Change of variables in partial derivatives

Questions
1

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:

chainrule.png
chainrule.png600×285 11 KB

generalchainrule.png
generalchainrule.png599×111 4.84 KB

generalchainrule2.png
generalchainrule2.png611×170 5.59 KB

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.

2

Setting up the derivative as an anonymous function might help.

newD = D[#, {{t, z}}] &;
newD@f[t u/L , z/L]
(*{(u*Derivative[1,0][f][(t*u)/L,z/L])/L,Derivative[0,1][f][(t*u)/L,z/\
L]/L}*)
3

It seems that my old code still works so you can do:

DChange[ D[f[t, z], {{t, z}}],  {z0 == z/L, t0 == t u /L}, {t, z}, {z0, t0}, f[t, z]]

{(u (f^(0,1))[z0,t0])/L,(f^(1,0))[z0,t0]/L}

You can read how to install DChange and use it from: Analogue for Maple’s dchange - change of variables in differential expressions

4

Maybe this helps

Regards M.I.

5

Dear users Tim Laska and Kuba Podkalicki:

Thank you for your answers. Now, a few comments:

  1. Tim Laska suggestion works very fine for rescaling transformations, but I’m not sure if it is enough for “mixed variables” transformations like:

           t0 = t - x/u ( t0, t = time, x = lenght, u = velocity)

As the new time coordinate (useful for example to simplify some wave equations) mixes the “original” x and t coordinates.

  1. The function DChange of Kuba Podkalicki looks just like what I was looking for, but I’m afraid of some software compatibility issues if I download the package. Could it cause some problems/damage to the original Mathematica language, specially for newer versions like Mathematica 12 (not to mention future versions in coming years)?
6

This package (“MoreCalculus`”) is a very simple one, it only introduces DChange function and there is not built-in DChange symbol so you don’t need to worry about it.

Will there ever be a built-in DChange symbol? Who knows, maybe, maybe not, I suppose WRI will pick a better name for such a function.