I'm working with transformations in orthogonal curvilinear systems and I'm trying to understand the relationships between the coefficients of vectors before and after the transformation.
Vectors remain invariant under transformation. Hence, we have: $$ \mathbf{B'} = \sum_i B_i' \mathbf{e}_i' = \sum_j B_j \mathbf{e}_j $$ Where $\mathbf{e}_i'$ are the unit vectors of the transformed (orthogonal curvilinear) system.
The relationship between the new and old basis vectors is given by: $$ \mathbf{e}_i' = \sum_j a_{ji} \mathbf{e}_j $$ Here, $a_{ji}$ represents the $ji$ component of the transformation matrix.
With the information above, I'm trying to deduce the following relation: $$ B_i' = \sum_{j=1}^3 a_{ij} B_j $$ Any insights or steps to arrive at this conclusion would be greatly appreciated.
Let summation over doubly-repeated indices be implicit. Then
$$ \begin{align} B_i'\hat{e}_i'&=B_j\hat{e}_j\\ B_i'a_{ji}\hat{e}_j&=B_j\hat{e}_j\\ 0&=\left(B_j-B_i'a_{ji}\right)\hat{e}_j \end{align} $$
By the linear independence of the $\{\hat{e}_i\}$
$$ B_j=a_{ji}B_i' \implies \vec{B}=A^T\vec{B}' $$
For an orthogonal transformation $AA^T=1$ and $\vec{B}'=A\vec{B}$.