分类： 高等数学

$F(x, y, z)$ $=$ $f(x, y, z)$ $+$ $\lambda$ $\varphi(x, y, z)$

$F(x, y)$ $=$ $f(x, y)$ $+$ $\lambda$ $\varphi(x, y)$

$\begin{cases} & A = f_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right) \\ & B= f_{x y}^{\prime \prime}\left(x_{0} \right., \left.y_{0} \right)\\ & C = f_{y y}^{\prime \prime}\left(x_{0}, y_{0} \right) \end{cases}$

$A C$ $-$ $B^{2}$ $>$ $0$ $\Rightarrow$ $\begin{cases} & A>0 \Rightarrow 极小值点 \\ & A<0 \Rightarrow 极大值点 \end{cases}$

$A C$ $-$ $B^{2}$ $>$ $0$ $\Rightarrow$ 点 $\left(x_{0}, y_{0} \right)$ 是极值点.

$A C$ $-$ $B^{2}$ $<$ $0$ $\Rightarrow$ 点 $\left(x_{0}, y_{0} \right)$ 不是极值点.

$A C$ $-$ $B^{2}$ $=$ $0$ $\Rightarrow$ 不确定点 $\left(x_{0}, y_{0} \right)$ 是否是极值点.

$\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $0$, $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $0$

在点 $(x_{0}, y_{0})$ 的邻域内任意一个异于点 $(x_{0}, y_{0})$ 的点 $(x, y)$, 都使得 $f(x, y) < f(x_{0}, y_{0})$（或 $f(x, y)$ $>$ $f(x_{0}, y_{0})$）成立 $\Rightarrow$ $f(x_{0}, y_{0})$ 是函数 $z$ $=$ $f(x, y)$ 的一个极大值（或极小值）

$\left\{\begin{array}{l} F_{x}^{\prime}+F_{y}^{\prime} \frac{\mathrm{d} y}{\mathrm{~d} x}+F_{z}^{\prime} \frac{\mathrm{d} z}{\mathrm{~d} x}=0 \\ G_{x}^{\prime}+G_{y}^{\prime} \frac{\mathrm{d} y}{\mathrm{~d} x}+G_{z}^{\prime} \frac{\mathrm{d} z}{\mathrm{~d} x}=0 \end{array}\right.$

$\frac{\partial z}{\partial x}$ $=$ $-$ $\frac{F_{x}^{\prime}(x, y, z)}{F_{z}^{\prime}(x, y, z)}$, $\frac{\partial z}{\partial y}$ $=$ $-$ $\frac{F_{y}^{\prime}(x, y, z)}{F_{z}^{\prime}(x, y, z)}$

$\frac{\mathrm{d} y}{\mathrm{~d} x}=$ $-\frac{F_{x}^{\prime}(x, y)}{F_{y}^{\prime}(x, y)}$

$\left\{\begin{array}{l} \frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}+\frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x} \\ \frac{\partial z}{\partial y}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}+\frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial y} \end{array}\right.$

$\left\{\begin{array}{l} \frac{\partial z} {\partial x}=\frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v} \cdot \frac{\partial v}{\partial x}, \\ \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v} \cdot \frac{\partial v}{\partial y} . \end{array}\right.$

$\frac{\mathrm{d} z}{\mathrm{~d} x}$ $=$ $\frac{\partial z}{\partial u}$ $\cdot$ $\frac{\mathrm{d} u}{\mathrm{d} x}$ $+$ $\frac{\partial z}{\partial v}$ $\cdot$ $\frac{\mathrm{d} v}{\mathrm{d} x}$

函数 $z$ $=$ $f(x, y)$ 在点 $(x, y)$ 处可微 $\Rightarrow$ 偏导数 $\frac{\partial z}{\partial x}$ 和 $\frac{\partial z}{\partial y}$ 必存在.

偏导数 $\frac{\partial z}{\partial x}$ 和 $\frac{\partial z}{\partial y}$ 存在 $\Rightarrow$ 函数 $z$ $=$ $f(x, y)$ 在点 $(x, y)$ 处不一定可微.

偏导数 $\frac{\partial z}{\partial x}$ 和 $\frac{\partial z}{\partial y}$ 在点 $(x, y)$ 的某邻域内存在且连续 $\Rightarrow$ 函数 $z$ $=$ $f(x, y)$ 在点 $(x, y)$ 处可微.

$f_{x y}^{\prime \prime}(x, y)$ $=$ $f_{y x}^{\prime \prime}(x, y)$