Examples
A common example of the star operator is the case n = 3, when it can be taken as the correspondence between the vectors and the skew-symmetric matrices of that size. This is used implicitly in vector calculus, for example to create the cross product vector from the wedge product of two vectors. Specifically, for Euclidean R3, one easily finds that
where dx, dy and dz are the standard orthonormal differential one-forms on R3. The Hodge dual in this case clearly relates the cross-product to the wedge product in three dimensions. A detailed presentation not restricted to differential geometry is provided next.
Read more about this topic: Hodge Dual
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