高數筆記 P05：多元函數微分學

1 多元函數的極限、連續、偏導數、全微分

極限

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

連續

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

極限和連續的多數性質與一元函數相同或類似。

偏導數

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

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

• 多元函數與一元函數復合：若函數\(u = \varphi(t), v = \psi(t)\)都在點\(t\)可導，函數\(z = f(x, y)\)在對應點\((u, v)\)具有連續一階偏導數，則復合函數\(z = f[\varphi(t), \psi(t)]\)在點\(t\)可導，且\(\cfrac {dz}{dt} = \cfrac {\partial z}{\partial u} \cfrac {du}{dt} + \cfrac {\partial z}{\partial v} \cfrac {dv}{dt}\)。
• 多元函數與多元函數復合：若函數\(u = \varphi(x, y), v = \psi(x, y)\)都在點\((x, y)\)有對\(x,y\)的偏導數，函數\(z = f(x, y)\)在對應點\((u, v)\)具有連續一階偏導數，則復合函數\(z = f[\varphi(x, y), \psi(x, y)]\)在點\((x, y)\)有對\(x, y\)的偏導數，且\(\cfrac {\partial z}{\partial x} = \cfrac {\partial z}{\partial u} \cfrac {\partial u}{\partial x} + \cfrac {\partial z}{\partial v} \cfrac {\partial v}{\partial x}, \quad \cfrac {\partial z}{\partial y} = \cfrac {\partial z}{\partial u} \cfrac {\partial u}{\partial y} + \cfrac {\partial z}{\partial v} \cfrac {\partial v}{\partial y}\)。
• 高階偏導數\(f_{xx}'', f_{yy}'', f_{xy}'', f_{yx}''\)
  • 若\(f_{xy}'', f_{yx}''\)在點\((x_0, y_0)\)處連續，則在該點\(f_{xy}'' = f_{yx}''\)。
• 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

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

• 可微的充分條件：函數\(z = f(x, y)\)的偏導數\(\cfrac{\partial z}{\partial x}, \cfrac{\partial z}{\partial y}\)在點\((x, y)\) 處連續，則函數在該點可微。
• 可微的必要條件：函數\(z = f(x, y)\)在點\((x, y)\) 處可微，則函數在該點偏導數必存在。
• 全微分形式不變性：若函數\(z = f(u, v)\)和\(u = \varphi(x, y), v = \psi(x, y)\)都具有連續的一階偏導數，則復合函數可微，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 多元函數的極值與最值

多元函數極值

• 極值點：若一點大於等於或者小於等於其某個鄰域內的所有的點，這個點就是一個極值點。
• 駐點：滿足偏導數全為 0 的點。
• 這里可以看出多元函數極值點不等價於駐點。極值點一定是駐點，但是駐點不一定是極值點。
• 取得極值點的充分條件：若\(z = f(x, y)\)在點\((x_0, y_0)\)的某鄰域內有連續的二階偏導數，且\(f_x'(x_0, y_0) = 0, f_y'(x_0, y_0) = 0\)。令\(A = f_{xx}''(x_0, y_0), B = f_{xy}''(x_0, y_0), C = f_{yy}''(x_0, y_0)\)，則
  1. \(AC - B^2 \gt 0\)時，點\((x_0, y_0)\)為極值點，且當\(A \gt 0\)時取極小值，當\(A \lt 0\)時取極大值。
  2. \(AC - B^2 \lt 0\)時，點\((x_0, y_0)\)不為極值點。
  3. \(AC - B^2 = 0\)時，不能確定，需進一步討論，比如使用極值的定義。

條件極值--拉格朗日乘子法

1. 二元：構造\(F(x, y, \lambda) = f(x, y) + \lambda \varphi(x, y)\)，解方程組\(\begin{cases} \cfrac {\partial F}{\partial x} &= \cfrac {\partial f}{\partial x} + \lambda \cfrac {\partial \varphi}{\partial x} = 0 \\ \cfrac {\partial F}{\partial y} &= \cfrac {\partial f}{\partial y} + \lambda \cfrac {\partial \varphi}{\partial y} = 0 \\ \cfrac {\partial F}{\partial \lambda} &= \varphi(x, y) = 0 \\ \end{cases}\)。所有滿足該方程組的解都是\(f(x, y)\)在\(\varphi(x, y) = 0\)下的條件極值。
2. 三元兩條件：構造\(F(x, y, z, \lambda, \mu) = f(x, y, z) + \lambda \varphi (x, y, z) + \mu \psi (x, y, z)\)，解方程組求解。

3 泰勒公式

若二元函數\(f(x, y)\)在點\(P_0(x_0, y_0)\)的某鄰域\(U(P_0)\)內具有二階連續偏導數，點\(P(x, y) \in U(P_0)\)，則

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

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

  其中\(P_1(x_0 + \theta(x - x_0), y_0 + \theta(y - y_0)), \theta \in (0,1)\)。\(R_1\)稱為拉格朗日余項。

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

  \(\omicron(\rho^2)\)稱為佩亞諾余項。