1.导数与微分
1.1.偏导数
对于二元函数\(z=f(x,y)\), 规定其在固定方向上的导数为偏导数。
其中在\(x\)方向上的偏导数记作\(\frac {\partial f}{\partial x}\)，在\(y\)方向上的偏导数记作\(\frac {\partial f}{\partial y}\)。
按照求导变量顺序的不同，存在如下偏导数：
\[
\frac {\partial ^{2} f}{\partial x^{2}}, \frac {\partial ^{2} f}{\partial y^{2}}, \frac {\partial ^{2} f}{\partial x \partial y}, \frac {\partial ^{2} f}{\partial y \partial x}
\]
有如下定理：
\[
若函数z=f(x,y)在区域D上连续且\frac {\partial ^{2} f}{\partial x \partial y}与\frac {\partial ^{2} f}{\partial y \partial x}在区域D上存在且连续，则\frac {\partial ^{2} f}{\partial x \partial y}=\frac {\partial ^{2} f}{\partial y \partial x}。
\]
1.2.全微分
考虑曲面\(z=f(x,y)\)在\((x_{0}, y_{0}, z_{0})\)存在切平面\(\pi :Ax+By+Cz+D=0\)，其法方向为：\(\vec {r}\)。则存在\(\vec {\alpha}=(1,0,\frac {\partial f}{\partial x}|_{x=x_{0}}), \vec {\beta}=(0,1,\frac {\partial f}{\partial y}|_{y=y_{0}})\)，使得\(\vec {\alpha} \parallel \pi, \vec {\beta} \parallel \pi\)，则有：
\[
\vec {r} = \vec {\alpha} \times \vec {\beta}=(\frac {\partial f}{\partial x}|_{x=x_{0}}, \frac {\partial f}{\partial y}|_{y=y_{0}}, -1)
\]
在\((x_{0}, y_{0}, z_{0})\)处的切平面\(\pi\)的点法式方程为：
\[
\vec {r} \cdot
\begin{pmatrix}
x-x_{0} \\
y-y_{0} \\
z-z_{0}
\end{pmatrix}=0 \Longrightarrow z-z_{0}=\frac {\partial f}{\partial x}|_{x=x_{0}} (x-x_0)+\frac {\partial f}{\partial y}|_{y=y_{0}}(y-y_{0})
\]
当\(x \to x_{0}, y \to y_{0}\)时：
\[
dz|_{z=z_{0}}=\frac {\partial f}{\partial x}|_{x=x_{0}}dx+\frac {\partial f}{\partial y}|_{y=y_{0}}dy
\]
因此有全微分公式：
\[
dz=\frac {\partial f}{\partial x}dx+\frac {\partial f}{\partial y}dy
\]
1.3.复合多元函数的导数与微分
如果多元函数函数\(z=f(\lambda , \mu)\)，其中\(\lambda=\phi(t), \mu=\psi(t)\)，则\(z\)关于\(t\)的导数：
\[
\frac {dz}{dt}=\frac {\partial f}{\partial \phi}\frac {d\phi}{dt}+\frac {\partial f}{\partial \psi}\frac {d\psi}{dt}
\]
若\(\lambda=\phi(x, y), \mu=\psi(x, y)\)，不妨令\(z=g(x, y)\)，则有：
\[
dz=\frac {\partial f}{\partial \phi}d\phi+\frac {\partial f}{\partial \psi}d\psi
\]
又：
\[
d\phi=\frac {\partial \phi}{\partial x}dx+\frac {\partial \phi}{\partial y}dy
\]
$$
d\psi=\frac {\partial \psi}{\partial x}dx+\frac {\partial \psi}{\partial y}dy
$$
所以：
\[
dz=(\frac {\partial f}{\partial \phi}\frac{\partial \phi}{\partial x}+\frac {\partial f}{\partial \psi}\frac{\partial \psi}{\partial x})dx+(\frac {\partial f}{\partial \phi}\frac{\partial \phi}{\partial y}+\frac {\partial f}{\partial \psi}\frac{\partial \psi}{\partial y})dy
\]
又：
$$
dz=\frac {\partial g}{\partial x}dx+\frac {\partial g}{\partial y}dy
$$
则\(z=g(x, y)\)偏微分：
\[
\frac {\partial g}{\partial x}=\frac {\partial f}{\partial \phi}\frac{\partial \phi}{\partial x}+\frac {\partial f}{\partial \psi}\frac{\partial \psi}{\partial x}
\]
\[
\frac {\partial g}{\partial y}=\frac {\partial f}{\partial \phi}\frac{\partial \phi}{\partial y}+\frac {\partial f}{\partial \psi}\frac{\partial \psi}{\partial y}
\]
1.4.隐函数求导
二元隐函数通常由方程\(F(x, y)=0\)确定。若在\((x_{0}, y_{0})\)邻域内具有连续偏导数，则有链式法则：
\[
dF(x, y(x)) = \frac {\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy
\]
又：
\[
\frac{dF(x, y)}{dx}=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}\frac{dy}{dx}=0
\]
故隐函数方程\(F(x, y)=0, y关于x的导数\)为：
\[
\frac{dy}{dx}=-\frac{{F_{x}}^{'}}{{F_{y}}^{'}}
\]
推广至多元函数，例如三元隐函数方程\(F(x, y, z(x, y))=0\)，有链式法则：
\[
dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz
\]
又有：\(z=z(x,y)，dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy\)，因此：
\[
dF=(\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x})dx+(\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial y})dy
\]
\[
\frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}=0
\]
因此有：\({F_{x}}^{'}+{F_{z}}^{'}\frac{\partial z}{\partial x}=0，即：\frac{\partial z}{\partial x}=-\frac{{F_{z}}^{'}}{{F_{x}}^{'}}\)。同理有：\(\frac{\partial z}{\partial y}=-\frac{{F_{z}}^{'}}{{F_{x}}^{'}}\)。
1.5.方向导数与梯度
讨论二元函数\(z=f(x,y)\)在\((x_0,y_0,z_0)\)处的切平面\(\pi\)(若此处偏导数存在)由向量\(\vec{\alpha}=(1,0,\frac{\partial f}{\partial x})，\vec{\beta}=(0,1,\frac{\partial f}{\partial y})\)确定。则平行于切平面\(\pi\)的任意方向向量\(\vec{l}\)都可由基向量\(\vec{\alpha}，\vec{\beta}\)的线性组合表示：
\[
\mathbf{l}=\lambda \mathbf{\alpha}+\mu \mathbf{\beta}=(\lambda,\mu,\lambda\frac{\partial f}{\partial x}+\mu \frac{\partial f}{\partial y})
\]
定义\(z=f(x,y)\)在\(\mathbf{l}\)方向的方向导数：
\[
\frac{\partial f}{\partial l}=\frac{\lambda}{\sqrt{\lambda^2+\mu^2}}\frac{\partial f}{\partial x}+\frac{\mu}{\sqrt{\lambda^2+\mu^2}}\frac{\partial f}{\partial y}
\]
由定义可知，\(z=f(x,y)\)在\(x\)轴方向的方向导数为\(\frac{\partial f}{\partial x}\)，在\(y\)轴方向的方向导数为\(\frac{\partial f}{\partial y}\)。描述了\(z=f(x,y)的(x_0,y_0,z_0)处\mathbf{l}\)方向上的变化量
不妨将函数\(z=f(x,y)\)的\((x_0,y_0,z_0)\)点在\(\mathbf{l}\)方向上的方向导数视作以下内积形式：
\[
\frac{\partial f}{\partial l}=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})\cdot(\frac{\lambda}{\sqrt{\lambda^2+\mu^2}},\frac{\mu}{\sqrt{\lambda^2+\mu^2}})^T=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})\cdot \mathbf{e_l}
\]
其中\(\vec{e_l}\)是\((x_0,y_0,z_0)\)处\(\vec{l}\)方向上的单位向量。定义函数\(z=f(x,y)\)的点\((x_0,y_0,z_0)\)处的梯度为：
\[
\nabla f(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})
\]
重新审视方向导数的内积形式：
\[
\frac{\partial f}{\partial l}=|\nabla f(x,y)|\cdot |\mathbf{e_l}|\cos{\theta}
\]
其中\(\theta\)是单位方向向量\(\mathbf{e_l}\)与梯度\(\nabla f(x,y)\)的夹角。
要使在\(\mathbf{l}\)方向上的变化率最大，\(\cos{\theta}=1\)，即方向单位向量\(\mathbf{e_l}\)与梯度\(\nabla f(x,y)\)平行。