Subsections

The weight Hessians of the ellipsoid

$ \mathfrak{O}_i$ - $ \mathfrak{O}_j$ partial derivative

The second partial derivatives with respect to the orientational parameters $ \mathfrak{O}_i$ and $ \mathfrak{O}_j$ are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \mathfrak{O}_...
...ial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}, \end{align}\end{subequations}\begin{subequations}\begin{multline}\frac{\partial^2 c_{-1}}{\partial \mathfrak{...
...ial \delta_y}{\partial \mathfrak{O}_j} \right), \end{multline}\end{subequations}\begin{subequations}\begin{multline}\frac{\partial^2 c_{0}}{\partial \mathfrak{O...
...ial \delta_x}{\partial \mathfrak{O}_j} \right), \end{multline}\end{subequations}\begin{subequations}\begin{multline}\frac{\partial^2 c_{1}}{\partial \mathfrak{O...
...ial \delta_x}{\partial \mathfrak{O}_j} \right), \end{multline}\end{subequations}\begin{subequations}\begin{align}\frac{\partial^2 c_{2}}{\partial \mathfrak{O}_i...
...ial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}, \end{align}\end{subequations}

where

$\displaystyle {\frac{{\partial^2 e}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{1}}{{\mathfrak{R}}}}$$\displaystyle \Bigg[$(1 + 3$\displaystyle \mathfrak{D}_r) \Bigg( \delta_x^2 \left( \delta_x \frac{\partial^...
...cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} \right) \phantom{\Bigg)}$      
+ δy2$\displaystyle \left(\vphantom{ \delta_z \frac{\partial^2 \delta_z}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_z}{\partial \mathfrak{O}_j} }\right.$δz$\displaystyle {\frac{{\partial^2 \delta_z}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_z \frac{\partial^2 \delta_z}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_z}{\partial \mathfrak{O}_j} }\right)$      
+ δz2$\displaystyle \left(\vphantom{ \delta_y \frac{\partial^2 \delta_y}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} }\right.$δy$\displaystyle {\frac{{\partial^2 \delta_y}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_y \frac{\partial^2 \delta_y}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} }\right)$      
+2δyδz$\displaystyle \left(\vphantom{ \frac{\partial \delta_y}{\partial \mathfrak{O}_i...
...mathfrak{O}_i} \cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} }\right.$$\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \frac{\partial \delta_y}{\partial \mathfrak{O}_i...
...mathfrak{O}_i} \cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} }\right)$$\displaystyle \Bigg)$      
+ (1 - 3$\displaystyle \mathfrak{D}_r) \Bigg( \delta_y^2 \left( \delta_y \frac{\partial^...
...cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} \right) \phantom{\Bigg)}$      
+ δx2$\displaystyle \left(\vphantom{ \delta_z \frac{\partial^2 \delta_z}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_z}{\partial \mathfrak{O}_j} }\right.$δz$\displaystyle {\frac{{\partial^2 \delta_z}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_z \frac{\partial^2 \delta_z}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_z}{\partial \mathfrak{O}_j} }\right)$      
+ δz2$\displaystyle \left(\vphantom{ \delta_x \frac{\partial^2 \delta_x}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right.$δx$\displaystyle {\frac{{\partial^2 \delta_x}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_x \frac{\partial^2 \delta_x}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right)$      
+2δxδz$\displaystyle \left(\vphantom{ \frac{\partial \delta_x}{\partial \mathfrak{O}_i...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right.$$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \frac{\partial \delta_x}{\partial \mathfrak{O}_i...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right)$$\displaystyle \Bigg)$      
-2$\displaystyle \Bigg($δz2$\displaystyle \left(\vphantom{ \delta_z \frac{\partial^2 \delta_z}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_z}{\partial \mathfrak{O}_j} }\right.$δz$\displaystyle {\frac{{\partial^2 \delta_z}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ +3$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_z \frac{\partial^2 \delta_z}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_z}{\partial \mathfrak{O}_j} }\right)$      
+ δx2$\displaystyle \left(\vphantom{ \delta_y \frac{\partial^2 \delta_y}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} }\right.$δy$\displaystyle {\frac{{\partial^2 \delta_y}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_y \frac{\partial^2 \delta_y}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_y}{\partial \mathfrak{O}_j} }\right)$      
+ δy2$\displaystyle \left(\vphantom{ \delta_x \frac{\partial^2 \delta_x}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right.$δx$\displaystyle {\frac{{\partial^2 \delta_x}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \delta_x \frac{\partial^2 \delta_x}{\partial \ma...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right)$      
+2δxδy$\displaystyle \left(\vphantom{ \frac{\partial \delta_x}{\partial \mathfrak{O}_i...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right.$$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_j}}}$ + $\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_i}}}$$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_j}}}$$\displaystyle \left.\vphantom{ \frac{\partial \delta_x}{\partial \mathfrak{O}_i...
...mathfrak{O}_i} \cdot \frac{\partial \delta_x}{\partial \mathfrak{O}_j} }\right)$$\displaystyle \Bigg)$ $\displaystyle \Bigg]$. (15.125)

$ \mathfrak{O}_i$ - τm partial derivative

The second partial derivatives with respect to the orientational parameter $ \mathfrak{O}_i$ and the geometric parameter τm are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \mathfrak{O}_...
...artial \mathfrak{O}_i \cdot \partial \tau_m} &= 0. \end{align}\end{subequations}

$ \mathfrak{O}_i$ - $ \mathfrak{D}_a$ partial derivative

The second partial derivatives with respect to the orientational parameter $ \mathfrak{O}_i$ and the geometric parameter $ \mathfrak{D}_a$ are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \mathfrak{O}_...
...mathfrak{O}_i \cdot \partial \mathfrak{D}_a} &= 0. \end{align}\end{subequations}

$ \mathfrak{O}_i$ - $ \mathfrak{D}_r$ partial derivative

The second partial derivatives with respect to the orientational parameter $ \mathfrak{O}_i$ and the geometric parameter $ \mathfrak{D}_r$ are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \mathfrak{O}_...
...ial \mathfrak{O}_i \cdot \partial \mathfrak{D}_r}, \end{align}\end{subequations}

where

$\displaystyle {\frac{{\partial^2 e}}{{\partial \mathfrak{O}_i \cdot \partial \mathfrak{D}_r}}}$ = $\displaystyle {\frac{{1}}{{\mathfrak{R}^3}}}$$\displaystyle \Bigg[$(1 - $\displaystyle \mathfrak{D}_r) \left( \delta_x^3 \frac{\partial \delta_x}{\parti...
...i} + \delta_z \frac{\partial \delta_y}{\partial \mathfrak{O}_i} \right) \right)$      
- (1 + $\displaystyle \mathfrak{D}_r) \left( \delta_y^3 \frac{\partial \delta_y}{\parti...
...i} + \delta_z \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) \right)$      
+2$\displaystyle \mathfrak{D}_r \left( \delta_z^3 \frac{\partial \delta_z}{\partia...
...i} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) \right)$ $\displaystyle \Bigg]$. (15.129)

τm - τm partial derivative

The second partial derivatives with respect to the geometric parameter τm twice are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{{\partial \tau_m}^2} &...
...\frac{\partial^2 c_{2}}{{\partial \tau_m}^2} &= 0. \end{align}\end{subequations}

τm - $ \mathfrak{D}_a$ partial derivative

The second partial derivatives with respect to the geometric parameters τm and $ \mathfrak{D}_a$ are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \tau_m \cdot ...
...artial \tau_m \cdot \partial \mathfrak{D}_a} &= 0. \end{align}\end{subequations}

τm - $ \mathfrak{D}_r$ partial derivative

The second partial derivatives with respect to the geometric parameters τm and $ \mathfrak{D}_r$ are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \tau_m \cdot ...
...artial \tau_m \cdot \partial \mathfrak{D}_r} &= 0. \end{align}\end{subequations}

$ \mathfrak{D}_a$ - $ \mathfrak{D}_a$ partial derivative

The second partial derivatives with respect to the geometric parameter $ \mathfrak{D}_a$ twice are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{{\partial \mathfrak{D}...
...artial^2 c_{2}}{{\partial \mathfrak{D}_a}^2} &= 0. \end{align}\end{subequations}

$ \mathfrak{D}_a$ - $ \mathfrak{D}_r$ partial derivative

The second partial derivatives with respect to the geometric parameters $ \mathfrak{D}_a$ and $ \mathfrak{D}_r$ are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{\partial \mathfrak{D}_...
...mathfrak{D}_a \cdot \partial \mathfrak{D}_r} &= 0. \end{align}\end{subequations}

$ \mathfrak{D}_r$ - $ \mathfrak{D}_r$ partial derivative

The second partial derivatives with respect to the geometric parameter $ \mathfrak{D}_r$ twice are

\begin{subequations}\begin{align}\frac{\partial^2 c_{-2}}{{\partial \mathfrak{D}...
...4} \frac{\partial^2 e}{\partial \mathfrak{D}_r^2}, \end{align}\end{subequations}

where

$\displaystyle {\frac{{\partial^2 e}}{{\partial \mathfrak{D}_r^2}}}$ = $\displaystyle {\frac{{1}}{{\mathfrak{R}^5}}}$$\displaystyle \bigg[$(6$\displaystyle \mathfrak{D}_r^2 - 9\mathfrak{D}_r - 1) \left(\delta_x^4 + 2\delta_y^2\delta_z^2\right)$      
+ (6$\displaystyle \mathfrak{D}_r^2 + 9\mathfrak{D}_r - 1) \left(\delta_y^4 + 2\delta_x^2\delta_z^2\right)$      
-2(6$\displaystyle \mathfrak{D}_r^2 - 1) \left(\delta_z^4 + 2\delta_x^2\delta_y^2\right)$ $\displaystyle \bigg]$. (15.136)

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