Chapter 10: Q12P (page 505)

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

Short Answer

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Answer

The equation has been proven.

Step by step solution

Given Information

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

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

Apply transformation law on A and B.

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

Let $S_{α} = (X'_{α β ω} - a_{α i} a_{β j} a_{γ k} a_{n l} X_{i j k l}) A'_{α β}$

Hence, $S^{T} A = 0$

The equation mentioned above is true for any matrix A.

Thus, X is a tensor.

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

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove that X is a tensor.

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XijklAkl=Bij.

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

Engineering Physics

Space Physics

Fundamentals of Physics

Magnetism and Electromagnetic Induction

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Fields in Physics

Study anywhere. Anytime. Across all devices.

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