曲面的切平面的存在性

在许多教科书中,曲面在一点处的切平面由曲面上过该点的曲线的切线定义.然后给出存在的充分条件:假定曲面由隐函数方程给出,如果函数在该点有连续的偏导数,则存在切平面.文中证明了对于隐函数方程给出的曲面,只要函数在该点可微就可以保证切平面存在.此外,还讨论了一些有关的问题.

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.