Subsections

The weight gradients of the ellipsoid

$\mathfrak{O}_i$ partial derivative

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

\begin{subequations}\begin{align}
\frac{\partial c_{-2}}{\partial \mathfrak{O}_i...
...ght) + \frac{\partial e}{\partial \mathfrak{O}_i},
\end{align}\end{subequations}

where

$\displaystyle {\frac{{\partial e}}{{\partial \mathfrak{O}_i}}}$ = $\displaystyle {\frac{{1}}{{\mathfrak{R}}}}$$\displaystyle \Bigg[$(1 + 3$\displaystyle \mathfrak{D}_r) \left(
\delta_x^3 \frac{\partial \delta_x}{\parti...
...i} + \delta_z \frac{\partial \delta_y}{\partial \mathfrak{O}_i} \right) \right)$      
+ (1 - 3$\displaystyle \mathfrak{D}_r) \left(
\delta_y^3 \frac{\partial \delta_y}{\parti...
...i} + \delta_z \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) \right)$      
-2$\displaystyle \left(\vphantom{
\delta_z^3 \frac{\partial \delta_z}{\partial \ma...
...} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) }\right.$δz3$\displaystyle {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}}$ + δxδy$\displaystyle \left(\vphantom{ \delta_x \frac{\partial \delta_y}{\partial \mathfrak{O}_i} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} }\right.$δx$\displaystyle {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_i}}}$ + δy$\displaystyle {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_i}}}$$\displaystyle \left.\vphantom{ \delta_x \frac{\partial \delta_y}{\partial \mathfrak{O}_i} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} }\right)$$\displaystyle \left.\vphantom{
\delta_z^3 \frac{\partial \delta_z}{\partial \ma...
...} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) }\right)$ $\displaystyle \Bigg]$. (15.119)

τm partial derivative

The partial derivatives with respect to the τm geometric parameter are

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

$\mathfrak{D}_a$ partial derivative

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

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

$\mathfrak{D}_r$ partial derivative

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

\begin{subequations}\begin{align}
\frac{\partial c_{-2}}{\partial \mathfrak{D}_r...
...{3}{4} \frac{\partial e}{\partial \mathfrak{D}_r},
\end{align}\end{subequations}

where

$\displaystyle {\frac{{\partial e}}{{\partial \mathfrak{D}_r}}}$ = $\displaystyle {\frac{{1}}{{\mathfrak{R}^3}}}$$\displaystyle \bigg[$(1 - $\displaystyle \mathfrak{D}_r) \left(\delta_x^4 + 2\delta_y^2\delta_z^2\right)
-...
...ght) + 2 \mathfrak{D}_r \left(\delta_z^4 + 2\delta_x^2\delta_y^2\right) \bigg].$ (15.123)

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