Derive the component manipulation rules (1.9g) and (1.9h). In component notation, the inner product of two vectors

Derive the component manipulation rules (1.9g) and (1.9h).

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In component notation, the inner product of two vectors and the value of a tensor when vectors are inserted into its slots are given by A.B=A; B₁, T(A, B, C) = Tijk AjBjCk› (1.9g) as one can easily show using previous equations. Finally, the contraction of a tensor [say, the fourth-rank tensor R(______)] on two of its slots (say, the first and third) has components that are easily computed from the tensor's own components: components of [1&3contraction of R] = Rijik. (1.9h) Note that Rijik is summed on the i index, so it has only two free indices, j and k, and thus is the component of a second-rank tensor, as it must be if it is to represent the contraction of a fourth-rank tensor.