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

Electrostatics

Astrophysics

Torque and Rotational Motion

Fluids

Magnetism

Thermodynamics

Study anywhere. Anytime. Across all devices.

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