I am a Microbiologist with no Math background.
I am looking for an equation or how to find equations that warp a normal XY plane in the Jacobian fashion. Like this https://youtu.be/CfW845LNObM?t=61 I want to end up with an f(x,y) that returns f(x',y'). In other words, I want an f(x,y) that will allow me to see how a point (x,y) moves to (x',y'). I want, in my hand, a function that when I plug in (x,y) it spits out x',y'.
Idea/Method.
A Jacobian matrix tracks or should track the movements of all points. If I can get the partial derivatives, then I can integrate to obtain the initial function.
In 1D, I'd like to think of a y=f(x) that returns a curve. So when I have the curve, I'd like to find f(x). I like to think of this https://youtu.be/CfW845LNObM?t=143 as a graph. If it can be seen as a graph then I'd like to obtain the function that graphs it. But I could be wrong.
I've heard of complex analysis but I'm yet to wet my feet in it.
Hmm it seems you still aren't convinced of my suggest, I'll use this answer as a 'sales pitch' of the book (*).
Page-71, how the function $e^z$ wraps the cartesian grid:
And here is how the book explains the complex derivative pg-196:
As simple as it looks, at a certain point on our input plane, imagine an infinitesimal vector emanating from that point. Now, the complex derivative acts on this vector and does an amplitwist, which is an amplification and a rotation.
Here is a worked out example on how a complex mapping works, I'll give you a sheet of paper and what you do is put flags at each point on the paper. Maybe use a ruler to mark the points on the graph (like a number line) , to determine a point on the plane we will need two numbers on this flag (why?).
Now, we can think of a complex function as taking in the location of the flag on the first grid (z-plane) and where you should put the flag on the output grid (w-plane). Let's take for example the complex function $f(z)=z^2$, we can expand this function in cartesian form as:
$$ f(z) = (x+iy)^2 = x^2 -y^2 + 2xiy$$
Setting $f(z)=u+iv$, our $u=x^2 - y^2 $ and $v=2xy$, think of $u$ as the horizontal in the new plane and $v$ as the vertical (in place of $x$ and $y$). Recall that $x^2 - y^2=c$ denotes the equation of hyperbola, this means that in our mapping hyperbolas gets mapped to straight lines as shown in page-191 of the book:
Left: z-plane, right: w- plane
There is a lot more in this book than I can show here. I took a long time to work through the first few chapters but when you start understanding the topics , I can tell you it's a real joy.
If you can wait for a few more months, then Tristan Needham (same prof who wrote his book) is writing a differential geometry book and it should probably contain all about the kind of mappings you are looking for.
*: I hope other MSE folk will be ok with this kind of posts