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1. (a) Remembering the distinction between summation indices and free indices, look at the following equations and state whether they conform to tensor notation, and if not why not:
(i) Tmn=Am^nB
(ii) Uij^i=Ai^kDk
(iii) Vjk^ii=Ajk
(iv) Ai^j=Xi^iC^j+Yi^j
(b) (i) Write out in full the equations bi=djgi^j (d-differential) in a 2-dimensional space.
(ii) If g^ij is the inverse of the metric tensor gij and di=d/dx^i, what are the values of the components of bi=djgi^j ?
(c) For this part, you should use the tensor transformation rules for a contravariant and covariant vector, and for a second-rank contravariant tensor:
P^i=(dx^i/dx^a)P^a, Pi=(dx^a/dx^i)Pa, T^ij=(dx^i/dx^a)(dx^j/dx^b)T^ab
, , .
(i) If A^i and B^j are contravariant vectors, prove that transforms as A^iB^j a contravariant second-rank tensor.
(ii) If A^i is a contravariant vector and Ci is a covariant vector, prove that A^iCi is a scalar field.
(iii) If T^ij is a skew-symmetric contravariant second-rank tensor, prove that its skew symmetry property is invariant under tensor transformations.
Anyone who can help me with that?
Many thanks
Mary
(i) Tmn=Am^nB
(ii) Uij^i=Ai^kDk
(iii) Vjk^ii=Ajk
(iv) Ai^j=Xi^iC^j+Yi^j
(b) (i) Write out in full the equations bi=djgi^j (d-differential) in a 2-dimensional space.
(ii) If g^ij is the inverse of the metric tensor gij and di=d/dx^i, what are the values of the components of bi=djgi^j ?
(c) For this part, you should use the tensor transformation rules for a contravariant and covariant vector, and for a second-rank contravariant tensor:
P^i=(dx^i/dx^a)P^a, Pi=(dx^a/dx^i)Pa, T^ij=(dx^i/dx^a)(dx^j/dx^b)T^ab
, , .
(i) If A^i and B^j are contravariant vectors, prove that transforms as A^iB^j a contravariant second-rank tensor.
(ii) If A^i is a contravariant vector and Ci is a covariant vector, prove that A^iCi is a scalar field.
(iii) If T^ij is a skew-symmetric contravariant second-rank tensor, prove that its skew symmetry property is invariant under tensor transformations.
Anyone who can help me with that?
Many thanks
Mary
