Chapter 10: Q7P (page 505)

As in problem 6, show that the sum of two $2^{n d}$ -rank tensors is a $2^{n d}$ -rank tensor; that the sum of two $4^{t h}$ -rank tensors is a $4^{t h}$ -rank tensor.

Short Answer

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Answer

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Step by step solution

Given Information

The two $2^{n d}$ -rank tensor and $4^{t h}$ - rank tensor.

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.

Let T and S be the two n-rank tensor.

$(T + S)'_{i_{1}, i_{2}, . . i_{n}} = T'_{i_{1}, i_{2}, . . . i_{n}} + S'_{i_{1}, i_{2}, . . i_{n}} (T + S)'_{i_{1}, i_{2}, . . i_{n}} = a_{i_{1} j_{1} . . .} a_{j_{n} j_{n}} T_{j_{1} . . . j_{n}} + a_{i_{1} j_{1}} . . . . a_{i_{n} j_{n}} S'_{j_{1} j_{n}, . . i_{n}} (T + S)'_{i_{1}, i_{2}, . . i_{n}} = T'_{j_{1}, i_{1} . . . .} a_{i_{n} j_{n}} {(T + S)}_{j_{1} . . . . j_{n}}$

Hence, sum of the two n-rank tensor is n-rank.

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As in problem 6, show that the sum of two $2^{n d}$ -rank tensors is a $2^{n d}$ -rank tensor; that the sum of two $4^{t h}$ -rank tensors is a $4^{t h}$ -rank tensor.

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

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As in problem 6, show that the sum of two 2nd-rank tensors is a 2nd-rank tensor; that the sum of two 4th-rank tensors is a 4th-rank tensor.

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

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Study anywhere. Anytime. Across all devices.

As in problem 6, show that the sum of two $2^{n d}$ -rank tensors is a $2^{n d}$ -rank tensor; that the sum of two $4^{t h}$ -rank tensors is a $4^{t h}$ -rank tensor.