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(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