I am given the quasi-linear PDE, $$u_t+uu_x=-2u,$$ with the boundary condition $u(0,t)=e^{-t}$.
I am trying to compute the transversality condition. I proceed as
$$\text{det}(J)=\begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial t}{\partial r}\\ \frac{\partial x}{\partial s }&\frac{\partial t}{\partial s}\\ \end{vmatrix}=\begin{vmatrix} 1 & 1 \\ 0 & 0 \\ \end{vmatrix}=0.$$ I was wondering if the condition is compute correctly ($J$ denotes the Jacobian matrix).