Subsections

The dot product gradient of the ellipsoid

The partial derivative of the dot product δi with respect to the orientational parameter $ \mathfrak{O}_j$ is

$\displaystyle {\frac{{\partial \delta_i}}{{\partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{\partial}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left(\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_i} }\right.$$\displaystyle \widehat{{XH}}$$\displaystyle \widehat{{\mathfrak{D}_i}}$$\displaystyle \left.\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_i} }\right)$
= $\displaystyle \widehat{{XH}}$$\displaystyle {\frac{{\partial \widehat{\mathfrak{D}_i}}}{{\partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \widehat{XH}}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \widehat{{\mathfrak{D}_i}}$.
(15.172)

Because $ \widehat{{XH}}$ is constant and not dependent on the Euler angles its derivative is zero. Therefore

$\displaystyle {\frac{{\partial \delta_i}}{{\partial \mathfrak{O}_j}}}$ = $\displaystyle \widehat{{XH}}$$\displaystyle {\frac{{\partial \widehat{\mathfrak{D}_i}}}{{\partial \mathfrak{O}_j}}}$. (15.173)

The Dx gradient

The partial derivatives of the unit vector $ \widehat{{\mathfrak{D}_x}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}\frac{\partial \widehat{\mathfrak{D}_x}}{\parti...
...lpha \cos \beta \cos \gamma \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}

The Dy gradient

The partial derivatives of the unit vector $ \widehat{{\mathfrak{D}_y}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}\frac{\partial \widehat{\mathfrak{D}_y}}{\parti...
...lpha \cos \beta \cos \gamma \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}

The Dz gradient

The partial derivatives of the unit vector $ \widehat{{\mathfrak{D}_z}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}\frac{\partial \widehat{\mathfrak{D}_z}}{\parti...
...a \\ \sin \beta \cos \gamma \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}

The relax user manual (PDF), created 2016-10-28.