J i is what we will call a flux vector. Right, in. Coordinate notation. 'Kay, and for coordinate notation i equals 1, 2, 3. For our purposes, this is just a way to specify explicitly, the components of a vector in three dimensions. Okay if, if we were doing a class in continuum physics. If we, we were doing lectures in continuum physics, this would be set up much more carefully than that, right? But for our purposes, it just denotes the components of a vector. Right? And when we explicitly refer to the components of vector that sort of notation is called coordinate notation. All right, so then what they're saying is that j if we wanted to write the vector and so called direct notation, okay? J is just the collection J1, J2, J3, okay? And this is what we call Direct notation. If properly the left-hand side of this last equation J equals J1, J2, J3 is direct notation for the same vector. Okay, so we will often switch back and forth between the, between these two type, between these two notations, though for what we need to develop we are going to use coordinate notation a little more than direct notation. Okay? But they are essentially just different ways of describing the same thing since J is a vector there, there is also a way of describing the J as a vector. To simply saying that J, being a three dimensional vector, j belongs to R3. Okay. Just this notation, all that means is that j is a 3D vector, okay? You know that exactly the same thing can be said for x as well, right? So, likewise, we have x equals x1, x2, x3, arranged together as a vector. Right, and x belongs to R3, and this thing has more meaning because not only is is x a vector, a three dimensional vector, but x also is a point in three dimensional space right, so this has even extra, even more meaning here really, more physical meaning. Okay. That's what we have. Now let me see. What else do we need to state here? Okay. In order to say more about this it's actually useful to go to a particular physical problem that this PDE could represent, okay? So in order to say more about this let's consider heat conduction. Heat conduction at steady state. Right? In 3D right. Let's suppose this is the problem we are talking about. One thing you recall of course that steady state simply means there is no time dependant right? The time dependent has been dropped from this description of heat conduction. Okay, so this is the keys what is u? What does u represent in the, in the problem of heat conduction? Can you recall from you recall from your study of heat conduction previously? Right, it's the temperature. Okay, J then is right the heat flux vector. Right, which is essentially the amount of heat. Crossing perpendicular to a unit area. Per unit time. Okay. That is J, right? Now, when we have the constitutive relation. Ji equals minus kappa ij u comma j, okay? This represents for us now, it essentially tells us that the the heat flux vector is driven by the temperature gradient. Right, so u comma j u comma j you recall cause this just partial of u with respect to xj. Right, so that's the temperature gradient. This constitutive law goes by the name of the Fourier law of heat conduction. Okay? Now, here well, let me just state this here, this is the temperature gradient. Temp as short for temperature, Okay. Do you remember what kappa ij represents here? What is it called? Right, in general, it's called the conductivity tensor right, or the heat conductivity conductivity tensor. Okay? Right. This a denser can be thought about again in our setting for the purposes we need here it can be taught off as say generalization of a vector and with the provisional basis that we have, what we find is kappa ij can be well that's not really right. With use of a base is that we have here, we can also write the heat conductivity tensor using direct notation, it would be kappa, right? Also with an under-bar, and like I said by context, we will understand whether something a vector or a tensor. Here it's a tensor, why do we know it's a tensor, because of the way it acts in this equation. Okay. So capital is direct notation for the heat conductivity tensor, and just as we could represent, the direct notation for a vector in terms of its components or related to coordinate notation, kappa can be written with the use of a basis, which we have, as a matrix. All right, and that matrix consists of the components kappa 11, kappa 12, kappa 13, kappa 21, kappa 22, kappa 23, kappa 31, kappa 32, kappa 33, okay? Now, it turns out that this tensor kappa is symmetric and their reasons for it to be symmetric so we would consider Kappa to be symmetric, which me, which we write as kappa equals kappa transpose, right? So this means that in general that the 1 2 component is the 2 1, 1 3 is equal to 3 1, and 2 3 equals 3 2. Right? Which is also written in coordinate notation as kappa ij equals kappa ji. All right, kappa has another property which is important for the physics of heat conduction, 'kay. So not only is it symmetric, right? It is also what we call, positive semi definite. Okay, and what this means is that if C belonging to R3 is vector. Right? Then. If we construct the following quadratic product, right, we allow kappa to act on C. Right? And the product of kappa acting on c gives us back the vector, right? Just like if you know, you've all, you've probably experienced linear algebra, a matrix acting on a vector gives you back a vector. All right, so that's what's happening. So capital T is a vector, if we dot that vector with c 'kay, c dot capital C, is greater than or equal to 0 right, for all c, right. For any c. Right? Well, actually, I, I, I realize I don't really need to say for all c. All right. So if c is a vector, c.kappa c is greater than or equal to 0. Okay? What this means? That there are actually some directions right, for which we are allowing the possibility that there is no conduction of heat in certain directions, right. So we are allowing the possibility that for some c, right, some vector, vector c this thing could be 0, all right, okay. So and this, this the fact that we're allowing the possibility for this quadratic product to equal 0 is what makes it semi definite, all right. In terms of Neumann condition, all right? Do you recall what we are allowing the possibility for by including that product being equal to 0? What physical possibility are we allowing? It allows the possibility that for some direction c this material actually acts as an insulator, all right, that there is no conduction in some direction. All right, so this basically corresponds to. If c.kappa c equals 0. Importantly, this has to be equal to 0 for c itself not equal to the 0 vector. Right of course if c's a 0 vector then that's a trivial result. Right? But if there exist some direction c not the 0 vector for which this quadratic product is itself equal to 0 then, there is no conduction, no heat conduction. Along c, right, in that direction. Okay. The very last, well, actually there's mo, not the very last thing, but there's one more thing I wanted to say, here. When we go back and look at this law, here. This, of course, the Fourier Law of heat conduction. What it tells us is that the heat flux vector is directed along the negative temperature reading, right. So tre, heat tends to flow from high temperature to low temperature, provided we also have this property of c being positive semi definite, okay. So let me state that as well. Heat flows down a temperature gradient. All right, that's what the formula of heat conduction tells us with the additional condition that c is positive semi dense. All right, the last thing we need to state here to wrap up our our introduction to this strong form of the problem, is the boundary conditions, right. So when we, when we return to the boundary conditions, the first one is fairly straightforward. When we have u equals ug on the Dirichlet boundary this is simply a temporary boundary condition. Right, so in the context of our continuum potato. Right, we have our basis here, the continuum potato here, the, the region of interest. Let's suppose that the maze part of the boundary is the Dirichlet boundary or the temperature boundary. What you're seeing is that on this part of the boundary we have the, we're, we are controlling the temperature, right. We're setting the temperature to be ug, ug of course could be a field, right. That's important here. So what we're seeing here is that this could be a nonuniform field, right, so that's possible. Okay, that's allowed. And then for the second part of the boundary this condition can be written as we're writing it as let me see how I wrote it actually, probably sorry, and let's go back here. Okay, I did it right. Okay, this part of the boundary, when we wrote it in direct notation, I think we wrote it as minus j.n equals j sub n. Okay. What that dot? What the n there is doing? Is that it is the unit outward normal to the boundary. Okay. It's a field. It's a vector field. Right, that's what n is. So, when we look at minus j.n, what we're seeing is that given the fact that n is a unit outward normal, what we're seeing is that we are controlling the heat influx. Boundary condition, okay? All right be, because n is the u, unit outward normal j.n would be the outward heat flux, and the minus sign essentially makes sure that it converts that into a heat influx. We're saying we're controlling the amount of heat flowing over the compliment to the Dirichlet boundary over the Neumann boundary, right, so over the blue part of the continued potato. We are controlling the heat, the heat flux, the heat influx, right? So we're controlling me, that of the heat influx. We're controlling the amount we're getting there, right. Okay. So, so we'll note that we're not controlling the vector itself, right? We cannot control the vector, and that the theory of PDE tells us that, in fact, physics also tells us that we cannot control the entire vector right? What we can control is just the normal component of the, of the heat flux vector here, okay? I should also write this in coordinate notation, in coordinate notation is just the dot product, right. So it is minus ji and i equals jn, j sub n. All right and this is coordinate notation. Okay, I believe that completes our basic introduction to the weak, to the, sorry, to the Strong form. We will end this segment here.
