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 5
 Problem Statement

Prove that if ##\mathbf{q}## is a point of a smooth surface ##S## in ##\mathbb{R}^m##, then the normal line to ##S## at ##\mathbf{q}## is independent of which smooth function ##f:D\subset\mathbb{R}^m\rightarrow\mathbb{R}## is used to define ##S##.
Hint: consider local graph like behaviour and follow the argument of Section##3.8##.
 Relevant Equations
 None.
Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that if ##S## is a level set of a function ##f## ##(grad\ f)(\mathbf{p})## is normal to the level set ##S##. I don't know how to use this, since this chapter is about the Implicit Function Theorem.
I've been trying to solve the problem, but I don't know how to begin, can you help me?
I've been trying to solve the problem, but I don't know how to begin, can you help me?