Calculations in Coordinates

  • Klaus Jänich
Part of the Undergraduate Texts in Mathematics book series (UTM)


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:


Vector Field Tensor Product Component Function Summation Convention Wedge Product 
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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Klaus Jänich
    • 1
  1. 1.NWF-I MathematikUniversität RegensburgRegensburgGermany

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