高数笔记 P05：多元函数微分学

1 多元函数的极限、连续、偏导数、全微分

极限

\(\displaystyle \lim_{x \to x_0, y \to y_0} f(x, y) = A\)，以任意方式趋向都成立，极限才存在。

连续

\(\displaystyle \lim_{x \to x_0, y \to y_0} f(x, y) = f(x_0, y_0)\)

极限和连续的多数性质与一元函数相同或类似。

偏导数

\(\displaystyle f_x'(x, y) = \lim_{\Delta x \to 0} \cfrac {f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x}\)

\(\displaystyle f_y'(x, y) = \lim_{\Delta y \to 0} \cfrac {f(x_0, y_0 + \Delta y) - f(x_0, y_0)}{\Delta y}\)

• 多元函数与一元函数复合：若函数\(u = \varphi(t), v = \psi(t)\)都在点\(t\)可导，函数\(z = f(x, y)\)在对应点\((u, v)\)具有连续一阶偏导数，则复合函数\(z = f[\varphi(t), \psi(t)]\)在点\(t\)可导，且\(\cfrac {dz}{dt} = \cfrac {\partial z}{\partial u} \cfrac {du}{dt} + \cfrac {\partial z}{\partial v} \cfrac {dv}{dt}\)。
• 多元函数与多元函数复合：若函数\(u = \varphi(x, y), v = \psi(x, y)\)都在点\((x, y)\)有对\(x,y\)的偏导数，函数\(z = f(x, y)\)在对应点\((u, v)\)具有连续一阶偏导数，则复合函数\(z = f[\varphi(x, y), \psi(x, y)]\)在点\((x, y)\)有对\(x, y\)的偏导数，且\(\cfrac {\partial z}{\partial x} = \cfrac {\partial z}{\partial u} \cfrac {\partial u}{\partial x} + \cfrac {\partial z}{\partial v} \cfrac {\partial v}{\partial x}, \quad \cfrac {\partial z}{\partial y} = \cfrac {\partial z}{\partial u} \cfrac {\partial u}{\partial y} + \cfrac {\partial z}{\partial v} \cfrac {\partial v}{\partial y}\)。
• 高阶偏导数\(f_{xx}'', f_{yy}'', f_{xy}'', f_{yx}''\)
  • 若\(f_{xy}'', f_{yx}''\)在点\((x_0, y_0)\)处连续，则在该点\(f_{xy}'' = f_{yx}''\)。
• 拉普拉斯方程：\(\cfrac {\partial^2 u}{\partial x^2} + \cfrac {\partial^2 u}{\partial y^2} + \cfrac {\partial^2 u}{\partial z^2} = 0\)

全微分

\(dz = A\Delta x + B\Delta y\)

• 可微的充分条件：函数\(z = f(x, y)\)的偏导数\(\cfrac{\partial z}{\partial x}, \cfrac{\partial z}{\partial y}\)在点\((x, y)\) 处连续，则函数在该点可微。
• 可微的必要条件：函数\(z = f(x, y)\)在点\((x, y)\) 处可微，则函数在该点偏导数必存在。
• 全微分形式不变性：若函数\(z = f(u, v)\)和\(u = \varphi(x, y), v = \psi(x, y)\)都具有连续的一阶偏导数，则复合函数可微，且\(dz = \cfrac {\partial z}{\partial x}dx + \cfrac {\partial z}{\partial y}dy = \cfrac {\partial z}{\partial u}du + \cfrac {\partial z}{\partial v}dv\)。

2 多元函数的极值与最值

多元函数极值

• 极值点：若一点大于等于或者小于等于其某个邻域内的所有的点，这个点就是一个极值点。
• 驻点：满足偏导数全为 0 的点。
• 这里可以看出多元函数极值点不等价于驻点。极值点一定是驻点，但是驻点不一定是极值点。
• 取得极值点的充分条件：若\(z = f(x, y)\)在点\((x_0, y_0)\)的某邻域内有连续的二阶偏导数，且\(f_x'(x_0, y_0) = 0, f_y'(x_0, y_0) = 0\)。令\(A = f_{xx}''(x_0, y_0), B = f_{xy}''(x_0, y_0), C = f_{yy}''(x_0, y_0)\)，则
  1. \(AC - B^2 \gt 0\)时，点\((x_0, y_0)\)为极值点，且当\(A \gt 0\)时取极小值，当\(A \lt 0\)时取极大值。
  2. \(AC - B^2 \lt 0\)时，点\((x_0, y_0)\)不为极值点。
  3. \(AC - B^2 = 0\)时，不能确定，需进一步讨论，比如使用极值的定义。

条件极值--拉格朗日乘子法

1. 二元：构造\(F(x, y, \lambda) = f(x, y) + \lambda \varphi(x, y)\)，解方程组\(\begin{cases} \cfrac {\partial F}{\partial x} &= \cfrac {\partial f}{\partial x} + \lambda \cfrac {\partial \varphi}{\partial x} = 0 \\ \cfrac {\partial F}{\partial y} &= \cfrac {\partial f}{\partial y} + \lambda \cfrac {\partial \varphi}{\partial y} = 0 \\ \cfrac {\partial F}{\partial \lambda} &= \varphi(x, y) = 0 \\ \end{cases}\)。所有满足该方程组的解都是\(f(x, y)\)在\(\varphi(x, y) = 0\)下的条件极值。
2. 三元两条件：构造\(F(x, y, z, \lambda, \mu) = f(x, y, z) + \lambda \varphi (x, y, z) + \mu \psi (x, y, z)\)，解方程组求解。

3 泰勒公式

若二元函数\(f(x, y)\)在点\(P_0(x_0, y_0)\)的某邻域\(U(P_0)\)内具有二阶连续偏导数，点\(P(x, y) \in U(P_0)\)，则

• \(f(x, y) = f(x_0, y_0) +f_x'(x_0, y_0)(x - x_0) + f_y'(x_0, y_0)(y - y_0) + R_1\)

  \(R_1 = \cfrac 1{2!}[\cfrac {\partial^2f(P_1)}{\partial x^2}(x - x_0)^2 + 2\cfrac {\partial^2 f(P_1)}{\partial x \partial y}(x - x_0)(y - y_0) + \cfrac {\partial^2 f(P_1}{\partial y^2}(y - y_0)^2]\)，

  其中\(P_1(x_0 + \theta(x - x_0), y_0 + \theta(y - y_0)), \theta \in (0,1)\)。\(R_1\)称为拉格朗日余项。

• \(f(x, y) = f(x_0, y_0) +f_x'(x_0, y_0)(x - x_0) + f_y'(x_0, y_0)(y - y_0) \\ + \cfrac 1{2!}[\cfrac {\partial^2f(x_0, y_0)}{\partial x^2}(x - x_0)^2 + 2\cfrac {\partial^2 f(x_0, y_0)}{\partial x \partial y}(x - x_0)(y - y_0) + \cfrac {\partial^2 f(x_0 , y_0)}{\partial y^2}(y - y_0)^2] + \omicron(\rho^2)\)

  \(\omicron(\rho^2)\)称为佩亚诺余项。