3D Laplacians
The Wolfram Language has information about many coordinate systems. Below, an interactive table is created displaying the Laplacian, using the standard coordinate names in each system.
systems = {{"BipolarCylindrical", {a}}, {"Bispherical", {a}}, "Cartesian", "CircularParabolic", {"Conical", {b, c}}, "Cylindrical", {"EllipticCylindrical", {a}}, {"OblateSpheroidal", {a}}, "ParabolicCylindrical", {"ProlateSpheroidal", {a}}, "Spherical", {"Toroidal", {a}}};Manipulate[Block[{f, vars = ToExpression /@ CoordinateChartData[{Coordinates, 3}, "StandardCoordinateNames"]},
Simplify@Laplacian[f@@vars, vars, {Coordinates}]], {{Coordinates, "Cartesian"}, systems}, FrameLabel -> "Famous 3D Laplacians"]