Subsections

The dot product Hessian of the spheroid

The second partial derivative of the single spheroidal dot product δz with respect to the orientational parameters $\mathfrak{O}_i$ and $\mathfrak{O}_j$ is

$\displaystyle {\frac{{\partial^2 \delta_z}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{\partial^2}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$$\displaystyle \left(\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_{\scriptscriptstyle \parallel}} }\right.$$\displaystyle \widehat{{XH}}$$\displaystyle \widehat{{\mathfrak{D}_{\scriptscriptstyle \parallel}}}$$\displaystyle \left.\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_{\scriptscriptstyle \parallel}} }\right)$
= $\displaystyle \widehat{{XH}}$$\displaystyle {\frac{{\partial^2 \widehat{\mathfrak{D}_{\scriptscriptstyle \parallel}}}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$.
(15.186)

The Dpar Hessian

The second partial derivatives of the unit vector $\widehat{{\mathfrak{D}_{\scriptscriptstyle \parallel}}}$ with respect to the spherical angles are
\begin{subequations}\begin{align}
\frac{\partial^2 \widehat{\mathfrak{D}_{\scrip...
...\
-\sin \theta \sin \phi \\
0 \\
\end{pmatrix}.
\end{align}\end{subequations}



The relax user manual (PDF), created 2019-03-08.