Chapter 10: Q4P (page 505)

Show that the contracted tensor $T_{i j k} V_{k}$ is a $2^{n d}$ -rank tensors.

Short Answer

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Answer

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Step by step solution

Given Information

The contracted tensor $T_{i j k} V_{k}$ .

Definition of a cartesian tensor.

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

Prove the statement.

The contracted tensor $T_{i j k} V_{k}$ .

$T_{i j k} V_{k} = a_{i l} a_{j m} a_{k n} T I m n a_{k p} V_{p} T_{i j k} V_{k} = a_{i I} a_{j m} a_{k n} a_{k p} T_{I m n} V_{p} T_{i j k} V_{k} = a_{i I} a_{i m} δ_{p n} T_{I m n} V_{p} T_{i j k} V_{k} = a_{i j} a_{j m} T_{I m n} V_{n}$

Hence, $T_{k j}$ is a $2^{n d}$ rank vector.

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Show that the contracted tensor $T_{i j k} V_{k}$ is a $2^{n d}$ -rank tensors.

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

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Show that the contracted tensor TijkVkis a 2nd-rank tensors.

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

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Show that the contracted tensor $T_{i j k} V_{k}$ is a $2^{n d}$ -rank tensors.