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Built-in Symbolic Tensors
3D Laplacians
Mathematica
9 has information about many coordinate systems. Below, a table is created displaying the Laplacian, using the standard coordinate names in each system.
In[1]:=
X
systems = {{"BipolarCylindrical", {a}}, {"Bispherical", {a}}, "Cartesian", "CircularParabolic", {"Conical", {b, c}}, "Cylindrical", {"EllipticCylindrical", {a}}, {"OblateSpheroidal", \ {a}}, "ParabolicCylindrical", {"ProlateSpheroidal", {a}}, "Spherical", {"Toroidal", {a}}};
In[2]:=
X
Grid[Map[{#, Block[{f, vars = ToExpression /@ CoordinateChartData[{#, 3}, "StandardCoordinateNames"]}, Simplify@Laplacian[f @@ vars, vars, {#}]]} &, systems], Dividers -> All, AllowScriptLevelChange -> False]
Out[2]=