$\hat{G}$ is a function of $u$ and $v$.
$u$ and $v$ are functions of $h_u, h_v$ .
$A$ in the forward function.
$A$ is a function of gaussian center $p_k$, scale factors $s_u,\;s_v$, and tangent basis $t_u,\; t_v$.
tangent basis $t$ are functions of the quaternions $q_0, \;q_1,\;q_2,\;q_3$ form of rotation.
The same as 3D gaussian, omit.
$$ \frac{dG}{du} = -ue^{-\frac{u^2+v^2}{2}} = -Gu $$
$$ \frac{dG}{dv} = -ve^{-\frac{u^2+v^2}{2}} = -Gv $$
$$ \frac{du}{dh_u} = [h_v^2\frac{h_u^3h_v^2-h_u^2h_v^3}{(h_u^1h_v^2−h_u^2h_v^1)^2}, h_v^2\frac{h_u^1h_v^3−h_u^3h_v^1}{(h_u^2h_v^1−h_u^1h_v^2)^2},\frac{h_v^2}{h_u^2h_v^1-h_u^1h_v^2}] $$
let the denominator $D = {h_{u1}h_{v2}-h_{u2}h_{v1}}$
Simplify：
$$ \frac{du}{dh_u} = [\frac{-uh_{v2}}{D}, \frac{-vh_{v2}}{D}, \frac{-h_{v2}}{D}] $$
$$ \frac{du}{dh_v} = [\frac{uh_{u2}}{D}, \frac{vh_{u2}}{D}, \frac{h_{u2}}{D}] $$
$$ \frac{dv}{dh_u} = [\frac{uh_{v1}}{D}, \frac{vh_{v1}}{D}, \frac{h_{v1}}{D}] $$
$$ \frac{dv}{dh_v} = [\frac{-uh_{u1}}{D}, \frac{-vh_{u1}}{D}, \frac{-h_{u1}}{D}] $$