Transformation of Differential Operators

Question

I would like to gain some knowledge about how to transform differential operators to different coordinate systems using Mathematica. I know that the Laplacian command has an optional parameter chart that does this for various coordinate systems but I would like to gain some insight and possibly intermediate solutions doing this. As a simple test example, I believe the laplacian in polar coordinates will do just fine. So, I would like to transform $$ \Delta u = u_{xx} + u_{yy} $$

using $u(x,y)=w(r,\phi)$ and the transformation $x=r \cos\phi, y=r\sin\phi$ into

$$ \Delta w = \color{red}{\frac{1}{r^2}}w_{rr} + \frac{1}{r}w_{r} + \frac{1}{r^2} w_{\phi\phi} $$

Without incorporating the specific transformation one can of course calculate

D[w[r[x, y], phi[x, y]], x, x] + D[w[r[x, y], phi[x, y]], y, y]

but how would you use the transformation from here? How would you do this in a "step by step" manner, e.g. how would you calculate the derivative(s) of $r$ w.r.t. $x$?

Edit: as was kindly pointed out by Daniel Huber, the red part in the above laplacian shouldn‘t be there.

Answer 1

A step by step derivation:

We have:

x[r_, ϕ_] = r Cos[ϕ];
y[r_, ϕ_] = r Sin[ϕ];
ϕ[x_, y_] = ArcTan[x, y];
r[x_, y_] = Sqrt[x^2 + y^2];

Then doing the derivatives formally by hand we have (I use u[x,y] and u[r, ϕ] instad of u and w):

We replace the formal derivatives by MMA syntax:

rep1 = {Subscript[u, r] -> D[u[r, ϕ], r], 
   Subscript[u, rr] -> D[u[r, ϕ], {r, 2}], 
   Subscript[u, rϕ ] -> D[u[r, ϕ], r, ϕ], 
   Subscript[u, ϕ] -> D[u[r, ϕ], ϕ], 
   Subscript[u, ϕϕ] -> D[u[r, ϕ], {ϕ, 2}]};
rep2 = {Subscript[r, x] -> D[r[x, y], x], 
   Subscript[r, xx] -> D[r[x, y], {x, 2}], 
   Subscript[ϕ, x] -> D[ϕ[x, y], x], 
   Subscript[ϕ, xx] -> D[ϕ[x, y], {x, 2}], 
   Subscript[r, xy] -> D[r[x, y], x, y], 
   Subscript[r, y] -> D[r[x, y], y], 
   Subscript[r, yy] -> D[r[x, y], {y, 2}], 
   Subscript[ϕ, y] -> D[ϕ[x, y], y], 
   Subscript[ϕ, yy] -> D[ϕ[x, y], {y, 2}]};
rep3 = {x -> x[r, ϕ], y -> y[r, ϕ]};

Finally we take the expression for Subscript[u, xx]+Subscript[u, yy] from the picture above, evaluate the replacements and simplify:

Subscript[u, rr] Subscript[r, x]^2 + 
     2 Subscript[u, rϕ ] Subscript[r, x] Subscript[ϕ, x] + 
     Subscript[u, r] Subscript[r, xx] + 
     Subscript[u, ϕϕ] Subscript[ϕ, x]^2 + 
     Subscript[u, ϕ] Subscript[ϕ, xx] + 
     Subscript[u, rr] Subscript[r, y]^2 + 
     2 Subscript[u, rϕ ] Subscript[r, y] Subscript[ϕ, y] + 
     Subscript[u, r] Subscript[r, yy] + 
     Subscript[u, ϕϕ] Subscript[ϕ, y]^2 + 
     Subscript[u, ϕ] Subscript[ϕ, yy] /. rep1 /. rep2 /. 
  rep3 // Simplify[#, {r >= 0}] &

This gives the result:

As you can see, your formula above for the Laplacian in spherical coordinates is wrong.