二阶微分的几何意义及其应用

THE GEOMETRICAL MEANING OF SECOND-ORDER DIFFERNTIAL AND ITS APPLICATION

设ｚ＝ｆ（ｘ，ｙ）是一个Ｃ２类二元函数，则它的图象是三维欧氏空间的一张曲面。本文书证明ｚ＝ｆ（ｘ，ｙ）的二阶微分与相应曲面的关于朝上法向量的第二基本齐式成比例，且比例因子为一正函数。作为一个应用，用此获得了关于极值的一个著名定理的几何证明。

Let z= f(x,y) be a binary function of class c￣2,then its gragh is a surface in the Euclidean three-space. In this paper, we will prove that the second-order differential of z = f(x,y) is proportional to the second fundamental quantic with respect to upward vector of the respeetive surface, and the proportional factor is a positive function. As an application,We have used this result to obtain a geometrical proof of a well-known theorem about extreme values.