is a plane smooth surface? let f(u,v)=a + u.p + v.q  : the equation of the plane
where p,q are unit vectors perpendicular to each other. a a point on the plane.
I do not understand how f can have partial derivatives of all orders, since derivative of wrt. u and v are p and q, respectively.
after this, aren't the derivatives zero?

The second order and higher partial derivatives of $f$ are indeed $0$, which is not a problem.
The zero function has partial derivatives of all orders.
