Welcome back. We'll continue with our development of the matrix vector weak form, for the linear elliptical PDE with scalar variables in three dimensions. So what we are going to do today is essentially not only completed assembling the matrix vector weak, the matrix vector weak form. But also talk a little about associated issues which arise mainly from the fact that we are looking at problems here in 3D. So so the topic of this segment and the next couple at least is the matrix vector weak form. Recall that we at the end of the last segment, we were working with the local element level integrals. Okay, And we had developed again the local element level matrix vector representations of these integrals. In particular we worked with the left hand side integral we, the, the bilinear term. We worked with the, the forcing function from the right hand side. And now that brings us to the, to the integral that imposes the Reimann boundary condition. Okay. So, let's start with this one, that one. So what we're doing now is to consider the integral integral over partial of omega e sub j of minus w h j n d s. Alright? And it's useful right away to recall for ourselves the sort of term we're working with here, right. So the, the situation we have is the following. We have our bases. We have our domain and we're looking here at an element that has an edge. Or in this case, a face really, that coincides with that, that coincides with an, with the face of the, of the body, of the body of interest itself. Right, so we have an element of that type, okay, and the whole point is that in this case, so this is omega e. Okay? And we may think of that face as being partial of omega e sub g. Okay? And we recall that partial of omega e sub g. Is the intersection of the boundary of that element omega e with the alignment boundary of the problem. Okay, so that's really the face of the element upon which we're imposing the Reimann boundary condition. Alright, using our finite dimensional basis function it takes on the following form, it is now minus integral over that of okay, a sum of NA CAe Jn dS. Now, one thing we've got to be careful about here is that we can indeed consider the sum to be a sum running over the entire set of element nodes. Right? And, and let's start out with writing it in that fashion. Okay, but now we recognize that in this picture that we've drawn here, not all the nodes have corresponding to themselves shape fun, basis functions. That are non-zero on the surface of interest. Right? On the interface of interest. If we say for instance, that this is the interface of interest that we've, that we've indicated, then let me highlight in a different color the nodes that lie on that Boundary. Let's suppose that those are the four nodes lying on that boundary. Okay? It should be pretty clear to us that it is only the basis functions that are one at those nodes that will contribute at all to this integral. Okay? So, having made that observation, there are. Th, th, there are at least a couple of ways in which we can process and this, the, the, the, the different ways of proceeding simply correspond to different ways of doing bookkeeping here. Okay? For one thing, one may say that one may define., Okay. When we define a subset, right? When we define a subset of nodes, a, right? Which A sub n shall we say okay? Which consists of all nodes A such that x a e, right? Which is the corresponding nodal point, okay? Right? Using the local node numbering such that x a e belongs to partial of omega e sub j, okay? Right. And then what one can do is simply restrict that sum to a lying in this set. Okay? In, in this case, I've used the subscript n, to suggest that this is the, set of nodes or degrees of freedom corresponding to the Neumann boundary data, okay. So let's use this approach, right? So what we see then is that minus integral over this surface, w h j n d S equals. Now again, I'll take several steps all together, right. I'll do this, I'll pull our summation out and I'll just say a belongs to the set we introduced, right, script A, sub n. Okay? We have your cAe, and I apologize, that c looks too much like an e. C a e integral over over partial of omega e sub j Na jn dS. Okay? Now we make another note, which is that we can of course convert from here to our parent sub domain. Right? The parent domain from which we construct every single element, okay? So in that setting, let's suppose that we are still talking on the same element here and so let me go to the next slide to get this done. Now we're going to elucidate a manner in which you carry out this integral, right? So for that purpose, let's suppose that we are now working with an element, right, some general element. Right. And on this element, let's suppose that. Surface of interest to us, is this one, right. Let's suppose that this one is the surface partial of omega e sub g. Okay, now this element of course is always constructed from the same parent domain. The nice, regular element in this volume domain. Okay. Right, we have that sort of a mapping. All right. So, let, let's suppose, just for the purpose of argument, that, the, that the face partial of omegas, omega e sub j is, is the mapping of that face. Okay? And the way I've, drawn things out, the face of interest here which I will mark just for the purpose of argument again I'm going to mark this as partial of omega c sub j. Okay? It is the face in the bi-unit domain in the parent subdomain that gets mapped onto the face of interest to us, which is the face on which we are imposing the Neumann boundary condition, okay? Essentially all we need to do here is now recognize that if we construct a sort of, a lower dimensional mapping, right? Which is the, the mapping that converts the area of this particular face, right, the one marked as partially forming e sub j, here, which, which obtains the, the, this particular face, from omega c sub j. Okay, what we observe here is that the integral that we need to carry out, okay, which is coming from the previous slide, it is minus, sum, AE belongs to A sub j, cAe integral over omega e sub j, NA jn dS. All right? Let's observe that this thing can essentially be constructed as. Integral over partial of omega z sub j NA jn, right. And, what we will do here is write determinant of let me just write this as J sub s, dS c. Okay? Well, what I'm talking about here is the idea that, for the mapping of the the faces, we can, we actually need to worry just about, we, we need to worry only about how to map from this phase, partial omega c sub g, to omega, to partial omega e sub g. Kay? And Js is that mapping. Okay, so Js, insert Js is what we may define as the area coordinates, right, that correspond to a that correspond to a, to a new set of variables. Maybe x tilde one comma. C2. X tilde one comma c3. X tilde two comma c2. X tilde two comma c3. Okay? And what we mean by this is is the following. If what we're referring to here is the following. If we look at the surface that we are working on in our physical domain, it is this one. Okay, so this is the surface partial of omega e sub j. Okay, what I'm suggesting is that we can now define local coordinates on the surface. Right? And these local coordinates are what I'm referring to as x tilde one, x tilde two. Okay? All right? And they, these ones are obtained from a mapping of. That face in the parent subdomain which is c2. C3. Okay? And this is easy enough to do because in terms of c2, sorry, in terms of c2 and c3, we can indeed express the local coordinate x tilda one, x tilda two. Okay? Okay, what we need to do here is define the map x tilda, as a function of c2 and c3. All right? Okay? And then the determinate that we're talking of is simply the determinate of this particular mapping, right? The detail of how to construct this can sometimes seem challenging, especially if the surface partial omega e, as I tried to represent here or here, is not a plain surface. Okay? That takes a little more work, but it can essentially be done. Often they will indeed be plain surfaces. And then the, the, the, this mapping is, is straightforward. In particular if partial of omega e sub j is a coordinate surface, the mapping is actually very, is, is almost trivial. Okay? So this is how you would we would go about doing it, right. So