偏导数与全微分

1 偏导数与全微分

1.1 方向导数与偏导数

$$\lim_{t \to 0} \frac{f(x_0 + t\cos\theta,\ y_0 + t\sin\theta) - f(x_0, y_0)}{t}$$ $$\frac{\partial f}{\partial \mathbf{e}}(x_0, y_0) \quad \text{或} \quad \partial_{\mathbf{e}} f(x_0, y_0).$$

• 对 $x$ 的偏导：$\dfrac{\partial f}{\partial x}(x_0, y_0)$, $f'_x(x_0, y_0)$, $f_x(x_0, y_0)$
• 对 $y$ 的偏导：$\dfrac{\partial f}{\partial y}(x_0, y_0)$, $f'_y(x_0, y_0)$, $f_y(x_0, y_0)$

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1.2 全微分

$$f(x_0 + \Delta x,\ y_0 + \Delta y) - f(x_0, y_0) = A\Delta x + B\Delta y + o\!\left(\sqrt{(\Delta x)^2 + (\Delta y)^2}\right),$$ $$\mathrm{d}f(x_0, y_0) = A\,\mathrm{d}x + B\,\mathrm{d}y.$$ $$\frac{\partial f}{\partial x}(x_0, y_0) = A, \qquad \frac{\partial f}{\partial y}(x_0, y_0) = B.$$

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1.3 连续可微函数

$$\begin{aligned} & f(x_0 + h,\ y_0 + k) - f(x_0, y_0) \\ = & \bigl[f(x_0 + h,\ y_0 + k) - f(x_0,\ y_0 + k)\bigr] + \bigl[f(x_0,\ y_0 + k) - f(x_0, y_0)\bigr] \\ = & f_x(x_0 + \theta h,\ y_0 + k)\,h + f_y(x_0,\ y_0 + \omega k)\,k, \qquad 0 < \theta,\ \omega < 1. \end{aligned}$$ $$f_x(x_0 + \theta h,\ y_0 + k) = f_x(x_0, y_0) + \alpha, \quad f_y(x_0,\ y_0 + \omega k) = f_y(x_0, y_0) + \beta,$$ $$f(x_0 + h,\ y_0 + k) - f(x_0, y_0) = f_x(x_0, y_0)h + f_y(x_0, y_0)k + \alpha h + \beta k.$$ $$\left|\frac{\alpha h + \beta k}{\sqrt{h^2 + k^2}}\right| \leq |\alpha| + |\beta| \to 0,$$ $$f(x_0 + h,\ y_0 + k) - f(x_0, y_0) = f_x(x_0, y_0)h + f_y(x_0, y_0)k + o\!\left(\sqrt{h^2 + k^2}\right).$$

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1.4 $m$ 元函数的一般情形

$$\lim_{t \to 0} \frac{f(\mathbf{x}^0 + t\mathbf{e}) - f(\mathbf{x}^0)}{t} = \lim_{t \to 0} \frac{f(x_1^0 + te_1, \dots, x_m^0 + te_m) - f(\mathbf{x}^0)}{t}$$ $$\frac{\partial f}{\partial x_i}(\mathbf{x}^0),\quad f_{x_i}(\mathbf{x}^0), \quad \text{或} \quad D_i f(\mathbf{x}^0).$$ $$f(\mathbf{x}^0 + \Delta \mathbf{x}) - f(\mathbf{x}^0) = \sum_{i=1}^{m} A_i \Delta x_i + o(\|\Delta \mathbf{x}\|),$$ $$\mathrm{d}f(\mathbf{x}^0) = \sum_{i=1}^{m} A_i \mathrm{d}x_i.$$

1. 可微蕴含连续：若 $f$ 在点 $\mathbf{x}^0$ 可微，则 $f$ 在该点连续。
2. 可微蕴含方向导数存在：若 $f$ 在点 $\mathbf{x}^0$ 可微，则 $f$ 在该点沿任意方向 $\mathbf{e}$ 的方向导数均存在。
3. 全微分系数即偏导数：若 $f$ 在点 $\mathbf{x}^0$ 可微，且 $\mathrm{d}f(\mathbf{x}^0) = \sum_{i=1}^{m} A_i \mathrm{d}x_i$，则 $$\frac{\partial f}{\partial x_i}(\mathbf{x}^0) = A_i, \quad i=1,2,\dots,m.$$ 因此，全微分可唯一地写为： $$\mathrm{d}f(\mathbf{x}^0) = \sum_{i=1}^{m} \frac{\partial f}{\partial x_i}(\mathbf{x}^0) \mathrm{d}x_i.$$

$$\begin{aligned} & f(x_1^0+h_1, \dots, x_m^0+h_m) - f(x_1^0, \dots, x_m^0) \\ = & \ [f(x_1^0+h_1, x_2^0, \dots, x_m^0) - f(x_1^0, \dots, x_m^0)] \\ + & \ [f(x_1^0+h_1, x_2^0+h_2, x_3^0, \dots, x_m^0) - f(x_1^0+h_1, x_2^0, \dots, x_m^0)] \\ + & \ \dots \\ + & \ [f(x_1^0+h_1, \dots, x_m^0+h_m) - f(x_1^0+h_1, \dots, x_{m-1}^0+h_{m-1}, x_m^0)]. \end{aligned}$$

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2 复合函数的偏导数与全微分

2.1 复合函数求导的链式法则

$$F(t)=f(\varphi^1(t),\varphi^2(t),...,\varphi^m(t))$$ $$\frac{\partial F}{\partial t^j}(t_0)=\sum_{i=1}^m \frac{\partial f}{\partial x^i}(x_0)\frac{\partial \varphi^i}{\partial t^j}(t_0).$$ $$f(x_0 + \Delta x) - f(x_0) = \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x_0) \Delta x^i + o(\|\Delta x\|),$$ $$\Delta \varphi^i = \varphi^i(t_0 + \Delta t) - \varphi^i(t_0),\quad i=1,\dots,m.$$ $$\Delta \varphi^i = \frac{\partial \varphi^i}{\partial t^j}(t_0) \Delta t^j + o(|\Delta t^j|).$$ $$\begin{aligned} F(t_0 + \Delta t) - F(t_0) &= f(\varphi^1(t_0 + \Delta t), \dots, \varphi^m(t_0 + \Delta t)) - f(\varphi^1(t_0), \dots, \varphi^m(t_0)) \\ &= f(x_0 + \Delta \varphi) - f(x_0), \end{aligned}$$ $$\begin{aligned} F(t_0 + \Delta t) - F(t_0) &= \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x_0) \Delta \varphi^i + o(\|\Delta \varphi\|) \\ &= \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x_0) \left[ \frac{\partial \varphi^i}{\partial t^j}(t_0) \Delta t^j + o(|\Delta t^j|) \right] + o(\|\Delta \varphi\|). \end{aligned}$$ $$F(t_0 + \Delta t) - F(t_0) = \left[ \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x_0) \frac{\partial \varphi^i}{\partial t^j}(t_0) \right] \Delta t^j + o(|\Delta t^j|).$$ $$\frac{\partial F}{\partial t^j}(t_0) = \lim_{\Delta t^j \to 0} \frac{F(t_0 + \Delta t) - F(t_0)}{\Delta t^j} = \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x_0) \frac{\partial \varphi^i}{\partial t^j}(t_0).$$

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2.2 微分表示的不变性

$$\mathrm{d}f(x)=\sum_{i=1}^m \frac{\partial f}{\partial x^i}(x)\,\mathrm{d}x^i.$$ $$\frac{\partial F}{\partial t^j} = \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x(t)) \frac{\partial x^i}{\partial t^j}.$$ $$\begin{aligned} \mathrm{d}F &= \sum_{j=1}^k \frac{\partial F}{\partial t^j} \,\mathrm{d}t^j \\ &= \sum_{j=1}^k \left( \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x(t)) \frac{\partial x^i}{\partial t^j} \right) \mathrm{d}t^j \\ &= \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x(t)) \left( \sum_{j=1}^k \frac{\partial x^i}{\partial t^j} \mathrm{d}t^j \right) \\ &= \sum_{i=1}^m \frac{\partial f}{\partial x^i}(x(t)) \,\mathrm{d}x^i, \end{aligned}$$

1. $\mathrm{d}(u(x)+v(x))=\mathrm{d}u(x)+\mathrm{d}v(x)$
2. $\mathrm{d}(\lambda u(x))=\lambda\,\mathrm{d}u(x)$
3. $\mathrm{d}(u(x)v(x))=v(x)\mathrm{d}u(x)+u(x)\mathrm{d}v(x)$
4. $\mathrm{d}\left(\dfrac{u(x)}{v(x)}\right)=\dfrac{v(x)\mathrm{d}u(x)-u(x)\mathrm{d}v(x)}{(v(x))^2}$，其中 $v(x)\neq 0$

$$\begin{aligned} \mathrm{d}(u(x)+v(x)) &= \sum_{i}\frac{\partial (u+v)}{\partial x^i}\mathrm{d}x^i \\ &= \sum_{i}\left(\frac{\partial u}{\partial x^i}+\frac{\partial v}{\partial x^i}\right)\mathrm{d}x^i \\ &= \sum_{i}\frac{\partial u}{\partial x^i}\mathrm{d}x^i + \sum_{i}\frac{\partial v}{\partial x^i}\mathrm{d}x^i \\ &= \mathrm{d}u(x)+\mathrm{d}v(x) \end{aligned}$$

$$\begin{aligned} \mathrm{d}(\lambda u(x)) &= \sum_{i}\frac{\partial (\lambda u)}{\partial x^i}\mathrm{d}x^i \\ &= \lambda\sum_{i}\frac{\partial u}{\partial x^i}\mathrm{d}x^i \\ &= \lambda\,\mathrm{d}u(x) \end{aligned}$$

$$\begin{aligned} \mathrm{d}(u(x)v(x)) &= \sum_{i}\frac{\partial (uv)}{\partial x^i}\mathrm{d}x^i \\ &= \sum_{i}\left(v\frac{\partial u}{\partial x^i} + u\frac{\partial v}{\partial x^i}\right)\mathrm{d}x^i \\ &= v\sum_{i}\frac{\partial u}{\partial x^i}\mathrm{d}x^i + u\sum_{i}\frac{\partial v}{\partial x^i}\mathrm{d}x^i \\ &= v\mathrm{d}u(x) + u\mathrm{d}v(x) \end{aligned}$$

$$\begin{aligned} \mathrm{d}\left(\frac{u(x)}{v(x)}\right) &= \sum_{i} \frac{\partial}{\partial x^i}\left(\frac{u(x)}{v(x)}\right)\mathrm{d}x^i \\ &= \sum_{i} \frac{v(x)\frac{\partial u}{\partial x^i} - u(x)\frac{\partial v}{\partial x^i}}{(v(x))^2}\mathrm{d}x^i \\ &= \frac{v(x)\mathrm{d}u(x) - u(x)\mathrm{d}v(x)}{(v(x))^2} \end{aligned}$$

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3 高阶偏导数

3.1 高阶偏导数的定义

1. 一阶偏导数：若对某个 $i \in \{1, \dots, m\}$，极限
   $$\frac{\partial f}{\partial x_i}(\mathbf{x}) = \lim_{h \to 0} \frac{f(x_1, \dots, x_i + h, \dots, x_m) - f(\mathbf{x})}{h}$$
   在 $\mathbf{x} \in D$ 处存在，则称 $f$ 在 $\mathbf{x}$ 处对 $x_i$ 可偏导，并称该极限值为 $f$ 在 $\mathbf{x}$ 处对 $x_i$ 的一阶偏导数。
2. 高阶偏导数（归纳定义）：假设对于某个正整数 $n \geq 2$，$f$ 的所有 $n-1$ 阶偏导数已在 $D$ 上定义。若某个 $n-1$ 阶偏导数函数
   $$g(\mathbf{x}) = \frac{\partial^{n-1} f}{\partial x_{i_1} \cdots \partial x_{i_{n-1}}}(\mathbf{x})$$
   在 $\mathbf{x} \in D$ 处对 $x_{i_n}$ 可偏导，则称该偏导数为 $f$ 在 $\mathbf{x}$ 处的一个 $n$ 阶偏导数，记作
   $$\frac{\partial^n f}{\partial x_{i_1} \cdots \partial x_{i_n}}(\mathbf{x}) = \frac{\partial}{\partial x_{i_n}} \left( \frac{\partial^{n-1} f}{\partial x_{i_1} \cdots \partial x_{i_{n-1}}} \right)(\mathbf{x}).$$

3.2 混合偏导数与求导顺序

$$f_{xy}(x_0,y_0)=f_{yx}(x_0,y_0).$$ $$\varphi(x)=f(x,y_0+k)-f(x,y_0), \quad \psi(y)=f(x_0+h,y)-f(x_0,y).$$ $$\begin{aligned} \varphi(x_0+h)-\varphi(x_0) &= \varphi'(x_0+\theta_1 h)h \\ &= (f_x(x_0+\theta_1h,y_0+k)-f_x(x_0+\theta_1h,y_0))h \\ &= f_{xy}(x_0+\theta_1 h,y_0+\theta_2 k)hk. \end{aligned}$$ $$\psi(y_0+h)-\psi(y_0) = f_{yx}(x_0+\theta_3 h,y_0+\theta_4 k)hk.$$ $$\begin{aligned} \varphi(x_0+h)-\varphi(x_0) &= f(x_0+h,y_0+k)-f(x_0+h,y_0)-f(x_0,y_0+k)+f(x_0,y_0) \\ &= \psi(y_0+k)-\psi(y_0). \end{aligned}$$ $$f_{xy}(x_0,y_0)=f_{yx}(x_0,y_0). \quad \square$$