All right, let's continue. So what we did in the previous segment was introduce the notion of the bi-unit domain and use that to define our basis functions, right? Specific-, and specifically, we are working with linear basis functions in this simplest of, finite element formulations. Right? And we've, we've gone ahead and defined those basis functions. And at the end of the segment, we also talked about. We observed that these basis functions have this Kronecker delta property. Okay? So, these basis functions also have another property which is useful to note. Also. Right? So, let's do the following. Let's consider N1 at some arbitrary point z in the Bi-Unit domain. Plus N two at Very trivially or very simply, this is one minus xi over two plus one plus xi over two. Which is one. Ok? So at any point in the domain these basis functions add up to one. This is important because it, makes sure that we can at the very least represent constants. Okay? So this, this, this is an important property. The, the other thing I should mention now is that the way we've written these functions can be generalized and we will do this, later. We can generalize these to a higher order Polynomials generalize, generalizable, because I'm not quite sure that's a word. Anyway, it's generalizable to, higher order polynomials Okay. And, in generalizing them to higher order polynomials we are going to do it. The, the time has not come to do it. We'll do it later after quite a few more, segments. But when, when we do, do that. We will do so by observing that these are linear basis functions of a class of polynomials called polynomials, okay. So the generalize-able to higher-order polynomial, and they are drawn from a family of polynomials that are called Lagrange polynomials. Okay? We do that later, not right now. Yet another thing, I want to point out about these basis functions. Let me see. That's a bullet point and this is another bullet point. Okay? Here's another bullet point. If we look at, these basis functions. And for just a minute, I'm going to go back to our physical domain, okay? So, let's suppose we have element here. This has, this is element e. And next to it we have element omega E plus one. Omega E plus two next to it. Right? And so on Now, we focused upon a specific, or and arbitrary element e in order to construct these, basis functions. And we, and we did that by going into the bi-unit domain. First of all, we need to, recognize that each of these elements in the physical domain is hooked in from this by unit domain, c equals minus one c equals one, and in this, this bayou domain we are going to denote as omega c. Okay, for, for obvious reasons. So note that each or, an arbitrary element, omega e or omega e plus one or whichever one we want, is always constructed as a mapping from the same Element in the bi-unit domain, right? The same parent element, as it is sometimes called, in the bi-unit domain, okay? Once we do this, we get our basis functions in the physical domain, and I want to draw them. which i want to do in a different color here. Ok, so in the physical domain, if we look at omega, element omega e, we've had this basis function, right? And that one. Ok, if we look at omega, at at that element omega e plus one We have also similar looking basis functions. But what happens here is that they are defined over element omega e plus one. Right, and so on. Right? Now If we look at, how this basis functions appear together, in, in the physical domain, let's focus on the fact that here we have no dome, x e plus one. We may choose to look at these basis functions in the physical domain as being located at global node X E plus one. When I say global node X E plus one, what I mean is that I'm looking at this node as being pa, one of the nodes in the overall domain omega. Right? So, I may look at x e plus 1 as a global node. Okay? And in fact it has global node number. E plus 1. Right? If you were to look at that same nodal point, in the context of element E. Okay? You would call it, local node number two For element E. Okay? Okay. It is also the local node number 2, 4. Omega E. Okay? And it is local node number one for element. >> Omega E plus one. Right, this is nomenclature that we'll use later on when we, when we, as we advance with our finite element formulation. Nevertheless, the point I wanted to make here is now when we look this collection of basis functions and we focus upon the individual nodes and view them as global no-, nodes on a sort of scale, right? But global node numbers What you observe is that we have for element sorry, for node x e plus 1, we have a basis function which I'm going to draw slightly off. I'm going to sort of emphasize it and I'm, I'm drawing it in red, but it's slightly off the actual line describing the basis functions, right? And, and I'm doing that just so you can see the underlying element. Level basis functions which when put together give us what may be called a, a basis function corresponding to node x e plus one. Okay? Now, the basis function corresponding to node x e plus one, which I will label like this, is the, basis function. For global node e plus 1. Observe that basis function corresponding to global node e plus one has a very local sort of sphere of influence. It is nonzero only on the elements that are immediately adjoining node number e plus one, global node number e plus one. Ok? So basis function for global node e equals one is nonzero. Only in elements, omega e And Omega e plus one. We may choose to regard the basis function correspond to global node number e plus 1, this red hat function I have, outlined here. We may choose to regard it as a globally defined basis function. However, its support is very local. Its support is local to the two elements in this case adjoining the node of interest, okay? So this property is referred to as local support. Sorry not local it's called compact support properly, sorry. Compact support. Compact support of Global basis functions. Okay? The idea here is that our In, in constructing our final element formulation, it is convenient to focus upon our element's subdomains. And by invoking this idea of the bi-unit parent domain, it gets to be very conve, very convenient and very clean to define basis functions over each element. However, we can step back and take the global picture, right? And recognize that, that by putting together these local functions, local basis functions from the elements adjoining each node, we can actually construct these global shape functions. Global basis functions, right? If we do that, what we observe is that the finance, that, that the type of basis functions we are using here are ones with compact support