Question

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

Short answer

The transformation equation is ${(\nabla \times V{)}_{\alpha }}^{\text{'}}=(detA)a_{\alpha }i(\nabla \times V{)}_i$

Step-by-step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$\left[\begin{array}{cc}cos\varphi & -sin\varphi \\ sin\varphi & cos\varphi \end{array}\right]$

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.

$\frac{\partial }{\partial {x_i}^{\text{'}}}=a_{i}j\frac{\partial }{\partial x_j}$

Continue evaluations.

Continue with evaluations.

$$\begin{gathered} {\left(\nabla \times V\right)}_{\alpha }\text{'}=\epsilon {\text{'}}_{\alpha \beta \gamma }\frac{\partial }{\partial X_{\beta }\text{'}}{V}_{\gamma }\text{'} \\ =\left(detA\right)a_{\alpha i}{a}_{\beta j}{a}_{\gamma k}{\epsilon }_{ijk}{a}_{\beta m}\frac{\partial }{\partial X_m}{a}_{\gamma n}{v}_n \\ =\left(detA\right)a_{\alpha i}{a}_{\beta j}{a}_{\beta m}{a}_{\gamma k}{a}_{\gamma n}{\epsilon }_{ijk}\frac{\partial }{\partial X_m}{v}_n \end{gathered}$$

Continue the simplification.

$$\begin{gathered} {\left(\nabla \times V\right)}_{\alpha }\text{'}=\left(detA\right)a_{\alpha i}{\partial }_{jm}{\partial }_{kn}{\epsilon }_{ijk}\frac{\partial }{\partial X_m}{v}_n \\ =\left(detA\right)a_{\alpha i}{\epsilon }_{ijk}\frac{\partial }{\partial X_m}{v}_k \\ =\left(detA\right)a_{\alpha i}{\left(\nabla \times V\right)}_i \end{gathered}$$

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

$$\begin{gathered} a_{\beta }j{a}_{\beta }m={\delta }_{j}m \\ {A}^{T}A=I \end{gathered}$$

Therefore, write the result.

${(\nabla \times V{)}_{\alpha }}^{\text{'}}=\left(detA\right)a_{\alpha }i(\nabla \times V{)}_i$