User:Eml4500.f08.gravy.bje/Notes7

$$\ \underline{\tilde{k}}^{(e)} \underline{\tilde{d}}^{(e)} = \underline{\tilde{f}}^{(e)} \,$$

$$\underline{k}^{(e)} = \underline{\tilde{T}}^{(e)T} \underline{\tilde{k}}^{(e)} \underline{\tilde{T}}^{(e)} $$

$$ \underline{\tilde{T}}^{(e)} = \begin{bmatrix} l^{(e)} & -m^{(e)} & 0 & 0 & 0 & 0\\ m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & l^{(e)} & -m^{(e)} & 0 \\ 0 & 0 & 0 & m^{(e)} & l^{(e)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}_{6x6}\,$$

$$ \underline{\tilde{T}}^{(e)T} = \begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0 & 0 & 0\\ -m^{(e)} & l^{(e)} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & l^{(e)} & m^{(e)} & 0 \\ 0 & 0 & 0 & -m^{(e)} & l^{(e)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}_{6x6}\,$$

$$\ l^{(e)}= \vec \tilde{i} \cdot \vec i= cos\theta^{(e)}\,$$

$$\ m^{(e)}= \vec i \cdot \vec j= cos\left (\frac{\pi}{2}- \theta^{(e)} \right )= sin\left (\theta^{(e)} \right )\,$$

$$ \underline{\tilde{T}}^{(e)T} * \underline{\tilde{T}}^{(e)} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}_{6x6} $$

Similarly

$$\underline{d}^{(e)} = \underline{\tilde{T}}^{(e)T} \underline{\tilde{d}}^{(e)} \underline{\tilde{T}}^{(e)} $$

And

$$\underline{f}^{(e)} = \underline{\tilde{T}}^{(e)T} \underline{\tilde{f}}^{(e)} \underline{\tilde{T}}^{(e)} $$

Therefore:

$$\underline{f}^{(e)} = \underline{\tilde{k}}^{(e)} \underline{\tilde{d}}^{(e)} \underline{\tilde{T}}^{(e)} \underline{\tilde{T}}^{(e)T} $$\

Which is equal to:

$$\ \underline{k}^{(e)} \underline{d}^{(e)} = \underline{f}^{(e)} \,$$