What is the curl of a vector field?

Question

I know that it is related to circulation density, but I have one book saying it's a single vector, and the other saying it's a vector field. So which is it?

Answer 1

The curl operator is a 3-dimensional operator on a vector function (that is why some books call is a vector field), made up of 3 vectors in 3 basis dimensions (which could be combined into a representation as single vector).

The result of the cross product of a 3-d vector (the curl operator) and a 3-d vector (the vector function) is a 3-d vector.

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