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.

$\begin{array}{l}{{\mathbf{\text{W}}}_{\mathbf{\text{i}}}}^{\mathbf{\text{'}}}{\mathbf{\text{=a}}}_{\mathbf{\text{ij}}}{\mathbf{\text{W}}}_{\mathbf{\text{j}}}\\ {{\mathbf{\text{S}}}_{\mathbf{\text{i}}}}^{\mathbf{\text{'}}}{\mathbf{\text{=(detA)a}}}_{\mathbf{\text{ij}}}{\mathbf{\text{S}}}_{\mathbf{\text{j}}}\end{array}$

Define the transformation laws.

Write the transformation laws assuming andare axial and polar vectors.

$\begin{array}{l}{W^{\text{'}}}_i={\text{a}}_{\text{ij}}{\text{W}}_{\text{j}}\\ {{\text{S}}_{\text{i}}}^{\text{'}}={\text{(detA)a}}_{\text{ij}}{\text{S}}_{\text{j}}\end{array}$

Continue the evaluations.

Write the implication of the transformation laws.

$\begin{array}{c}{(W^{\text{'}}\times S^{\text{'}})}_{\alpha }={ϵ}_{\text{αβγ}}^{\text{'}}{\text{W}}_{\text{β}}^{\text{'}}{\text{S}}_{\text{γ}}^{\text{'}}\\ =(\det {\text{A)a}}_{\text{αi}}{\text{a}}_{\text{βj}}{\text{a}}_{\text{γk}}{ϵ}_{\text{ijk}}{\text{a}}_{\text{βm}}{\text{W}}_{\text{m}}{\text{(detA)a}}_{\text{γm}}{\text{S}}_{\text{n}}\\ ={\text{a}}_{\text{ci}}{ϵ}_{\text{ijk}}{\text{W}}_{\text{j}}{\text{S}}_{\text{k}}\\ =\text{a}{\left(w\times S\right)}_{\text{i}}S\end{array}$

This implies that W Sis a polar vector