【高等数学】多元微分学 （一）

偏导数

偏导数定义

• 如果二元函数 $f$ 在 $x_0,y_0$ 的某邻域有定义, 且下述极限存在
  $$\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$$
  其极限称作 $f$ 关于 $x$ 的偏导数, 记为

$$\frac{\partial f}{\partial x}{|}_{(x_0,y_0)}=\underset{\Delta x}{\lim }\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$$

类似的
$$\frac{\partial f}{\partial y}{|}_{(x_0,y_0)}=\underset{\Delta y}{\lim }\frac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}$$

• $n$ 元函数的偏导数 $u=f(x_1,\cdots ,x_n)$,
  $$\frac{\partial f}{\partial x_i}=\underset{\Delta x_i}{\lim }\frac{f(x_1,\cdots ,x_i+\Delta x_i,\cdots ,x_n)-f(x_1,\cdots ,x_i,\cdots ,x_n)}{\Delta x_i}$$

导数性质 $\to$ 偏导数性质

$$加:\frac{\partial (f(x,y)+g(x,y))}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial g}{\partial x}$$
$$减:\frac{\partial (f(x,y)-g(x,y))}{\partial x}=\frac{\partial f}{\partial x}-\frac{\partial g}{\partial x}$$
$$乘:\frac{\partial (f(x,y)g(x,y))}{\partial x}=\frac{\partial f}{\partial x}g+f\frac{\partial g}{\partial x}$$
$$除:\frac{\partial (\frac{f(x,y)}{g(x,y)})}{\partial x}=\frac{1}{g^2}\left(\frac{\partial f}{\partial x}g-f\frac{\partial g}{\partial x}\right)$$

高阶偏导数

$$\frac{{\partial }^{2}f}{\partial x^2}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial x}\right)$$
$$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial x}\right)$$
$$\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)$$
$$\frac{{\partial }^{2}f}{\partial y^2}=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial y}\right)$$

• 当 $\frac{{\partial }^{2}f}{\partial y\partial x}$, $\frac{{\partial }^{2}f}{\partial x\partial y}$ 是连续函数时, $\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{{\partial }^{2}f}{\partial x\partial y}$.

全微分

$u=f(x,y)$, $\Delta u= f(x+\Delta x, y+\Delta y)-f(x,y) $

定义 存在 $A,B$, $\Delta z=A\Delta x+B\Delta y+o(\rho )$, $\rho =\sqrt{(\Delta x{)}^2+(\Delta y{)}^2}$, 称函数 $f$ 在 $(x,y)$ 处可微, $dz$ 称为全微分, $(A,B)$ 称为梯度.

当函数 $f$ 可微时, $dz=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$, 梯度计算公式为 $\nabla f(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}{)}^{\top }$

微分性质 $\to$ 全微分性质

可微函数:
$$加:d(f+g)=df+dg$$
$$减:d(f-g)=df-dg$$
$$乘:d(fg)=gdf+fdg$$
$$除:d\left(\frac{f}{g}\right)=\frac{gdf-fdg}{g^2}$$

偏导数性质 $\to$ 梯度性质

可微函数:
$$加:\nabla (f+g)=\nabla f+\nabla g$$
$$减:\nabla (f-g)=\nabla f-\nabla g$$
$$乘:\nabla (fg)=g\nabla f+f\nabla g$$
$$除:\nabla \left(\frac{f}{g}\right)=\frac{g\nabla f-f\nabla g}{g^2}$$

当 $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ 是连续函数时, $f(x,y)$ 可微.

复合函数的微分法

双层复合偏导

一元内嵌一元函数 (全导数)

• $\frac{d}{dx}f(u(x))=\frac{df}{du}\frac{du}{dx}$

一元内嵌二元函数

• $\frac{\partial }{\partial x}f(u(x,y))=\frac{df}{du}\frac{\partial u}{\partial x}$
• $\frac{\partial }{\partial y}f(u(x,y))=\frac{df}{du}\frac{\partial u}{\partial y}$

一元内嵌三元函数

• $\frac{\partial }{\partial x}f(u(x,y,z))=\frac{df}{du}\frac{\partial u}{\partial x}$
• $\frac{\partial }{\partial y}f(u(x,y,z))=\frac{df}{du}\frac{\partial u}{\partial y}$
• $\frac{\partial }{\partial z}f(u(x,y,z))=\frac{df}{du}\frac{\partial u}{\partial z}$

二元内嵌一元函数 (全导数)

• $\frac{\partial }{\partial x}f(u(x),v(x))=\frac{\partial f}{\partial u}\frac{du}{dx}+\frac{\partial f}{\partial v}\frac{dv}{dx}$

二元内嵌二元函数

• $\frac{\partial }{\partial x}f(u(x,y),v(x,y))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}$
• $\frac{\partial }{\partial y}f(u(x,y),v(x,y))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}$

二元内嵌三元函数

• $\frac{\partial }{\partial x}f(u(x,y,z),v(x,y,z))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}$

• $\frac{\partial }{\partial y}f(u(x,y,z),v(x,y,z))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}$

• $\frac{\partial }{\partial z}f(u(x,y,z),v(x,y,z))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial z}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial z}$

三元内嵌一元函数 (全导数)

• $\frac{\partial }{\partial x}f(u(x),v(x),w(x))=\frac{\partial f}{\partial u}\frac{du}{dx}+\frac{\partial f}{\partial v}\frac{dv}{dx}+\frac{\partial f}{\partial w}\frac{dw}{dx}$

三元内嵌二元函数

• $\frac{\partial }{\partial x}f(u(x,y),v(x,y),w(x,y))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial x}$

• $\frac{\partial }{\partial y}f(u(x,y),v(x,y),w(x,y))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial y}$

三元内嵌三元函数

• $\frac{\partial }{\partial x}f(u(x,y,z),v(x,y,z),w(x,y,z))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial x}$

• $\frac{\partial }{\partial y}f(u(x,y,z),v(x,y,z),w(x,y,z))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial y}$

• $\frac{\partial }{\partial z}f(u(x,y,z),v(x,y,z),w(x,y,z))=\frac{\partial f}{\partial u}\frac{\partial u}{\partial z}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial z}+\frac{\partial f}{\partial w}\frac{\partial w}{\partial z}$

(选看) 三层复合偏导

二元内嵌二元内嵌二元函数

• $\frac{\partial }{\partial s}f(u(x(s,t),y(s,t)),v(x(s,t),y(s,t)))=\frac{\partial f}{\partial u}(\frac{\partial u}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial s})+\frac{\partial f}{\partial v}(\frac{\partial v}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial v}{\partial y}\frac{\partial y}{\partial s})$

• $\frac{\partial }{\partial t}f(u(x(s,t),y(s,t)),v(x(s,t),y(s,t)))=\frac{\partial f}{\partial u}(\frac{\partial u}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial t})+\frac{\partial f}{\partial v}(\frac{\partial v}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial v}{\partial y}\frac{\partial y}{\partial t})$