Is there any simple expression for $(\dot{F}\phi_u)\times(\frac{\partial \dot{F}}{\partial u} \phi_v)$

Question

$F$ is an isometry of two dimensional regular surfaces $A$ to $B$. $\varphi$ and $F\circ\varphi$ are the coordinate chart for $A$ and $B$ respectively. Let $\varphi_u$ and $\varphi_v$ generates the tangent plane of $A$ at a point $p$ and $\dot{F} $ denote the differential map of $F$. I am trying to simplify $(\dot{F}\varphi_u)\times(\frac{\partial \dot{F}}{\partial u} \varphi_v)$, where $\frac{\partial \dot{F}}{\partial u}$ is a matrix obtained by differentiating each elements of $\dot{F}$ partially with respect to $u$.
Is there any simple expression for $(\dot{F}\varphi_u)\times(\frac{\partial \dot{F}}{\partial u} \varphi_v)$.
Thank you

Answer 1

The following cross products with $r_1 , r_2$ may be helpful in simplifying using standard notation ; $r$ is position vector. Suffix indep. parameters $(1,2)=(u,v)$

$$ r_1 \times N = r_1\times \frac{r_1\times r_2}{H}= \frac{F r_1 -E r_2}{H}$$

$$ r_2 \times N = r_2\times \frac{r_1\times r_2}{H}= \frac{G r_1 - F r_2}{H}$$

$H$ anyhow is the isometric scalar triple product of $ (N,r_1,r_2) $ .. is all I can say for now.