定义1
设开集\(D \subset \mathbb{R}^n\)，\(f: D \to \mathbb{R}\)，\(\boldsymbol{u}\)是一个方向，\(x_0 \in D\)，如果极限
\[ \lim \limits_{t \to 0} \frac{f(\boldsymbol{x}_0 + t\boldsymbol{u}) - f(\boldsymbol{x})}{t} \]
存在且有限，则称这个极限为函数\(f\)在\(\boldsymbol{x}_0\)处沿方向\(u\)的方向导数，记为\(\dfrac{\partial f}{\partial \boldsymbol{u}}(\boldsymbol{x}_0)\)。
定义2
记单位坐标向量
\[ \begin{aligned} & \boldsymbol{e}_1 = (1, 0, 0, \cdots, 0) \\ & \boldsymbol{e}_2 = (0, 1, 0, \cdots, 0) \\ & \cdots \\ & \boldsymbol{e}_n = (0, 0, 0, \cdots, 1) \end{aligned} \]
称函数\(f\)在点\(\boldsymbol{x}_0\)处沿方向\(\boldsymbol{e}_i\)的方向导数为\(f\)在\(\boldsymbol{x}_0\)处的第\(i\)个一阶偏导数，记作
\[ \frac{\partial f}{\partial x_i} (\boldsymbol{x}_0) 或 \mathrm{D}_if(\boldsymbol{x}_0) \]
并称\(\mathrm{D}_i = \dfrac{\partial}{\partial x_i}\)为第\(i\)个偏微分算子\((i=1,2,\cdots,n)\)；令
\[ \boldsymbol{J}f(\boldsymbol{x}) = (\mathrm{D_1}f(\boldsymbol{x}),\mathrm{D_2}f(\boldsymbol{x}),\cdots,\mathrm{D_n}f(\boldsymbol{x})) \]
并称它为函数\(f\)在点\(\boldsymbol{x}\)处的Jacobi矩阵\(1 \times n\)矩阵。Jacobi矩阵也常记作\(\mathrm{grad} f\)或\(\nabla f\)，也称为数量函数\(f\)的梯度。
设开集\(D \subset \mathbb{R}^n\)，\(f: D \to \mathbb{R}\)，取定一点\(\boldsymbol{x}_0 \in D\)，\(\boldsymbol{h} \in \mathbb{R}^n\)。由于\(\boldsymbol{x}_0\)是\(D\)的一个内点，故当\(\Vert \boldsymbol{h} \Vert\)充分小时，可以使\(\boldsymbol{x}_0+\boldsymbol{h}\)完全在\(D\)之内。
定义3
设\(\boldsymbol{h}=(h_1,h_2,\cdots,h_n)\)，如果成立
\[ f(\boldsymbol{x}_0 + \boldsymbol{h}) - f(\boldsymbol{x}_0) = \sum_{i=1}^n \lambda_i h_i + o(\Vert \boldsymbol{h} \Vert) \quad (\Vert \boldsymbol{h} \Vert \to 0) \]
其中\(\lambda_i(i=1,2,\cdots,n)\)是不依赖于\(\boldsymbol{h}\)的常数，那么称函数\(f\)在点\(\boldsymbol{x}_0\)处可微，并称\(\sum \limits_{i=1}^n \lambda_i h_i\)为\(f\)在\(\boldsymbol{x}_0\)处的微分，记作
\[ \mathrm{d}f(\boldsymbol{x}_0) (\boldsymbol{h}) = \sum_{i=1}^n \lambda_i h_i \]
如果\(f\)在开集\(D\)上的每一点处都可微，则称\(f\)是\(D\)上的可微函数。
定理1
为了方便，将\(\boldsymbol{x}_0,\boldsymbol{h}\)表示列向量形式，设函数\(f\)在\(\boldsymbol{x}_0 = (x_1,x_2,\cdots,x_n)^T\)处可微，则
\[ \mathrm{d}f(\boldsymbol{x}_0) (\boldsymbol{h}) = \boldsymbol{J}f(\boldsymbol{x_0}) \boldsymbol{h} \]
证：定义3中令\(\boldsymbol{h} = (h_1,0,\cdots,
0)^T\)，此时
\[
f(x_1+h_1, x_2, \cdots, x_n) - f(x_1,x_2,\cdots,x_n) = \lambda_1 h_1
+ o(|h_1|)
\]
从而
\[
\frac{f(x_1+h_1, x_2, \cdots, x_n) - f(x_1,x_2,\cdots,x_n)}{h_1} =
\lambda_1 + o(1)
\]
令\(h_1 \to 0\)，得
\[
\lambda_1 = \mathrm{D}_1f(\boldsymbol{x}_0)
\]
一般地，有
\[
\lambda_i = \mathrm{D}_if(\boldsymbol{x}_0) \quad (i=1,2,\cdots,n)
\]
所以有
\[
\mathrm{d}f(\boldsymbol{x}_0) (\boldsymbol{h}) =
\boldsymbol{J}f(\boldsymbol{x_0}) \boldsymbol{h}
\]
Q.E.D.
定理2
设\(f\)在\(\boldsymbol{x}_0\)处可微，则\(f\)必在\(\boldsymbol{x}_0\)处连续。
证：由于\(f\)在\(\boldsymbol{x}_0\)处可微，当\(\boldsymbol{h} \to
\boldsymbol{0}\)时，有\(h_i \to 0
(i=1,2,\cdots,n)\)，此时\(\mathrm{d}f(\boldsymbol{x}_0) (\boldsymbol{h}) =
\sum \limits_{i=1}^n \lambda_i h_i \to 0\)，从而\(f(\boldsymbol{x_0} + \boldsymbol{h}) -
f(\boldsymbol{x}_0) \to 0\)，所以\(f\)在\(\boldsymbol{x}_0\)处连续。
Q.E.D.
定理3
函数\(f\)在\(\boldsymbol{x}_0\)处可微当且仅当等式
\[ f(\boldsymbol{x}_0 + \boldsymbol{h}) - f(\boldsymbol{x}_0) = \boldsymbol{J}f(\boldsymbol{x}_0) \boldsymbol{h} + \sum_{i=1}^n \beta_i(\boldsymbol{h}) h_i \]
成立。其中，当\(\Vert \boldsymbol{h} \Vert \to 0\)时，
\[ \beta_i(\boldsymbol{h}) \to 0 \quad (i=1,2,\cdots,n) \]
证：充分性。当\(\boldsymbol{h} \to
\boldsymbol{0}\)时，有
\[
\frac{1}{\Vert \boldsymbol{h} \Vert} |\sum_{i=1}^n
\beta_i(\boldsymbol{h})h_i| = \left| \sum_{i=1}^n
\beta_i(\boldsymbol{h}) \frac{h_i}{\boldsymbol{\Vert h \Vert}} \right|
\le \left| \sum_{i=1}^n \beta_i(\boldsymbol{h}) \right| \to 0
\]
即
\[
\sum_{i=1}^n \beta_i(\boldsymbol{h}) h_i = o(\Vert \boldsymbol{h}
\Vert)
\]
由定义3可知，函数\(f\)在\(\boldsymbol{x}_0\)处可微。
必要性。记
\[
r(\boldsymbol{h}) = f(\boldsymbol{x}_0 + \boldsymbol{h}) -
f(\boldsymbol{x}_0) - \boldsymbol{J}f(\boldsymbol{x_0}) \boldsymbol{h}
\]
可知当\(\Vert \boldsymbol{h} \Vert \to
0\)时，有\(r(\boldsymbol{h}) = o(\Vert
\boldsymbol{h} \Vert)\)，由于
\[
r(\boldsymbol{h}) = \left(\sum_{i=1}^n \frac{h_i}{\Vert
\boldsymbol{h} \Vert} h_i \right)\frac{r(\boldsymbol{h})}{\Vert
\boldsymbol{h} \Vert}
\]
故令
\[
\beta_i(\boldsymbol{h}) = \frac{r(\boldsymbol{h})}{\Vert
\boldsymbol{h} \Vert} \frac{h_i}{\Vert \boldsymbol{h} \Vert}
\]
由于
\[
\frac{h_i}{\Vert \boldsymbol{h} \Vert} \le 1 \quad (i=1,2,\cdots,n)
\]
从而可知\(\beta_i(\boldsymbol{h}) \to
0\)，且
\[
f(\boldsymbol{x}_0 + \boldsymbol{h}) - f(\boldsymbol{x}_0) =
\boldsymbol{J}f(\boldsymbol{x}_0) \boldsymbol{h} + \sum_{i=1}^n
\beta_i(\boldsymbol{h}) h_i
\]
Q.E.D.
定理4
设开集\(D \subset \mathbb{R}^n\)，\(f: D \to \mathbb{R}\)，\(\boldsymbol{x}_0 \in D\)，如果\(\mathrm{D}_if(\boldsymbol{x}) (i=1,2,\cdots,n)\)在\(\boldsymbol{x}_0\)的一个邻域中存在且在点\(\boldsymbol{x}_0\)处连续，则\(f\)在点\(\boldsymbol{x}_0\)处可微。
证：使用数学归纳法。当\(n=1\)时自然成立，因为单变量函数的导数存在即可微。设定理对\(n-1\)维成立，令
\[
f(\boldsymbol{x}_0 + \boldsymbol{h}) - f(\boldsymbol{x}_0) = K_1 +
K_2
\]
其中
\[
\begin{aligned}
& K_1 = f(x_1+h_1,x_2+h_2,\cdots,x_n+h_n) -
f(x_1+h_1,\cdots,x_{n-1}+h_{n-1},x_n) \\
& K_2 = f(x_1+h_1,\cdots,x_{n-1}+h_{n-1},x_n) -
f(x_1,x_2,\cdots,x_n)
\end{aligned}
\]
对\(K_1\)运用一元微分中值定理，得到
\[
K_1 = \frac{\partial f}{\partial
x_n}(x_1+h_1,\cdots,x_{n-1}+h_{n-1},x_n+\theta h_n) h_n
\]
其中\(\theta \in (0, 1)\)，可以令
\[
K_1 = \frac{\partial f}{\partial x_n}(\boldsymbol{x}_0) h_n + r_1
\]
其中
\[
\begin{aligned}
r_1 & = \left( \frac{\partial f}{\partial
x_n}(x_1+h_1,\cdots,x_{n-1}+h_{n-1},x_n+\theta h_n) - \frac{\partial
f}{\partial x_n}(\boldsymbol{x}_0) \right) h_n \\
& = \beta_n(\boldsymbol{h}) h_n
\end{aligned}
\]
由\(\dfrac{\partial f}{\partial
x_n}\)函数的连续性可知，当\(\Vert
\boldsymbol{h} \Vert \to 0\)时，\(\beta_n(\boldsymbol{h}) \to 0\)，从而
\[
K_1 = \frac{\partial f}{\partial x_n}(\boldsymbol{x}_0)h_n +
\beta_n(\boldsymbol{h})h_n
\]
对\(K_2\)使用\(n-1\)维的归纳假设，可知
\[
K_2 = \sum_{i=1}^{n-1} \frac{\partial f}{\partial x_i} h_i +
\sum_{i=1}^{n-1} \beta_i(\boldsymbol{h}) h_i
\]
当\(\Vert \boldsymbol{h} \Vert \to
0\)时，\(\beta_i(\boldsymbol{h}) \to 0
(i=1,2,\cdots,n-1)\)，从而
\[
f(\boldsymbol{x}_0 + \boldsymbol{h}) - f(\boldsymbol{x}_0) = K_1+K_2
= \sum \limits_{i=1}^n \frac{\partial f}{\partial x_i}
(\boldsymbol{x}_0) h_i + \sum_{i=1}^{n-1} \beta_i(\boldsymbol{h})h_i
\]
其中\(\beta_i(\boldsymbol{h}) \to 0 (\Vert
\boldsymbol{h} \Vert \to 0)(i=1,2,\cdots,n)\)。所以\(f\)在\(\boldsymbol{x}_0\)处可微。即对\(n\)维定理也成立。
Q.E.D.
定理5
若\(f\)在\(\boldsymbol{x}_0\)处可微，则\(f\)在\(\boldsymbol{x}_0\)处的任意方向\(\boldsymbol{u} = (u_1,u_2,\cdots,u_n)\)的方向导数都存在，且
\[ \frac{\partial f}{\partial \boldsymbol{u}}(\boldsymbol{x}_0) = \frac{\partial f}{\partial x_1}(\boldsymbol{x}_0) u_1 + \frac{\partial f}{\partial x_2}(\boldsymbol{x}_0) u_2 + \cdots + \frac{\partial f}{\partial x_n}(\boldsymbol{x}_0) u_n \]
证：由于\(f\)在\(\boldsymbol{x}_0\)处可微，从而
\[
f(\boldsymbol{x}_0 + t \boldsymbol{u}) - f(\boldsymbol{x}_0) =
\sum_{i=1}^n \frac{\partial f}{\partial x_i} (\boldsymbol{x}_0) tu_i +
o(t)
\]
而
\[
\frac{\partial f}{\partial \boldsymbol{u}}(\boldsymbol{x}_0) = \lim
\limits_{t \to 0} \frac{f(\boldsymbol{x}_0+t \boldsymbol{u}) -
f(\boldsymbol{x})}{t} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} u_i
\]
Q.E.D.