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【高等数学】多元微分学（一）

news 2026/2/8 12:30:16

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
$\lim_{\Delta x\to 0} \frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$
其极限称作 $f$ 关于 $x$ 的偏导数, 记为

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

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

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

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, $d z$ 称为全微分, $(A, B)$ 称为梯度.

当函数 $f$ 可微时, $\frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y}dy$ , 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:
$加 : d (f + g) = df + d g$
$减 : d (f - g) = df - d g$
$乘 : d (f g) = g df + fd g$
$除:d\left(\frac{f}{g}\right)=\frac{g df- f dg}{g^2}$

偏导数性质 $\to$ 梯度性质

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

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)

$\frac{d }{d x}f(u(x)) =\frac{d f}{d u}\frac{d u}{d x}$

一元内嵌二元函数

$\frac{\partial }{\partial x}f(u(x,y)) =\frac{d f}{d u}\frac{\partial u}{\partial x}$
$\frac{\partial }{\partial y}f(u(x,y)) =\frac{d f}{d u}\frac{\partial u}{\partial y}$

一元内嵌三元函数

$\frac{\partial }{\partial x}f(u(x,y,z)) =\frac{d f}{d u}\frac{\partial u}{\partial x}$
$\frac{\partial }{\partial y}f(u(x,y,z)) =\frac{d f}{d u}\frac{\partial u}{\partial y}$
$\frac{\partial }{\partial z}f(u(x,y,z)) =\frac{d f}{d u}\frac{\partial u}{\partial z}$

二元内嵌一元函数 (全导数)

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

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)

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

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

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

偏导数