Jacobian and depending questions

Lets understand what is Jacobian.

Imagine you have a function that takes several variables as input and gives you a vector (a list of numbers) as output. For example, it could take in the coordinates of a point in space and give you the velocities of particles at that point.

Now, let's say you want to understand how small changes in the input variables affect the output vector. The Jacobian matrix helps with exactly that. It's a matrix where each entry represents how much each component of the output vector changes with respect to each input variable.

The determinant of this matrix, called the Jacobian determinant or simply the Jacobian, tells you how the volume changes when you transform from the input space to the output space. In simpler terms, it gives you a measure of how much the function "stretches" or "shrinks" space around a particular point.

So, in essence, the Jacobian is a way to quantify how a function distorts space when you apply it to a set of variables. It's a fundamental tool in calculus and has many applications, especially in fields like physics, engineering, and computer graphics.