Homework Set # 6 - Due October 14

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     1)  a) Express the components of a cross-product vector C, C=AXB, in terms of εijk and the components of A and B.

b) Use the antisymmetry of εijk to show that A.AXB=0.

Solution (pdf file)

     2)  4.2.3

     3)  Expressing cross products in terms of Levi-Civita symbols (
εijk), derive the BAC-CAB rule, Eq.(3.18).

Solution (pdf file)

     4)  4.2.7

     5) Generalize the cross product of two vectors to n-dimensional space for n=4, 5, .... Check the consistency of your construction and discuss concrete exemples.

  Solution (pdf file)

     6) Prove that the contravariant tensor is given by g
ij=εi.εj.

Solution (pdf file)

     7) Derive the covariant and contravariant metric tensors in spherical coordinates.