EXISTENCE OF TANGENT PLANES OF A SURFACE

In many textbooks, the tangent plane of a surface S at a point P is defined as the set of all tangent lines of curves at P which are on S and pass through P. Then a sufficient condition for the existence of the tangent plane is followed. Suppose that S is defined by an implicit function F, if the partial derivatives of F are continuous at P, then there is a tangent plane of S at P. In this paper, we prove that if S is defined by an implicit function F, then the differentiation of F at P is sufficient to the existence of tangent plane of S at P. We also discuss some related problems in this paper.