J Contravariant Transformation

12 The vector representation in both systems must be the same

V = Vqbq = Vkbk = Vk (bkbq) ^ (v* - Vk b* )bq = 0 (1.23)

since the base vectors bq are linearly independent, the coefficients of bq must all be zero hence

showing that the forward change from components vk to vq used the coefficients bqk of the backward change from base bq to the original bk. This is why these components are called contravariant.

13 Generalizing, a Contravariant Tensor of order one (recognized by the use of the superscript) transforms a set of quantities rk associated with point P in xk through a coordinate transformation into a new set rq associated with xq _

14 By extension, the Contravariant tensors of order two requires the tensor components to obey the following transformation law

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