Let $\mathbf{r}:(a,b)\times (0,1)\to\mathbb{R}^2$ be a injective application, given by:
$$\mathbf{r}(u,v)=A(u)+v\cdot B(u), \forall\ (u,v)\in (a,b)\times (0,1)$$
where $A,B:(a,b)\to\mathbb{R}^2$ are two functions of class $C^1((a,b))$.
Is it true that $\dfrac{\partial\mathbf{r}}{\partial u} (u_0,v_0)\times\dfrac{\partial\mathbf{r}}{\partial v} (u_0,v_0)\neq (0,0), \ \forall\ (u_0,v_0)\in (a,b)\times (0,1)$?