# 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?

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What prevents zero from being a derivative? – azarel Nov 21 '13 at 0:30

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