The Total Derivative of Multivariate Functions and Its Applications

In this paper, we consider the concept of total derivatives of functions and its relationship with continuity, differentiability, and other concepts. The results indicate that: first, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the function's continuity; second, in general, there is no mutually implicative relationship between the existence of the total derivative and the existence of partial derivatives for bivariate functions. However, if the total derivative of a bivariate function equals zero, then the partial derivatives with respect to both independent variables exist and are also zero; third, generally, there is no implicative relationship between the existence of the total derivative and differentiability for bivariate functions. Nevertheless, the total derivative being equal to zero is equivalent to the function being differentiable at that point and the total differential also being equal to zero; fourth, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the existence of the directional derivative along any direction at that point. Finally, we presents an application of total derivatives——the mean value theorem, enriching the theory of differential calculus of multivariate functions. 讨论多元函数的全导数概念以及与连续、可微等概念之间的关系。结果表明：首先，二元函数全导数存在是函数连续的充分条件而非必要条件；其次，在通常情况下，二元函数全导数存在与偏导数存在之间没有蕴含关系，但若二元函数的全导数等于零，则关于两个自变量的偏导数都存在且均为零；第三，一般来说，二元函数全导数存在和可微之间也没有蕴含关系，但全导数等于零等价于函数在该点可微且全微分也等于零；第四，二元函数全导数存在是该点处沿任意方向的方向导数存在的充分条件而非必要条件。最后，给出全导数的一个应用——微分中值定理，丰富了多元函数微分学的理论。