Vector Laplacian Identity 

The Laplacian of a vector field in -dimensional flat space can be computed via the formula . This formula is well known in three dimensions.

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X Laplacian[{v1[x, y, z], v2[x, y, z], v3[x, y, z]}, {x, y, z}] == Grad[Div[{v1[x, y, z], v2[x, y, z], v3[x, y, z]}, {x, y, z}], {x, y, z}] - Curl[ Curl[{v1[x, y, z], v2[x, y, z], v3[x, y, z]}, {x, y, z}], {x, y, z}]

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The following creates a table that automates the verification of the identity in different dimensions and coordinate systems.

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X Laplacian[{v1[x, y, z], v2[x, y, z], v3[x, y, z]}, {x, y, z}] == Grad[Div[{v1[x, y, z], v2[x, y, z], v3[x, y, z]}, {x, y, z}], {x, y, z}] - Curl[ Curl[{v1[x, y, z], v2[x, y, z], v3[x, y, z]}, {x, y, z}], {x, y, z}]; labels = {HoldForm[vec], HoldForm[Laplacian[vec, vars, sys]], HoldForm[% == Grad[Div[vec, vars, sys], vars, sys] + (-1)^k Curl[Curl[vec, vars, sys], vars, sys]]};

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X systems = {{"Cartesian", "Polar", "Bipolar", "PlanarParabolic"}, {"Cartesian", "Cylindrical", "Spherical", "CircularParabolic", "BipolarCylindrical"}, {"Cartesian", "Hyperspherical"}, {"Cartesian"}};

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X Manipulate[ If[Not@MemberQ[systems[[dim - 1]], system], system = "Cartesian"]; Block[{vars = Array[Subscript[x, #] &, dim], k = dim, vec, lap}, Grid[Transpose[{labels, {vec = Array[Subscript[v, #] @@ vars &, k], lap = (Laplacian[vec, vars, system] // Simplify), Simplify[ lap == Grad[Div[vec, vars, system], vars, system] + (-1)^k Curl[Curl[vec, vars, system], vars, system]]}}], Dividers -> Center, ItemSize -> {{20, 20}, Automatic}, Alignment -> Left]], {dim, Range[2, 5]}, {{system, "Cartesian"}, systems[[dim - 1]], ControlType -> SetterBar}, SaveDefinitions -> True]

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