I am starting a book on PDE and there is the following sentence:
"The fundamental idea is that $u(x,y)$ is a surface in $\mathbb{R}^3$, as remembered the direction of the normal of the surface is given by the vector $(u_x,u_y,-1)$
How did they get to this vector?
Note that a vector that is normal to the surface parametrized by $\Psi:(x,y) \mapsto (x,y,u(x,y))$ has to be orthogonal to the tangent vectors $$ \Psi_x=(1,0,u_x) \quad \text{and} \quad \Psi_y = (0,1,u_y), $$ and hence parallel to their cross product $$ \Psi_x \times \Psi_y = (1,0,u_x) \times (0,1,u_y) = (-u_x, -u_y, 1) = -(u_x, u_y, -1). $$