Question

Show that a necessary and sufficient condition for the set of quantities pi...k (with P suffixes) to form a covariant tensor of rank P is that pij...ka'b'...ck should be invariant for every set of P contravariant vectors a, bV ,... , ck

Answer 1

any enty arbitrary tensor Ccovariant or According to auotient laa ahose inner product contavaiant) is a tensor, is itself a tensor So, here if invariant in every say that ij.-form the inner prYoduct is then a show that he product a'bi c co-ordinate System inner We C K ís -then We Covaiant tensor. Because scalar .e tensor of rank zero a can t0-OYdinate system Noid in anolhen innerproduct the can be wDYitten as

K ij JAy.. Y a j Le x xe e or a b Y Since dummy inde we Can onelude that the inner prouct 4i ab.k is obrdinate Aystem ie it is invariant M, and Y independant of