Dear Mathematica team:

I am trying witout success to make a change of variables in a partial derivative of a function of 2 variables (for example the time coordinate "t" and the lenght coordinate "z"), like

fu:= f[t,z]
dfu:= D[fu,{{t,z}}]

Then I want to rescale the t and z coordinates (something that is useful for example to simplify equations in fluid mechanics), defining the new variables: z0=z/L; t0= t*u/L

z0:=z/L
t0:= t*u/L

Putting all together:

{t0,z0}:={t*u/L,z/L}
fu:= f[t0,z0]
dfu:= D[fu,{{t0,z0}}]

Unfortunately, this substitution does not work because I am actually differentiating with respect to functions of the original t and z variables, so the program send and error message like: "z/L is not a valid variable"

After some reading of calculus textbooks, I found that I need to apply the chain rule. The problem with this is twofold:

1) The formulae are a bit messy, like these in the Wilfred Kaplan (2002) textbook:

Those formulae have either explicit sums or products of Jacobian matrices, and while constructing Jacobians from substitution rules (like the reascaling ones above) is rather easy, I still don't undestand how to implement the full formulae in Mathematica.

I would greatly appreciate you help.

Nota bene: maybe someone else posted something similar before (my apologies if it is the case) , I just don't know yet how to seach by keywords in the forum.