Calculations in Coordinates
In this last chapter we examine how to calculate with the star operator and coderivative in local coordinates on semi-Riemannian manifolds. But first we pick up where we left off in Chapter 10 and consider the simple but important example M = ℝ3 with the usual coordinates x 1, x 2, x 3, the usual orientation, and the usual scalar product (denoted by the multiplication symbol ·). The index is zero and k(3 − k) is always even, so by Note 3 in Section 12.3 the star operator is an involution: ** = Id. Note 1 in the same section gives the following:
KeywordsVector Field Tensor Product Component Function Summation Convention Wedge Product
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