kaoyan2advanced 高等数学 第210题
📝 题目
### 第210题
设 $u=f(x, y, z)$ ，其中 $z=\int_{0}^{x y} \mathrm{e}^{t^{2}} \mathrm{~d} t, f$ 有二阶连续偏导数，求 $\displaystyle \frac{\partial u}{\partial x}, \frac{\partial^{2} u}{\partial x \partial y}$ ．
💡 答案解析
**答案**：$\displaystyle \frac{\partial u}{\partial x} = f_x + f_z \cdot y \mathrm{e}^{x^{2}y^{2}}$，$\displaystyle \frac{\partial^{2} u}{\partial x \partial y} = f_{xy} + f_{xz} \cdot x \mathrm{e}^{x^{2}y^{2}} + (f_{zy} + f_{zz} \cdot x \mathrm{e}^{x^{2}y^{2}}) \cdot y \mathrm{e}^{x^{2}y^{2}} + f_z \cdot (\mathrm{e}^{x^{2}y^{2}} + 2x^{2}y^{2} \mathrm{e}^{x^{2}y^{2}})$ **解析**： 步骤1：$z = \int_{0}^{xy} \mathrm{e}^{t^{2}} \mathrm{d} t$，则 $\displaystyle \frac{\partial z}{\partial x} = \mathrm{e}^{(xy)^{2}} \cdot y = y \mathrm{e}^{x^{2}y^{2}}$，$\displaystyle \frac{\partial z}{\partial y} = x \mathrm{e}^{x^{2}y^{2}}$。 步骤2：$\displaystyle \frac{\partial u}{\partial x} = f_x + f_z \cdot \frac{\partial z}{\partial x} = f_x + f_z \cdot y \mathrm{e}^{x^{2}y^{2}}$。 步骤3：$\displaystyle \frac{\partial^{2} u}{\partial x \partial y} = \frac{\partial}{\partial y}(f_x + f_z \cdot y \mathrm{e}^{x^{2}y^{2}}) = f_{xy} + f_{xz} \cdot \frac{\partial z}{\partial y} + (f_{zy} + f_{zz} \cdot \frac{\partial z}{\partial y}) \cdot y \mathrm{e}^{x^{2}y^{2}} + f_z \cdot (\mathrm{e}^{x^{2}y^{2}} + y \cdot 2x^{2}y \mathrm{e}^{x^{2}y^{2}}) = f_{xy} + f_{xz} \cdot x \mathrm{e}^{x^{2}y^{2}} + (f_{zy} + f_{zz} \cdot x \mathrm{e}^{x^{2}y^{2}}) \cdot y \mathrm{e}^{x^{2}y^{2}} + f_z \cdot \mathrm{e}^{x^{2}y^{2}}(1+2x^{2}y^{2})$。 **难度**：★★★☆☆