Coordinate transformations and Jacobians

47:32to understand the Jacobian conjecture we have to first understand functional maps which in physics language we like to call coordinate transformations. Here's the idea. A map in this mathematical sense is when you take a coordinate plane or uh hyper plane or many dimensional plane and all of the points in your original space get mapped to a new space. In this case, what we're seeing is a 2D plane being mapped to another 2D plane. Okay, there's some kind of nonlinear transformation where the new x coordinate and the new y-coordinate depend on the old x

48:12coordinate and the old ycoordinate. Right? In this case, a square is taken into a parallelogram. Notice that the new x can depend on both the old x and y, not just the x. And the new y can depend on both sets of coordinates. This can happen in higher dimensions as well. The Jacobian is the ratio between the new area and the old area >> and which can also scale from 2D. >> Yes. You can have volumes. For example, you can have a volume in 3D that goes to a new volume in 3D, right? A cube can become like a weird >> kind of uh romboid what whatever 3D

48:55prism is what it's called, I And then you're comparing the ratio between the volume or the area depending on what dimensional space you're talking about between the two is where the Jacobian comes in. >> Yes. Exactly. And in this case the I think I think from from my eye the area is about the same. So the Jacobian in this transform would be one. Got it. >> Okay. Right. >> That's what the Jacobian means. Effectively it's saying okay I have a a map and that map is preserving area if the Jacobian is one. If the Jacobian is greater than one, then it's going to larger cases, right? And like when I think about maps, the reason why it's called a map is because it it really starts from this concept of like maps of the earth, right? You can think about

49:36the Mercada projection or these other projections that change the area and change the shape but keep the sorry, they change the shape but they keep the area the same. For example, with the Mercada projection, right? We all know that areas near the poles get larger. Mhm. >> So the Jacobian near the poles is greater than one. >> Right? That's why Africa looks small and Greenland looks the size of Africa >> because the Jacobian near Greenland is much larger than the Jacobian near Africa when it comes to the Mercada projection. But there's other projections where they try to preserve the Jacobian everywhere. But then that's going to distort the shape in some sense. >> So and people would argue which is

50:18better when you talk about geographic maps. Is it better to keep the Jacobian closer to one? >> Yeah. >> Uh where you're going to get some shape distortion or is it better to have it have shape uh clarity but you have distortions at some extremes >> as it rel Okay. >> Yeah. So that that's what we're talking about when we talk about Jacobians. Okay. It's really it's a it's a ratio of how my transformation is changing little tiny areas, infinite decimal areas, right? Because in calculus we think about like there's continuous transformations. So the Jacobian here can be different from the Jacobian there and so on and so forth. >> It's not a family in the game of thrones is what you're saying. >> No, not at all. And so let let's let's do a very simple case. Okay. A linear

51:00transformation. >> Yes. >> Now in this case this is a very simple matrix 2112. Okay. When we multiply that by any 2D vector, we're going to get a new vector. For example, the red vector is getting slightly skewed like that. So the red vector is really a one zero. That's the x-axis. just uh one on the x- axis, zero on the on the y axis that gets transformed to two on the x axis and one on the y on the y- axis. Right? Similarly, the y unit vector which is 01 that gets transformed to one comma 2 which means one on the x axis, two on the y axis. That's why the green arrow is slanted more towards the y- ais, the red arrow is slanted more towards the x

51:41axis. And in this case, the determinant of that matrix, if you were to do it, it's 4 - 1. So that's three. You can see that the square becomes a rhombus that's the size of three and it it has to do with the the determinant of that matrix. So when we say a Jacobian determinant, that's what we're talking about. Okay. >> Yes. >> Okay. So this is a very simple case. Now let's give you um kind of a more complicated case but one of my favorite coordinate transformations as a physicist which is the Lorent transformation. This is the transformation that we know and love from relativity. >> Right. >> Right. In special relativity, we have the Lorent transformation. You might have heard that when we move

52:22fast, length contracts and time dilates. So the clock runs slower and length gets shorter. That's two different things,