I have always thought that we may obtain information about the vorticity of a vector field $f$ by considering the cross product $\nabla \times f$. In higher level texts on fluid mechanics I see that they instead define the vorticity using wedge products, for example $\nabla \wedge f$.
I do not know much about differential forms or the wedge product, except that there is some link between the wedge product and the cross product in three dimensions. Why is the wedge-product used here instead of the cross-product? Is there some benefit?
