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}}{\part...
...ha \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}}{\part...
...ha \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}}{\part...
...\
\sin \beta \cos \gamma \\
0 \\
\end{pmatrix}.
\end{align}\end{subequations}

The relax user manual (PDF), created 2019-06-14.