# Replacement rules in combination with pure functions to make a change of variables

I have defined my function in this way

w = f[r,Θ] ;


After some calculations i obtained my results with respect to the previous function. For instance:

Set[lapla1, -(1/(2 μ)) (1/r D[w r, {r, 2}] - 1/r^2 (D[w, {Θ, 2}] + Cot[Θ] D[w, {Θ, 1}]))] // ExpandAll


In this moment i don't need to work with the previous function $w$, instead i need to keep my previous result, multiply lapla1 by $r$ and work with a new $w$

w = f[r, Θ]/r ;


I'm trying to use pure functions to accomplish this, however i don't know if i'm proceeding correctly.

Set[lapla11,lapla1*r /. f -> (f[#, #2]/r &) // ExpandAll ]


EDITED:

When you make this transformation, the term $\frac{1 }{rμ }\frac{\partial f}{\partial r}$ must disappear from lapla1. This is the only way i found to prove it

Set[lapla12, -(r/(2 μ)) (1/r D[w/r r, {r, 2}] - 1/r^2 (D[w/r, {Θ, 2}] + Cot[Θ] D[w/r, {Θ, 1}]))] // ExpandAll

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If I understand your problem correctly, an immediate solution would be use SetDelayed instead of Set in the expression Set[lapla1, ...] // TrigExpand // ExpandAll. Then when you change $w_1$ the change would reflect next time you evaluated lapla1. That said, this isn't a great way to write code - perhaps you should make lapla1 a function of $w_1$. Also note that the single quote ' isn't a valid character for a variable name, assuming that's what you're going for in the last command. –  Aky Mar 30 at 12:48
@kuba I eliminated all subscripts and expressed my function in term of two variables instead of six. –  shadraws Mar 30 at 12:59
I understand that you want to multiply lapla1 with (ra*rb) but 'protect" the terms including an 'f[]' from this multiplication. Your code works, except that you apply the replacement rule outside the Set statement instead of inside. Same goes for the first use of ExpandAll. –  Wouter Mar 30 at 13:05
@Wouter I can't exactly understand what you means with protect, at least in terms of code. –  shadraws Mar 30 at 13:26
With 'protect from' I just mean 'but not multiply' the terms etc. –  Wouter Mar 30 at 14:31

If you want the term $\frac{\partial f}{\partial r}$ to disappear you need to introduce new function which would be:

w2 = f[r, θ] r


which means that you have to make a substitution f -> w2/r, this way:

lapla1 /. f -> (w2[#, #2]/# &) // Simplify // ExpandAll


If you once used f or w, don't change theirs definitions, use a new one, you will less likely make a mistake.

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is this what you're after? –  Kuba Mar 30 at 19:03
The answer looks more easier than i thought. Thank you very much @Kuba. In order to get the result i was expecting, I didn't define a new variable, instead i use the previous one for w lapla1*r /. f -> (f[#, #2]/# &) // Simplify // ExpandAll // TraditionalForm Regarding to your advise, in this case is ease to see the new definition, however i'll take it into account for future. –  shadraws Mar 30 at 19:22