Chapter 10: Q5P (page 505)

Show that $T_{i j k l m} S_{l m}$ is a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).

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Step by step solution

Given Information

The tensor $T_{ijklm} S_{lm}$
.

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.

Prove the statement.

The tensor $T_{i j k l m} S_{I m}$

The number of dummy index pairs in $T_{i j k l m} S_{I m}$ is 2, double contraction.

Each contraction reduces the rank of tensor by 2.

So, the rank of the tensor be $5 + 2 - 2 \times 2 = 3$ .

Prove that the rank is 3.

$(T_{i' j' l' m'} S_{l' m'} = a_{i' i'} a_{j' j'} a_{k' k'} a_{l' l'} a_{m' m'} a_{l' l'} a_{m' m'} T_{i j k l m} S_{l m} (T_{i' j' l' m'} S_{l' m'} = a_{i' i'} a_{j' j'} a_{k' k'} δ_{l' l'} δ_{m' m'} a_{l' l'} a_{m' m'} T_{i j k l m} S_{l m} (T_{i' j' l' m'} S_{l' m'} = a_{i' i'} a_{j' j'} a_{k' k'} T_{i j k l m} S_{l m j_{1} . . . j_{n}}$

Hence, the rank of the tensor is 3.

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Show that $T_{i j k l m} S_{l m}$ is a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

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Step by step solution

Given Information

Definition of a cartesian tensor.

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Show that $T_{i j k l m} S_{l m}$ is a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).