comma u of x y u x of x y plus b x y b x y u x y u y of x y equal to c of x y u of x y is satisfied at every x y in D that is a meaning of solution. So, the integral surface associated to u is defined as set of all triples x y z in x y belongs to D right and z belongs to R. So, x y z such that belonging to D cross R so, it is a subset of R 3 such that the third coordinate z is u of x y it is also denoted often in short notation as s equal s colon z equal to u of x y in practice we say s colon z equal to u of x y is an integral surface caution very important caution remember that the function always comes with its domain function always comes with domain. So, if you say that yes z equal to u x y is an integral surface it means that you have a function u defined on some domain all this needs to be mentioned or you know it therefore, you are not mentioning. But if you change the domain to some other domain it can be a still a solution but it will be a different integral surface we will come to these aspects in future lectures. The integral surface associated to a solution of the Cauchy linear equation is actually the graph of the function we are saying set of all x y z such that z equal to u x y just means the third coordinate is a function of first two coordinates which means it is a graph of a function u and any point on the integral surface looks like x y u x y for some x y in D. Therefore, this map is 1 1 x y going to x y u x y is 1 1 graph is always 1 1 the moment x y is different from x dash y dash because the first two coordinates x y and x dash y dash are different the triples x y u x y will be different from x dash y dash u of x dash y dash they are different if x y is different from x dash y dash because this itself is different that is why the function is 1 2 1 now projection of the integral surface to x y plane equals D. Now a fact from differential geometry let S be an integral surface where u is a differentiable function okay u is a solution right therefore it must be differentiable function actually in this fact we need not have integral surface written it is true for any surface that is why wrote this let this be a surface and use a differentiable function and why we write integral surface here is because we are going to discuss we are worried about only integral surfaces in this lecture. In fact in this PDE we want to find solution therefore we are interested in surfaces described by solution of the equation. So, take a point of P naught on the surface okay then normal to the surface S normal to the surface at the point P naught has this direction u x u y minus 1 okay u x u y minus 1 there is a more general result I will write only roughly okay phi of x y z equal to 0 if this is a surface then what you have is grad phi okay what is that namely phi x phi y phi z that will represent a normal of course this should be nonzero. So, all that conditions will be there and in our case what is phi phi of x y z is equal to let us write in this form u of x y minus z and its gradient is nothing but u x u y and minus 1. So, that is true at every point so this is a more general fact that this is normal. Therefore, quasi linear equation is nothing but the dot product of this u x u y minus 1 with a b c equal to 0. What you get is a u x plus b u y equal to c that is the meaning of saying it satisfies the PDE at the point x 0 y 0. In other words the a b c which defined our quasi linear PDE is a direction in the tangent space to the surface s at the point p naught. Always you have to remember this phrasing tangent space or tangent plane if you are in plane to something at some point okay. So, this is a tangent direction we called hold of a tangential direction in a what would have be a search for it integral surface we know that I do not know that surface in the sense that I do not know solution but I know a vector which is in the tangent space at a point on such a possible surface. This idea will be useful in finding solution later on. So, by definition of tangent space there is a curve on the surface with what property. So, imagine this is a surface okay this is a point p 0 then through this you have a curve with gamma passing through that point and at this point the tangent vector is a of p 0 b of p 0 c of p 0 that is the meaning of somebody being a direction in the tangent space. So, tangent to the curve at the point p 0 is in the direction of a p naught b p naught c p naught. Now, what do you mean by that there is a curve with this property this is the meaning there is a delta positive and a curve gamma described by a function r from some small interval minus delta delta what all you need is 0 should be there in the interval such that gamma lies on the surface this curve is on the surface and it passes through the point p 0 and the tangential direction is this this is a tangential direction that coincides with a b c at that point p 0. These geometrical considerations motivate definitions of characteristic direction, characteristic vector field and characteristic curve. For a clear understanding of these geometrical objects you may consult books on differential geometry written by Docarmo or another book by Millman and Parker where these are very clearly explained. Now, we are going to define characteristic direction and vector field the direction of the vector a b c is called the characteristic direction at the point x y z in omega 3. Now, the association to every point in omega 3 take a point in omega 3 associate at that point an infinitesimal line element having this direction a b c this association is called a characteristic vector field. Basically vector field means you are associating a lot of you are associating a vector to each point. So, it would look like I mean we cannot write things in 3 dimensions but let us see this is imagine this is your omega I will not write what is maybe you can write 3 any point you just put some lines like that. These are the directions of a b c at that point. So, of course it varies from point to point and so on. So, this is called characteristic vector field associated to the quasi linear equation q l if you change the equation a b c will change therefore this association will also change. Now, characteristic curve a curve gamma lying in omega 3 is called a characteristic curve for the quasi linear equation q l if at each point on the curve gamma the tangential direction to the curve is same as the characteristic direction both directions coincide that is called a characteristic curve. Now, if you project the characteristic curve gamma which is in omega 3 to x y plane that will be called base characteristic curve it is called a base characteristic curve. So, base characteristic curve lies in omega 2 which is the projection of omega 3 to x y plane characteristic curve lies in omega 3. Now, we have a characteristic system of ODE's the characteristic system of ODE for the equation q l is the system of ordinary differential equations dx by dt equal to a dy by dt equal to b dz by dt equal to c. So, it is an autonomous system of ordinary differential equations with the right hand side a b c coming from the equation q l. Now, let us look at an example for this PDE u x equal to 0 very simple PDE let us say it is PDE in two variables x and y the characteristic system of ODE is dx by dt equal to a in this example a is 1 b is 0 and c is 0 therefore dy by dt is 0 dz by dt equal to 0 we can integrate them x of t equal to t plus constant y of t equal to another constant z of t equal to another constant where c 1 c 2 c 3 are real numbers. Now, what is it geometrically what we have to look at is t plus c 1 this three tuple t plus c 1 comma c 2 comma c 3 as t varies in r t belongs to r this is what we have to look at of course to write down this fix c 1 c 2 c 3 fix c 1 c 2 c 3 and look at this okay this is a line right second coordinate third coordinate are fixed first coordinate will give you entire r so this is precisely what you get is r cross singleton as c 1 cross sorry c 2 cross c 3 this is precisely that second coordinate is fixed third coordinate is fixed first coordinate can be any real number so that is what we get. So, characteristic curves are the family of straight lines which are given by the intersection of the plane y equal to c 2 and z equal to c 3 which are nothing but the translates by points of the y z plane of x axis. So, this is a picture specific to some particular quasi linear equation u x equal to 0 equation is u x equal to 0 and u x y equal to sin y is a solution given you can verify it does not depend on y u x equal to 0 means it is constant right with respect to x. So, it is a function of y only I have taken that to be sin y now this is a graph the sin is going like that right the curve you can see the sin curve like that okay. Now in this red axis red is always x axis green is y axis blue is z axis. So, this is x axis green is y axis blue is z axis now characteristic curves are in black which are straight lines we know that straight lines and characteristic vector field is in red color which is this this okay because characteristic vector field is 1 0 0 at every point. So, it is in the direction of E 1 every point and normal to the solution surface is in magenta. So, it keeps changing the normal from point to point. So, blue surface this just an observation and this is in fact also a general situation we will show the theorem later on blue surface okay that is what integral surface corresponding to the solution u x y equal to sin y. So, z equal to sin y is this okay that is union of these black lines. So, you can think one line this line keeps on moving and you get the surface okay remark on characteristic curves for the quasi-linear equation characteristic curve lies in omega 3 and base characteristics lies in omega 2 this is very important or many times this word is blurred. So, people refer to both of them as characteristics. So, you have to be very careful therefore to clear this problem or confusion we use the word base characteristic curve. Now if a, b, c are c 1 functions then a unique characteristic passes through every point of omega 3 what is the characteristic curve? It is the solution of x dx by dt equal to a dy by dt equal to b dz by dt equal to c its image. This is guaranteed by Cauchy-Picard-Lipschitz existence and uniqueness theorem in ODE's. Therefore, two distinct characteristics do not intersect by otherwise uniqueness will be violated at the intersection unless one of them lies completely on the other unless one is on the other. In other words, we may ask what do you mean by one lies on the other. So, it may be that you have a solution which is that long if you look at a small piece of that that is all solution. So, this is the only way uniqueness is violated okay. So, therefore we do not consider such things as different solutions. So, therefore two distinct characteristics do not intersect unless one of them lies completely on the other. Base characteristic curves corresponding to two non-intersecting characteristic curves may intersect. This can happen. Characteristic curves do not but their projections can intersect. Intersecting base characteristic curves prevent Cauchy problems from having global solutions. We will discuss them further but this is a fundamental obstacle. Now, this is an exercise for you to think over. Base characteristics, they also arise as solutions to ODE's. If base characteristic intersect does it contradict Cauchy-Lipschitz-Picard theorem? Does it contradict the theorem? Of course, things are working well. It should not contradict but then this confusion is there right because there are also solutions to initial value problems of ODE's. Okay. We will clear that later. So, when the Cauchy-Linear equation reduces to a semi-linear equation, what happens? a, b are not functions of the third variable anymore. They depend only on the x and y variables. Then neither distinct characteristic curves intersect that we know even for Cauchy-Linear equations nor the corresponding base characteristic curves intersect because what is the base characteristic curve? It is a projection of the characteristic curve to omega 2. But solutions of which equations? Because it is semi-linear, the equations are dx by dt equal to axy dy by dt equal to bxy. Okay. And a, b are C1 functions still. So, by Cauchy-Lipschitz-Picard theorem, there is exactly one base characteristic passing through every point of omega 2. That is why. Semi-linear base characteristics do not intersect because the equations governing the base characteristic curves, the right hand side is a and b and they are functions of xy only. If they are also functions of xyz, you cannot claim uniqueness because you cannot solve. Problem does not make sense. Okay. Now, a surface in R3, okay, how does it look? It looks like a plane but slightly deformed, right? That is a feeling. Therefore, it looks like there are 2 degrees of freedom to move on a surface. If you are a surface like this, you can move like this, right? 2 degrees of freedom. It is not like one single straight line, right? Like a road on which you cannot overtake. Okay. On the other hand, we also know the following. The family of characteristic curves are determined using solutions of initial value problem, which by design have 3 degrees of freedom because a system of 3 equations, being solutions to a system of 3 ODE's. Now, some question, do you think both statements are correct? Answer is yes. They are both are correct. Are they contradictory? Because you are saying on one hand, there are 3 degrees of freedom for these curves, which are lying on the surface. Surface itself is 2 degrees of freedom. Are they contradictory? Answer is no. Clarification to this is the following. Solutions to the characteristic system of ODE, they form a 3-parameter family. Why? Because they are parameterized by the points of omega 3. So, take a point in omega 3, x0, y0, z0 and then for any given x0, y0, z0 in omega 3, there is a solution to the initial value problem, which we call Cara ODE. This stands for dx by dt equal to a, dy by dt equal to b and dz by dt equal to c with these initial conditions. At t equal to 0, x0, y0, z0 is x0, y0, z0. Now, observe the difference between a function and its image. Image is often called trace or trajectory, just image. Trace is not a graph. On trace, no information of j is retained. We will look at an example. Look at this t belongs to R going to cos t sin t. It is a function. Trace is a circle. That means, set of all cos t sin t as t varies. This is a circle, which is in R2. Graph will be a subset of R3. Graph will look like t, cos t, sin t. That is not a circle. So, this is the difference between solutions of the system Cara ODE and the characteristic curves, which are images of this or trace or trajectories of the solutions described by solutions of the Cara ODE. For ease of presentation, let us assume that A is never 0 in omega 3. Once it is never 0, you know what I am planning to do, I am planning to divide. Now, I will eliminate the parameter t from this Cara ODE system. Okay, t will go. Then I will get dy by dx equal to b by a and dz by dy equal to c by, dz by dx, I think. We will see. We conclude that that is a satisfy. Yes, dy by dx and dz by dx. I eliminate t. It means that I am assuming something about it. Let us not discuss that right now. So, note that solutions of one forms a two-parameter family system of two equations. So, two-parameter family, justification exactly same that we gave for system of characteristic ODE, system of three equations, three-parameter family. Okay, similar explanation gives you that it is a two-parameter family. Now, you see here the t has vanished in this right. There is no t in this. Okay, re-parameterization of a characteristic curve. What is a re-parameterization? Let gamma be a curve that is gamma is the image of a function r from some interval j to omega 3 where j is an interval in r. This is a curve given. Now, we are going to define what do we mean by a re-parameterization of this. It is a function phi defined on some interval j tilde to j. Okay, j tilde to j where j tilde is an interval. That is it. With what property? It is a diffeomorphism. That means the function is one to one and onto. So, that is function is bijective and therefore the function and inverse makes sense. Both of them are differentiable. Now, image of this function r circle phi which is now defined on j tilde to r 3 coincides with gamma. So, the image of r circle phi and r are the same. Both are gamma. So, that is a re-parameterization. Now, if a curve is described by two parameterizations then is it true that one appears as a re-parameterization of the other. In other words, you are given r from j to omega 3 and r tilde from j tilde to omega 3 such that images are same. Now, the question is is there a function phi from j tilde to j which is a diffeomorphism? Answer is almost true, but you have to assume some more properties about the parameterization being regular. So, this you can once again look up books of Docarmo or Millman Parker on differential geometry. But our question is not about this. We are not asking this question. Now, we are interested in re-parameterization of a characteristic curve. So, a curve can be parameterized in infinite number of ways simply because the function that we have here phi tilde, the function phi from j tilde to j, you can take any interval j tilde and you can always define any interval means the open interval j tilde then you can always find a diffeomorphism between j tilde and j. So, therefore, parameterizations are infinitely many. So, curve is a geometric object the image is a curve that is a geometric object while parameterization is a function whose trace is the curve. So, at any point on the curve the tangential direction does not change with parameterization should not change because tangent is a geometric object associated with curve. So, characteristic curves fit the characteristic vector field that means you have a characteristic vector field, characteristic curves are precisely those curves at each of whose points the tangent is the characteristic direction. This is not coming under re-parameterization but we wrote this because we will see this. We will use this. So, the following lemma is a restatement of the geometric fact discussed on the previous slide in the context of characteristic curves. Any re-parameterization of a characteristic curve is also a characteristic curve. Remember it is not about a curve it is a characteristic curve it means the tangent should be the characteristic direction in this context is this statement made here the last one. You know that the characteristic vector field is the same and tangents are geometric objects characteristic curves are somebody who fits this therefore the next theorem is expected this lemma and therefore one advantage of this is you take J tilde to be R to that particular J you can always define it if you have morphism therefore you can always re-parameterize by R. It is very useful we will see that when we prove existence theorems for the Cauchy problem of Quasinia equations and as well as general non-linear equations. So, proof of the lemma let gamma be a characteristic curve that means first of all it is the image of a function. Let us write because it is sitting in omega 3 there are 3 components. Let us write RT equal to XTYT ZT and XTYT ZT are solutions to the characteristic system of ODE. Let phi be a re-parameterization of gamma given to us phi is given J tilde is given such that this phi is a diffeomorphism this is a meaning of diffeomorphism. Now R tilde is R circle phi and R tilde let us use a different running parameter in J tilde for J we use T for J tilde let us use tau R tilde of tau is X tilde of tau comma Y tilde of tau comma Z tilde of tau. We want to show this is also a characteristic curve it means we have to find the tangent to this curve and show that it is a characteristic has the characteristic direction. So, X tilde of tau by definition is this R tilde is R circle phi. So, X tilde of tau equal to X of phi of tau Y tilde of tau equal to Y of phi of tau Z tilde of tau equal to Z of phi of tau. So, differentiating each of the equations in 2 with respect to tau gives us the following equations in 3 which is actually chain rule. Now XTYT ZT solve the curve ODE therefore we know that dx by dt dy by dt dz by dt are ABC respectively at the point X of phi of tau Y of phi of tau Z of phi of tau. So, this is the equation we get. Now in view of the relations 2 what are those in relation to it is the definition X tilde Y tilde Z tilde gives us this equations. Since phi is a diffeomorphism we have d phi by d tau is nonzero at every point. So, that means this quantity which is multiplying ABC is a nonzero quantity therefore the direction of this right hand side is same as the direction of this ABC because it is nonzero if it is zero we cannot say this is nonzero because of the diffeomorphism. So, proof of the lemma is complete on observing that there is a diffeomorphism between any open interval and R. Therefore, we can re-parameterize a characteristic curve by R. Importance of this lemma is it gives us a consistent usage of the terminology. A characteristic curve for QL should be an intrinsic property of the PDE. It should not depend on the parameterization used. Remember the definition you was using parameterization. This is confirmed by this lemma and parameter T is artificial. So, maximal interval of existence for any solutions need not be R even for autonomous systems. For example, look at dy by dt equal to y square and y of 0 equal to any number you take y0. Solution will not be defined on R even though the ODE makes sense for all T in R. Therefore, there is a concept of maximal interval of existence it would not be R. Even if it is an autonomous system as we just saw in this example. But thanks to this lemma we may assume that solutions are defined on whole of R. Equation is different. Our interest is only in the trace, in the trace of the solution, in the image of the solution that is same. So, it is not surprising that the lemma holds since the parameter T itself was introduced by us. Equation gave us ABC. We said yes, there is a curve whose tangent is this. Therefore, we said curve means R of T, X of T, Y of T, Z of T its derivative equal to ABC. We did that, we introduced the T. Therefore, it is already artificial, it is confirmed. In fact, this usage is there in Fritz John's book. He also says this parameter is artificial which is this lemma confirms. Now characteristic curves fit the characteristic vector field. For autonomous system of ODE is the RHS may be thought of as a tangent vector field. Solving this is a means to find curves which fit the vector field. Thus, as long as we are interested in the trajectories of solutions, we may assume that trajectories are parameterized by R. This observation reconfirms that the parameter T describing a characteristic curve is artificial. The above interpretation that is parameter T is artificial does not hold for trajectories of the non-autonomous systems of ODE. The proof of the lemma suggests where the proof will fail and hence that is the reason why this will not hold. Let us summarize what we did now today. We introduced the notion that will be used throughout the discussion on quiesce linear equation, notations. We have introduced the notations and then we discussed the geometry of quiesce linear equations which led to the introduction of concepts of characteristic vector field and characteristic curves. We analyzed the degrees of freedom for solutions of a caravodi and characteristic curves. We proved that a characteristic curve may be parameterized by R. So, using this geometry of quiesce linear equations, we try to solve Cauchy problem for quiesce linear equations in future lectures. Thank you.