Chapter 10: Q6P (page 517)

Write the transformation equations to show that $\nabla \times V$ is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).

Short Answer

Expert verified

The transformation equation is ${(\nabla \times V)_{α}}^{'} = (d e t A) a_{α} i (\nabla \times V)_{i}$

Step by step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$[\begin{matrix} c o s ϕ & - s i n ϕ \\ s i n ϕ & c o s ϕ \end{matrix}]$

Define a vector and an orthogonal matrix.

Define to be a vector. Define an orthogonal matrix denoting proper or improper rotation. Write the result for proper and improper rotations.

$\frac{\partial}{\partial {x_{i}}^{'}} = a_{i} j \frac{\partial}{\partial x_{j}}$

Continue evaluations.

Continue with evaluations.

${(\nabla \times V)}_{α}' = ε'_{α β γ} \frac{\partial}{\partial X_{β}'} V_{γ}' = (d e t A) a_{α i} a_{β j} a_{γ k} ε_{i j k} a_{β m} \frac{\partial}{\partial X_{m}} a_{γ n} v_{n} = (d e t A) a_{α i} a_{β j} a_{β m} a_{γ k} a_{γ n} ε_{i j k} \frac{\partial}{\partial X_{m}} v_{n}$

Continue the simplification.

${(\nabla \times V)}_{α}' = (d e t A) a_{α i} \partial_{j m} \partial_{k n} ε_{i j k} \frac{\partial}{\partial X_{m}} v_{n} = (d e t A) a_{α i} ε_{i j k} \frac{\partial}{\partial X_{m}} v_{k} = (d e t A) a_{α i} {(\nabla \times V)}_{i}$

This implies that $\nabla \times V$ is a pseudo vector. Since A is defined to be orthogonal, write its result.

$a_{β} j a_{β} m = δ_{j} m A^{T} A = I$

Therefore, write the result.

${(\nabla \times V)_{α}}^{'} = (d e t A) a_{α} i (\nabla \times V)_{i}$

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Write the transformation equations to show that $\nabla \times V$ is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Define a vector and an orthogonal matrix.

Continue evaluations.

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Write the transformation equations to show that ∇×Vis a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Define a vector and an orthogonal matrix.

Continue evaluations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

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Study anywhere. Anytime. Across all devices.

Write the transformation equations to show that $\nabla \times V$ is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).