Chapter 10: Q9P (page 505)

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

Short Answer

Expert verified

Answer

The equation has been proven.

Step by step solution

Given Information

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

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

Apply transformation law on A and B.

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

Let $S_{α} = (X'_{α} - a_{α i} X_{i}) A'_{α β}$

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

The equation mentioned above is true for any matrix A.

Thus, X is a tensor.

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

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove that X is a tensor.

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XiAij=Bj.

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

Physics of Motion

Wave Optics

Medical Physics

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Electrostatics

Study anywhere. Anytime. Across all devices.

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