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第六章 多元函数微分学
多元复合函数求导举例 Part3
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多元复合函数求导举例 Part3
多元复合函数
## 多元复合函数求导举例 > 多元复合函数求导拿到试题后,首先就是画图,标注变量之间的关系,然后找到对应的路径,这在复杂的题目里,尤其有用。阅读本文前最好已经阅读了上一节内容 `例` 设 $z=f\left(\sin x, x^2-y^2\right) , f$ 具有一阶连续的偏导数,求 $\frac{\partial z}{\partial x} , \frac{\partial z}{\partial y}$. 解 设 $u=\sin x, v=x^2-y^2$ , 则函数 $z=f\left(\sin x, x^2-y^2\right)$ 是由函数 $z=f(u, v)$ , $u=\sin x, v=x^2-y^2$ 复合而成. 由函数的结构图  从图中可以看到 **$z$到$x$ 有两条路** 所以可得 $\quad \frac{\partial z}{\partial x}=\frac{\partial f}{\partial u} \frac{\mathrm{d} u}{\mathrm{~d} x}+\frac{\partial f}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} x}=\cos x \cdot f_u+2 x f_v $. 而 **$z$到$y$有1条路** ,所以 $\frac{\partial z}{\partial y}=\frac{\partial f}{\partial v} \frac{\partial v}{\partial y}=-2 y f_v$ 注 有时,为表达简便起见,引入以下记号: $$ f_1^{\prime}(u, v)=f_u(u, v), \quad f_2^{\prime}(u, v)=f_v(u, v), $$ > **这里下标 1 表示对第一个变量求偏导数,下标 2 表示对第二个变量求偏导 数,利用这样的记号,上例的结果可以表示为** $$ \frac{\partial z}{\partial x}=\cos x \cdot f_1^{\prime}+2 x f_2^{\prime}, \quad \frac{\partial z}{\partial y}=-2 y f_2^{\prime} $$ 同理也可以引入 $f_{11}^{\prime \prime} 、 f_{12}^{\prime \prime} 、 f_{22}^{\prime \prime}$ 等记号 `例`(1) 设 $u=f(x+y+z, x y z)$ ,其中 $f$ 具有二阶偏导数,求 $\frac{\partial u}{\partial x} , \frac{\partial^2 u}{\partial x \partial z}$; (2)设 $u=f(x, x y, x y z)$ ,其中 $f$ 具有二阶连续偏导数,求 $\frac{\partial^2 u}{\partial x \partial y}$. 解 (1) $$ \begin{aligned} \frac{\partial u}{\partial x}=f_1^{\prime}+f_2^{\prime} y z, \frac{\partial^2 u}{\partial x \partial z} & =\frac{\partial}{\partial x}\left(f_1^{\prime}\right)+\frac{\partial}{\partial z}\left(f_2^{\prime} y z\right) \\ & =f_{11}^{\prime \prime}+f_{12}^{\prime \prime} x y+\left(f_{21}^{\prime \prime}+f_{22}^{\prime \prime} x y\right) y z+f_2^{\prime} y \end{aligned} $$ > 注 本题条件中并没有二阶偏导数连续,因此 $f_{12}^{\prime \prime}$ 与 $f_{21}^{\prime \prime}$ 末必相等,因此不要将 其合并. (2) $\frac{\partial u}{\partial x}=f_1^{\prime}+y f_2^{\prime}+y z f_3^{\prime}$, $$ \begin{aligned} \frac{\partial^2 u}{\partial x \partial y} & =\frac{\partial}{\partial x}\left(f_1^{\prime}\right)+\frac{\partial}{\partial x}\left(y f_2^{\prime}\right)+\frac{\partial}{\partial x}\left(y z f_3^{\prime}\right) \\ & =x f_{12}^{\prime \prime}+x z f_{13}^{\prime \prime}+y\left(x f_{22}^{\prime \prime}+x z f_{23}^{\prime \prime}\right)+f_2^{\prime}+y z\left(x f_{32}^{\prime \prime}+x z f_{33}^{\prime \prime}\right)+z f_3^{\prime} \\ & =f_2^{\prime}+z f_3^{\prime}+x f_{12}^{\prime \prime}+x z f_{13}^{\prime \prime}+x y f_{22}^{\prime \prime}+2 x y z f_{23}^{\prime \prime}+x z f_{33}^{\prime \prime} . \end{aligned} $$ > 注 本题条件中有二阶偏导数连续,因此 $f_{23}^{\prime \prime}=f_{32}^{\prime \prime}$ , 因此需要将其合并. `例` 设函数 $u=f(x, y)$ 具有二阶连续偏导数,将下列表达式转换为极坐标的形式: (1) $\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2$; (2) $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}$. 解 (1) 直角坐标系与极坐标系的关系 : $\left\{\begin{array}{l}x=r \cos \theta \\ y=r \sin \theta\end{array}\right.$ ,或 $\left\{\begin{array}{l}r=\sqrt{x^2+y^2} \\ \theta=\arctan \frac{y}{x}\end{array}\right.$ , $u=f(x, y)=f(r \cos \theta, r \sin \theta)=F(r, \theta)$, $$ \begin{aligned} \frac{\partial r}{\partial x}= & \frac{x}{\sqrt{x^2+y^2}}=\frac{x}{r}=\cos \theta, \frac{\partial \theta}{\partial x}=\frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot\left(-\frac{y}{x^2}\right)=-\frac{y}{x^2+y^2}=-\frac{x}{r^2}-\frac{\sin t}{r} \\ & \frac{\partial r}{\partial y}=\frac{y}{\sqrt{x^2+y^2}}=\frac{y}{r}=\sin \theta, \frac{\partial \theta}{\partial y}=\frac{1}{1+\left(\frac{y}{x}\right)^2} \cdot \frac{1}{x}=\frac{x}{x^2+y^2}=\frac{\cos }{r} \end{aligned} $$ $\frac{\partial r}{\partial x}=\cos \theta, \frac{\partial \theta}{\partial x}=-\frac{x}{r^2}-\frac{\sin \theta}{r}$ $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial r} \cdot \frac{\partial r}{\partial x}+\frac{\partial u}{\partial \theta} \cdot \frac{\partial \theta}{\partial x}=\frac{\partial u}{\partial r} \cdot \frac{x}{r}-\frac{\partial u}{\partial \theta} \cdot \frac{y}{r^2}=\frac{\partial u}{\partial r} \cos \theta-\frac{\partial u}{\partial \theta} \cdot \frac{\sin \theta}{r} $$ $$ \frac{\partial r}{\partial y}=\sin \theta, \frac{\partial \theta}{\partial y}=\frac{\cos \theta}{r} $$ 因此平方相加得 $$ \begin{aligned} \left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2 & =\left(\frac{\partial u}{\partial r} \cos \theta-\frac{\partial u}{\partial \theta} \cdot \frac{\sin \theta}{r}\right)^2+\left(\frac{\partial u}{\partial r} \sin \theta+\frac{\partial u}{\partial \theta} \cdot \frac{\cos \theta}{r}\right)^2 \\ & =\left(\frac{\partial u}{\partial r}\right)^2+\frac{1}{r^2}\left(\frac{\partial u}{\partial \theta}\right)^2 \end{aligned} $$ (2) $$ \begin{aligned} \left(\frac{\partial^2 u}{\partial x^2}\right)= & \frac{\partial}{\partial r}\left(\frac{\partial u}{\partial x}\right) \cdot \frac{\partial r}{\partial x}+\frac{\partial}{\partial \theta}\left(\frac{\partial u}{\partial x}\right) \cdot \frac{\partial \theta}{\partial x} \\ = & \frac{\partial}{\partial r}\left(\frac{\partial u}{\partial r} \cos \theta-\frac{\partial u}{\partial \theta} \cdot \frac{\sin \theta}{r}\right) \cos \theta-\frac{\partial}{\partial \theta}\left(\frac{\partial u}{\partial r} \cos \theta-\frac{\partial u}{\partial \theta} \cdot \frac{\sin \theta}{r}\right) \frac{\sin \theta}{r} \\ = & \left(\frac{\partial^2 u}{\partial r^2} \cos \theta-\frac{\partial^2 u}{\partial \theta \partial r} \sin \theta+\frac{\partial u}{\partial \theta} \cdot \frac{\sin \theta}{r^2}\right) \cos \theta \\ & -\left(\frac{\partial^2 u}{\partial r \partial \theta} \cos \theta-\frac{\partial u}{\partial r} \sin \theta-\frac{\partial^2 u}{\partial \theta^2} \cdot \frac{\sin \theta}{r}-\frac{\partial u}{\partial \theta} \frac{\cos \theta}{r}\right) \frac{\sin \theta}{r} \\ = & \frac{\partial^2 u}{\partial r^2} \cos ^2 \theta-2 \frac{\partial^2 u}{\partial \theta \partial r} \cdot \frac{\sin \theta \cos \theta}{r}+2 \frac{\partial u}{\partial \theta} \frac{\sin \theta \cos \theta}{r^2}+\frac{\partial u}{\partial r} \frac{\sin ^2 \theta}{r}+\frac{\partial^2 u}{\partial \theta^2} \cdot \frac{\sin ^2 \theta}{r^2} \end{aligned} $$ $\frac{\partial u}{\partial y}=\frac{\partial u}{\partial r} \sin \theta-\frac{\partial u}{\partial \theta} \cdot \frac{\cos \theta}{r}$ 同理, $$ \frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 u}{\partial r^2} \sin ^2 \theta+2 \frac{\partial^2 u}{\partial \theta \partial r} \cdot \frac{\sin \theta \cos \theta}{r}-2 \frac{\partial u}{\partial \theta} \frac{\sin \theta \cos \theta}{r^2}+\frac{\partial u}{\partial r} \frac{\cos ^2 \theta}{r}+\frac{\partial^2 u}{\partial \theta^2} \cdot \frac{\sin ^2 \theta}{r^2} $$ 因此, $$ \dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=\dfrac{\partial^2 u}{\partial r^2}+\dfrac{1}{r} \frac{\partial u}{\partial r}+\dfrac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}=\dfrac{1}{r^2}\left(r \frac{\partial}{\partial r}\left(r \dfrac{\partial u}{\partial r}\right)+\dfrac{\partial^2 u}{\partial \theta^2}\right) $$ 第(2)问里, $\left(\frac{\partial^2 u}{\partial x^2}\right)=\frac{\partial}{\partial r}\left(\frac{\partial u}{\partial x}\right) \cdot \frac{\partial r}{\partial x}+\frac{\partial}{\partial \theta}\left(\frac{\partial u}{\partial x}\right) \cdot \frac{\partial \theta}{\partial x}$ 注 也可将 $r 、 \theta$ 看作是自变量, $x 、 y$ 看作是中间变量. $$ \begin{aligned} & \frac{\partial u}{\partial r}=\frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial r}+\frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial r}=\frac{\partial u}{\partial x} \cos \theta+\frac{\partial u}{\partial y} \sin \theta \\ & -=-{ }^2-{ }^{\frac{\partial u}{\partial \theta}}=\frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial \theta}+\frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial \theta}=\frac{\partial u}{\partial x} \cdot(-\sin \theta)+\frac{\partial u}{\partial y} r \cos \theta \end{aligned} $$ 以上两式联立,解出 $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial r} \cos \theta-\frac{\partial u}{\partial \theta} \cdot \frac{\sin \theta}{r}, \frac{\partial u}{\partial y}=\frac{\partial u}{\partial r} \cdot \sin \theta+\frac{\partial u}{\partial \theta} \cdot \frac{\cos \theta}{r} $$
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