I want to insert a note on (gradient of ).
1. It is not perpendicular to everything in the surface. Most proofs only go as far as to prove it is perpendicular to continuous parameterized curves. Nothing more. Stop reading too deeply into it.
2. It is mostly useful for finding perpendiculars to all parameterized curves, rather than the parameterized curves themselves. The tangent is an exception, as the normal vector to a straight line can easily be used to find the straight line. For example, if we know that is perpendicular to a straight line passing through point , we can easily determine the straight line. Non-straight line curves do not in general lend themselves to such determination with knowledge of just a perpendicular vector and a point through which the curve passes.