一. 简答题（共1题，100分）

1. (简答题) 求函数\begin{aligned} z=x^3+y^3-3xy \end{aligned}的全微分.

解：计算函数\begin{aligned} z=x^3+y^3-3xy \end{aligned}关于x和y的偏导数.

对于x的偏导数，把y视为常数，得到\begin{aligned} \frac{\partial z}{\partial x}=3x^2-3y \end{aligned}.

对于y的偏导数，把x视为常数，得到\begin{aligned} \frac{\partial z}{\partial y}=3y^2-3x \end{aligned}.

由\begin{aligned} {\rm d}z=\frac{\partial z}{\partial x}{\rm d}x+\frac{\partial z}{\partial y}{\rm d}y \end{aligned},代入得到

\begin{aligned} {\rm d}z=(3x^2-3y){\rm d}x+(3y^2-3x){\rm d}y \end{aligned}

\therefore函数\begin{aligned} z=x^3+y^3-3xy \end{aligned}的全微分为\begin{aligned} {\rm d}z=(3x^2-3y){\rm d}x+(3y^2-3x){\rm d}y \end{aligned}.