Subsections

The dot product Hessian of the ellipsoid

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

$\displaystyle {\frac{{\partial^2 \delta_i}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$ = $\displaystyle {\frac{{\partial^2}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$$\displaystyle \left(\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_i} }\right.$$\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^2 \widehat{\mathfrak{D}_i}}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$.
(15.177)

The Dx Hessian

The second partial derivatives of the unit vector $\widehat{{\mathfrak{D}_x}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}
\frac{\partial^2 \widehat{\mathfrak{D}_x}}{\pa...
...ha \cos \beta \sin \gamma \\
0 \\
\end{pmatrix}.
\end{align}\end{subequations}

The Dy Hessian

The second partial derivatives of the unit vector $\widehat{{\mathfrak{D}_y}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}
\frac{\partial^2 \widehat{\mathfrak{D}_y}}{\pa...
...ha \cos \beta \sin \gamma \\
0 \\
\end{pmatrix}.
\end{align}\end{subequations}

The Dz Hessian

The second partial derivatives of the unit vector $\widehat{{\mathfrak{D}_z}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}
\frac{\partial^2 \widehat{\mathfrak{D}_z}}{\pa...
...
-\sin \beta \sin \gamma \\
0 \\
\end{pmatrix}.
\end{align}\end{subequations}

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