Nonlinear maps and calculus

1:05:32that bad when you like really just think about it it's like yeah >> kind of makes sense it's when we get to general when [laughter] it's like I don't whoa whoa whoa whoa slow down I I just got here [laughter] like so so that's that's special relativity right and the jacobian there is one Mhm. >> Meaning that the volume in spaceime is preserved no matter what reference frame you're in. >> Okay. >> And when you say reference frame, you mean both position and speed. >> Mhm. Uh and time. >> And time. So both position and time. >> Yeah. Yeah. Whatever volume in your position and time versus my position and time are going to be exactly the same. It's just that the each individual thing is going to be changed, but the product

1:06:13of the two are going to remain the same. Yep. Right. Because the Lorren can the Lorren factor is going to cancel out. Makes sense. >> Okay. And that happens in we showed a 1D case where we were only considering like >> you know forward and backwards where the boost is but this this applies in all three. In that case you get like the manowsky tensor and all and all sorts of stuff but that that that's effectively what's happening. So this is a coordinate transformation right spacetime gets distorted but the jacobian is one. So the volume of spaceime remains the same. Now let's talk about nonlinear transformations. Okay, this is an example of a nonlinear transformation. Here the x-axis goes to

1:06:53x + sin of y / 2 and the y-axis goes to y + sin of x over2. This is what I mean by now you can start mixing >> stuff right in the other one you were also mixing but here you're doing it in a nonlinear way. You're not just like adding uh some lorren factor times the constant. This is like sign which is an oscilly thing. And so you're seeing like stuff shift, right? Things are getting wiggly. >> Okay. In this case, what is the Jacobian going to do? Well, the Jacobian now is actually different >> in different spots. You can see in some cases the areas are getting squished. In others cases, the areas are remaining about the same. >> Okay. And if we go

1:07:33>> and see and zoom in on a central area element >> of this transformation, what happens? Well, let's look at let's look at those let's look at those that that sector that's right in the middle of the one one box. Okay. Now the transformation is nonlinear. So it's curved right in every sense. But the beauty of calculus is the following. If I take a small enough element stuff is mostly going to be a line. That's that's all calculus is >> like the the essence of calculus is >> weird stuff is happening but if I take a small enough element it's mostly just linear

1:08:14>> calculus just zoom in >> just zoom yeah no exactly just zoom in and the square becomes a parallelogram it's not becoming a weird uh curved shape >> it goes back to the example we just talked about previously but it's all about what is your uh perspective of the >> or or or context by which you're making your calculation. >> Yeah, exactly. And in this case, we can understand what the Jacobian is because the Jacobian at that point is going to be what is the ratio of the square area to the new parallelogram area. Okay, in this case it looks like one because like it's getting squished and this like the rhombus looks about the same. So in that locality the Jacobian is one but somewhere else it could be two,

1:08:54somewhere else it could be less than one and so on and so forth. That's the beauty of calculus is that you can zoom in and you can figure out a well- definfined Jacobian at every given point. >> We have a larger complex system but at sufficient zoom in the that complex system still comes back to the fundamental that we just talked about previously. >> Okay. Yeah. Exactly. And so now we've got this core concept which is the Jacobian matrix. The Jacobian matrix tells you how each of the directions sort of swish and become one another. And the determinant of that matrix tells you the ratio of the areas before and after or I should say it's the ratio of it's the ratio of areas after divided by before. Okay. So two means that the

1:09:35ratio went up. Less than one means the ratio went down. There's also negative Jacobians which means that the the axis got flipped. Okay. So it's like I mean >> it's just yeah it's it's like it got it got flipped, right? >> Okay. So if the determinant at a given point is non zero as we just saw then it signifies a linear approximation right and it doesn't collapse. Now here we've got an example that is at the center of the Jacobian conjecture. Okay we got to consider situations where the determinant is a nonzero constant. It's a constant everywhere. >> Meaning no matter where I am in my map

1:10:16>> the area is actually the same. Even though the upper part is getting more squished, you would think the actual area, the ratio of the areas is actually exactly the same. This is a particular map that um I came up with in Desmos. There's a lot of really cool Desmos um >> pre-made applications where you can plug in your own custom linear map and you can visualize how the map is going to change the axis. So in this case the new xaxis is x + 1/4 y^2. >> Mhm. >> And the y- ais is just a y y y y y y y y y y y y y y y y y y y y y ais. So every y gets plotted to it it's it its own y but the x gets shifted by a parabola.

1:10:58That's why the these things are becoming a parabola but like the the horizontal lines are remaining horizontal lines. >> It's going from lanes on a highway to a track. The elliptical of a track >> and so and the lanes on the highway is is the X, right? So that's why that's getting shifted into a parabola. But the the horizontal like lines of latitude so to speak are remaining the same because the y just gets plotted to the new y. Nothing happens. Okay. In this case, if you were to plot the if you were to plot that area, that little green area, >> that little green area becomes a parallelogram, but the area is preserved. >> The jacobian here is one. If I were to take that area and put it up top, >> the parallelogram might get more

1:11:39squished, right? It might have a different shape but the area is still going to be one. This particular mapping is one where the Jacobian determinant is constant meaning that the transformation of area everywhere >> to all of infinity in X and Y the area is preserved. >> Okay. Now given this one can ask suppose I was