Gradient vectors

So at a point on a contour, the gradient vector is always perpendicular to any tangent vector to the contour i.e. perpendicular to the surface.

But if you have any surface is it true that grad f is always perpendicular to the surface at that point when f is defined on the surface?

So to find a normal to the surface given by z=f(x,y) then F(x,y,z) = z - f(x,y) = 0 and gradF(x,y,z) is a normal to surface at (x,y,z)?

Thanks