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Chapter 10: Q5P (page 505)

Show that TijklmSlmis a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).

Short Answer

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Answer

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Step by step solution

01

Given Information

The tensor TijklmSlm
.

02

Definition of a cartesian tensor.

The first rank tensor is just a vector. A tensor of the second rank has nine components (in three dimensions) in every rectangular coordinate system.

03

Prove the statement.

The tensor TijklmSIm

The number of dummy index pairs in TijklmSImis 2, double contraction.

Each contraction reduces the rank of tensor by 2.

So, the rank of the tensor be 5+2-2脳2=3.

Prove that the rank is 3.

(Ti'j'l'm'Sl'm'=ai'i'aj'j'ak'k'al'l'am'm'al'l'am'm'TijklmSlm(Ti'j'l'm'Sl'm'=ai'i'aj'j'ak'k'未l'l'未m'm'al'l'am'm'TijklmSlm(Ti'j'l'm'Sl'm'=ai'i'aj'j'ak'k'TijklmSlmj1...jn

Hence, the rank of the tensor is 3.

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