 Hi everyone. I wanted to make some quick notes on one of the homework sets that is out right now, the one on surfaces and the first fundamental form. I just wanted to give you guys some notes that only help you get started with the problems. So I will share my screen here. Okay. Right, so the point of this section is that we're talking about these regular parameterizations, subsets in R3, essentially ways that we can represent them with two coordinates. One thing to note about this definition, this is just coming straight from Jason's notes. There are two things here that will be important to us momentarily in this definition. One is that it is a one-to-one function, and then two is this condition on cross-product of these tangent vectors not being zero. So I guess at least my motivation for the reason we do this is that the tangent vectors at each point will span a plane. In other words, the dimension of this tangent space is going to everywhere be two. And what's nice about that is like right now over here off to the side is that you can, well, this cross-product of tangent vectors here appears when you are integrating a function, a real-valued function, over a surface. And it sort of takes the place of the Jacobian when you change variables. So we can now integrate functions over sort of more exotic structures. So the first question asks us, while the setup for it is that we have this sphere, it is sitting one unit above the x, y plane. And we are defining this map in a stereographic projection, which essentially takes a line here. I am trying to picture here. For any point on the sphere, which I have denoted here with this red x, we take the point 0, 0, 2 sitting at the top of the sphere. We take a line through 0, 0, 2 and our point. And we look at where that intersects the x, y plane. And that is our projection. And so the first problem gives us this inverse map, pi inverse, which is taking something in the UV plane, projecting it up into S2 as a subset of R3. And so the way I would approach this problem is first, well, it's asking us to show that this map is indeed the inverse of pi, but we don't actually know pi offhand. So I think probably the first thing is this bit is what is the actual map pi in coordinates. This should be something that takes S2 minus this one point and gives you a two vector out of it, two component vector. So this will be something that takes a P and S2 here and three coordinates x, y, z and should give you something with two coordinates. So the way I would recommend doing this is to actually find the equation of this line. I've written it as LP here. It's this line through 0, 0, 2 and your arbitrary point x, y, z on the sphere. And just check where this line where the z coordinate. So when you write that parameterization of this line, you'll get a three vector that should be a function in each component. And just check where that third component is equal to zero. And so that should give you this map pi. Right. And it will send you, right, because the third component we know is going to be zero for any of these. So we can see what the first two components of that function look like. And that will be our pi. And we'll just define the first component to be our U and the second component to be our V. Okay, so once we have pi, we can actually check what the question is asking, namely if this is the inverse map. And at least one way to do this, the way I recommend going about it is just to take pi compose pi inverse evaluated on a pair UV, and just check that that gives you UV back. And this will just be algebra. It'll be sort of a long, somewhat nasty expression, but it should simplify really nicely in the end. Similarly, you just check the, so that's the first line here. Then you just sort of do the opposite thing where now we compose pi inverse with pi evaluated on a vector with three components x, y, z, and just check that that gives you x, y, z back. And once you've done that, then you've proved that this is indeed the map pi inverse. So the second part asks us now to show that this inverse map is a regular parameterization of S two minus this point. And so now this definition up near the top will be important, right? So we need to check that it's one to one, and we need to check this condition on the cross product. So going back down here. So here's what you need to show. We're taking pi inverse as our map. And this condition right here says, well, you know, just take the partial with respect to you. You'll get some vector out of that. Take the partial with respect to me. You'll get some other vector. Take the cross product of these two vectors. And now just somehow argue that this function is never zero. Okay? That's kind of what I'm saying down here at this part. And then just remember too that the other part of being a regular parameterization is that you need to have injectivity. The way I like to remember this is that it's something like the horizontal line test. If you have a sort of one y value that's mapped to by two different x values, that's a failure of injectivity. So what you need to check is that if two y values are the same, then they really did come from the x value. And that's essentially what this condition is down here. And again, this should, I think, just be, this should just reduce test some algebra. Third part of this now asks us to compute the first fundamental form ISMP of this parameterization as a matrix. And this is, I think, covered pretty well in the notes. This ends up essentially being a computation. But just remember that the first fundamental form is this matrix where we're just taking all of these inner products of the partial of your map with respect to one variable against the partial of your map with respect to another variable. And so this thing should be a symmetric matrix because the partial is commute. So the top right corner, in this case, right, we only have two variables to worry about you and me. So like the top left corner, we're taking partial of you against partial of you, the inner product of that top right corner, we're taking partial of you against partial of the and so on. And you can compute this as just one big, it'll be a matrix of functions. And there should be some nice simplification at the end of this. And I think once you have this part C where you've computed the first fundamental form of this particular matrix, most of part two, or sorry, most of question two will follow almost analogously, I think. It's really just giving you a bunch of different surfaces with different parameterizations given in equations. And it's asking you to just compute the first fundamental forms. And so it's going to be the same game you played up here, you're going to take partials with respect to some two variables here, for example, for two a, this is going to be partial of x with respect to you, and then with respect to V, you're going to take a bunch of inner products of these guys, put them in a matrix, and then get something out of it. And I will just mention that for the remainder of these problems, you might get some sort of long expressions involving trig functions, cosine, sines, hyperbolic, cosines and sines. So it may just be easier to compute just E, F, and G as stated in the notes. And I think it's right, then you just get a matrix that's E in the upper left, F in the upper right and bottom left corners, and then G at the bottom left. And so I think it's just maybe easier to compute just E by itself, F by itself, and G by itself, rather than try and fit them into some giant matrix. Alright, so hopefully that's helpful. Feel free to email me or reach out if you need any help with anything else.