[!J. This exercise is used in Section ell. 7. Let F : R3 --+ R be continuously differentiable at

(a,

b, c) with 'V F

(a,

b, c) :f:. O.

(a) Prove that the graph of the relation F(x, y, z) = 0; Le., the set 9 :=

{(x,y,z): F(x,y,z) = O}, has a tangent plane at (a,b,c).

(b) Prove that a normal of the tangent plane to 9 at

(a,

b, c) is given by 'V F

(a,

b, c).