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To implement your first identity you can use: vec /: Cross[vec[x_], Cross[vec[y_], vec[z_]]] := Cross[Cross[vec@x, vec@y], vec@z] -Cross[Cross[vec@x, vec@z], vec@y] Resulting in: In[1] := vec[x]\[Cross](vec[y]\[Cross]vec[z]) Out[1] = (vec[x]\[Cross]vec[y])\[Cross]vec[z] - (vec[x]\[Cross]vec[z])\[Cross]vec[y] The other identity is implemented ...


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In my opinion, this is not a problem that is easy to solve by reading the docs, because anything less than very thorough and careful reading of the docs on the subject of converting spherical to cartesian coordinates can easily lead the user astray. A casual reading of the docs will lead one to think map = CoordinateTransformData["Spherical" -> ...



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