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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:
Wolfram Language code: 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}]
The following creates a table that automates the verification of the identity in different dimensions and coordinate systems:
Wolfram Language code: 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]]};
Wolfram Language code: systems = {{"Cartesian", "Polar", "Bipolar", "PlanarParabolic"}, {"Cartesian", "Cylindrical", "Spherical", "CircularParabolic", "BipolarCylindrical"}, {"Cartesian", "Hyperspherical"}, {"Cartesian"}};
Wolfram Language code: 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]