Laplacian 2-D polar form

• To: mathgroup at smc.vnet.net
• Subject: [mg6906] Laplacian 2-D polar form
• From: sergio at scisun.sci.ccny.cuny.edu (Sergio Rojas)
• Date: Fri, 25 Apr 1997 14:00:37 -0400 (EDT)
• Organization: City College Of New York - Science
• Sender: owner-wri-mathgroup at wolfram.com

```(****
Hello folks,

Using Mathematica 2.2 for DEC OSF/1 Alpha, I wrote a few lines
to find the Laplacian 2-D polar form starting from its expression
in  rectangular coordinates:
****)

R = Sqrt[ x^2 + y^2];
THETA = ArcTan[y/x];

Dx[f_] := Dt[f,x];
DfDx = Expand[Together[ Dx[f[r /. r -> R,theta /. theta -> THETA]]
/. R -> r /. THETA -> theta /. Dt[y, x] -> 0 ]];

DDfDxDx = Expand[Together[ Dx[ DfDx /. r -> R /. theta -> THETA]
/. R -> r /. THETA -> theta /. Dt[y, x] -> 0 ]];
Dy[f_] := Dt[f,y];
DfDy = Expand[Together[ Dy[f[r /. r -> R,theta /. theta -> THETA]]
/. R -> r /. THETA -> theta /. Dt[x, y] -> 0 ]];

DDfDyDy = Expand[Together[ Dy[ DfDy /. r -> R /. theta -> THETA]
/. R -> r /. THETA -> theta /. Dt[x, y] -> 0 ]];
Laplacian2Dpolarform =
(Expand[Simplify[DDfDxDx + DDfDyDy /. {x -> r*Cos[theta], y -> r*Sin[theta]}]]
/. Sqrt[r^2] -> r);
Print["Laplacian2Dpolarform = ", Laplacian2Dpolarform];

(****
However, I am not satisfied with this code. Can somebody show a better way
of doing it?

On the other hand, let's say one want to start from the last mathematica
output :

(0,2)              (1,0)
f     [r, theta]   f     [r, theta]    (2,0)
---------------- + ---------------- + f     [r, theta]
2                 r
r
with InputForm[%] =
****)

Derivative[0, 2][f][r, theta]/r^2 + Derivative[1, 0][f][r, theta]/r + \
Derivative[2, 0][f][r, theta];

(****
How can one tells to mathematica: now substitute f[r,theta] by the function
g[r,theta] :=  r*Cos[theta^2 + r^2]. I tried the form:
****)

g[r_,theta_] :=  r*Cos[theta^2 + r^2];

Laplacian2Dpolarform /. [f] -> [g]

(****
but the error:

Syntax::sntxf: "Laplacian2Dpolarform /. " cannot be followed by "[f] -> [g]".

is outputed

Salut,

Sergio
e-mail: sergio at scisun.sci.ccny.cuny.edu

****)

```

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