I am teaching myself tensor calculus from Pavel Grinfeld's Introduction to Tensor Analysis and the Calculus of Moving Surfaces. I am having trouble understanding the Jacobian property in tensor notation. From his book:
Consider two alternative coordinate systems $Z^i$ and $Z^{i'}$ in an $N$-dimensional space. Notice that we placed the prime next to the index rather than the letter $Z$. Let us call the coordinates $Z ^i$ $\mathit{unprimed}$ and the coordinates $Z^{i'}$ $\mathit{primed}$. We also use the symbols $Z^i$ and $Z^{i'}$ to denote the functions that express the relationships between the coordinates: $$Z^{i'}(Z) = Z^{i'}$$ $$Z^i = Z^i (Z')$$
As sets of functions, $Z^i$ and $Z^{i'}$ are inverses of each other. This fact can be expressed by the identity: $$Z^{i'}(Z(Z')) = Z^{i'}$$
The above identity represents $N$ relationships, and each of the $N$ relationships can be differentiated with respect to each of the $N$ independent variables of the functions $Z^i$ and $Z^{i'}$. This will yield $N^2$ relationships for the first-order partial derivatives of the functions $Z^i$ and $Z^{i'}$. With the help of tensor notation, all of the operations can be carried out in a single step. We differentiate $Z^i (Z' (Z)) = Z^i$ with respect to $Z^j$. The result of this differentiation reads $$\frac{\partial{Z^i}}{\partial{Z^{i'}}} \frac{\partial{Z^{i'}}}{\partial{Z^j}} = \frac{\partial{Z^{i}}}{\partial{Z^j}}$$
How can we differentiate $Z^i$ with $Z^j$?
This is what is throwing me off. If I think about $Z$ as the classical Cartesian system $(x,y)$ and $Z'$ as $(r,\theta)$, shouldn't $x$ and $y$ be independent variables? Meaning, $\frac{\partial{y}}{\partial{x}} = 0$? My question is, why would you differentiate two independent variables?