The weight gradients of the ellipsoid

$\mathfrak{O}_i$ partial derivative

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

where

$${\frac{{\partial e}}{{\partial \mathfrak{O}_i}}} {\frac{{1}}{{\mathfrak{R}}}} \Bigg[ \mathfrak{D}_r) \left( \delta_x^3 \frac{\partial \delta_x}{\parti... ...i} + \delta_z \frac{\partial \delta_y}{\partial \mathfrak{O}_i} \right) \right)$$

$$\mathfrak{D}_r) \left( \delta_y^3 \frac{\partial \delta_y}{\parti... ...i} + \delta_z \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) \right)$$

$$\left(\vphantom{ \delta_z^3 \frac{\partial \delta_z}{\partial \ma... ...} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) }\right. {\frac{{\partial \delta_z}}{{\partial \mathfrak{O}_i}}} \left(\vphantom{ \delta_x \frac{\partial \delta_y}{\partial \mathfrak{O}_i} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} }\right. {\frac{{\partial \delta_y}}{{\partial \mathfrak{O}_i}}} {\frac{{\partial \delta_x}}{{\partial \mathfrak{O}_i}}} \left.\vphantom{ \delta_x \frac{\partial \delta_y}{\partial \mathfrak{O}_i} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} }\right) \left.\vphantom{ \delta_z^3 \frac{\partial \delta_z}{\partial \ma... ...} + \delta_y \frac{\partial \delta_x}{\partial \mathfrak{O}_i} \right) }\right) \Bigg]$$ (15.119)

τ_m partial derivative

The partial derivatives with respect to the τ_m geometric parameter are

$\mathfrak{D}_a$ partial derivative

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

$\mathfrak{D}_r$ partial derivative

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

where

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