Question

Write the transformation equations for$\mathit{W}\mathbf{\times }\mathit{S}$ to verify the results of Example 3.

Short answer

This answer proves that $W\times S$ is a polar vector.

Step-by-step solution

Given information.

Matrix definitions are given.

Definition of transformation laws.

Define the transformation laws.

$$\begin{gathered} {{\mathit{W}}_{\mathbf{i}}}^{\mathbf{\text{'}}}\mathbf{=}{\mathit{a}}_{\mathbf{i}\mathbf{j}}{\mathit{W}}_{\mathbf{j}} \\ {{\mathit{S}}_{\mathbf{i}}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{\left(}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{\right)}{\mathit{a}}_{\mathit{i}\mathit{j}}{\mathit{S}}_{\mathbf{j}} \end{gathered}$$

Define the transformation laws.

Write the transformation laws assuming and are axial and polar vectors.

$$\begin{gathered} {W_{\mathrm{i}}}^{\text{'}}=a_{\mathrm{ij}}{W}_{\mathrm{j}} \\ {S_{\mathrm{i}}}^{\text{'}}=\left(detA\right)a_{ij}{S}_{\mathrm{j}} \end{gathered}$$

Continue the evaluations.

Write the implication of the transformation laws.

$$\begin{gathered} {\left(W\text{'}\times S\text{'}\right)}_{\alpha }=\epsilon {\text{'}}_{\alpha \beta \gamma }{W}_{\beta }\text{'}{S}_{\gamma }\text{'} \\ =\left(detA\right)a_{\alpha i}{a}_{\beta j}{a}_{\gamma k}{\epsilon }_{ijk}{a}_{\beta m}{w}_m\left(detA\right)a_{\gamma m}{S}_n \\ ={\left(detA\right)}^2{a}_{ci}{\epsilon }_{ijk}wj{S}_k \\ =a_{\alpha i}{\left(W\times S\right)}_{I_{}} \end{gathered}$$

This implies that $W\times S$ is a polar vector.