题目
若函数 z = z(x, y) 的全微分 dz = cos y dx - x sin y dy，则二阶偏导数 (partial^2 z)/(partial x partial y)=A. -sin yB. sin xC. cos xD. cos y
若函数 $z = z(x, y)$ 的全微分 $dz = \cos y dx - x \sin y dy$，则二阶偏导数 $\frac{\partial^2 z}{\partial x \partial y}=$
A. $-\sin y$
B. $\sin x$
C. $\cos x$
D. $\cos y$
题目解答
答案
A. $-\sin y$
解析
步骤 1：确定一阶偏导数
由全微分 $dz = \cos y \, dx - x \sin y \, dy$，可得一阶偏导数： \[ \frac{\partial z}{\partial x} = \cos y, \quad \frac{\partial z}{\partial y} = -x \sin y \]
步骤 2：计算二阶偏导数 $\frac{\partial^2 z}{\partial x \partial y}$
求二阶偏导数 $\frac{\partial^2 z}{\partial x \partial y}$： \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right) = \frac{\partial}{\partial y} (\cos y) = -\sin y \] 或 \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right) = \frac{\partial}{\partial x} (-x \sin y) = -\sin y \]
由全微分 $dz = \cos y \, dx - x \sin y \, dy$，可得一阶偏导数： \[ \frac{\partial z}{\partial x} = \cos y, \quad \frac{\partial z}{\partial y} = -x \sin y \]
步骤 2：计算二阶偏导数 $\frac{\partial^2 z}{\partial x \partial y}$
求二阶偏导数 $\frac{\partial^2 z}{\partial x \partial y}$： \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right) = \frac{\partial}{\partial y} (\cos y) = -\sin y \] 或 \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right) = \frac{\partial}{\partial x} (-x \sin y) = -\sin y \]