Homework Set # 6 - Due October 15
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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 gij=εi.εj.
Solution
(pdf file)
7) Derive
the covariant and contravariant metric tensors in spherical
coordinates.