Chapter 10: Q11P (page 505)

$X_{i j} A_{k} = B_{i j k}$ .

Short Answer

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Answer

The equation has been proven.

Step by step solution

Given Information

The tensor $X_{i j} A_{k} = B_{i j k}$ .

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 that X is a tensor.

Let $X_{i j} A_{k} = B_{i j k}$ where B is a non-zero tensor.

Apply transformation law on A and B.

$0 = X'_{α β} A'_{γ} - A'_{α β γ} 0 = X'_{α β} A'_{γ} - a_{α j} a_{β j} a_{γ k} B_{i j k} 0 = X'_{α β} A'_{γ} - a_{α i} a_{α j} a_{β j} a_{γ k} X_{i j} A_{n} 0 = X'_{α β} A'_{γ} - a_{α i} a_{β j} a_{γ k} a_{n k} X_{i j} A'_{n}$

$0 = X'_{α β} A'_{γ} - a_{α i} δ_{γ k} a_{γ n} X_{i j} A'_{n} 0 = (X'_{α β} - a_{α i} a_{β i} X_{i j}) A'_{n}$

A is an arbitrary vector.

Hence $(X'_{α β} - a_{α i} a_{β j} X_{i j}) = 0$

Thus, X is a tensor.

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$X_{i j} A_{k} = B_{i j k}$ .

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove that X is a tensor.

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XijAk=Bijk.

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove that X is a tensor.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

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Study anywhere. Anytime. Across all devices.

$X_{i j} A_{k} = B_{i j k}$ .