# 偏导数

## 简介

f = x2 + xy + y2的图像。我们希望求出函数在点(1, 1)的对x的偏导数；对应的切线与xOz平面平行。

${\displaystyle z=f(x,y)=x^{2}+xy+y^{2}}$

${\displaystyle {\frac {\partial f}{\partial x}}=2x+y}$

${\displaystyle {\frac {\partial f}{\partial x}}=3}$

## 定义

${\displaystyle f(x,y)=f_{x}(y)=\,\!x^{2}+xy+y^{2}}$

${\displaystyle f_{x}(y)=x^{2}+xy+y^{2}}$

${\displaystyle f_{a}(y)=a^{2}+ay+y^{2}}$

${\displaystyle f_{a}'(y)=a+2y}$

${\displaystyle {\frac {\partial f}{\partial y}}(x,y)=x+2y}$

${\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{n})}{h}}}$

${\displaystyle {\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{1},\ldots ,a_{n})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})}$

${\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right)}$

${\displaystyle \nabla ={\bigg [}{\frac {\partial }{\partial x}}{\bigg ]}\mathbf {\hat {i}} +{\bigg [}{\frac {\partial }{\partial y}}{\bigg ]}\mathbf {\hat {j}} +{\bigg [}{\frac {\partial }{\partial z}}{\bigg ]}\mathbf {\hat {k}} }$

${\displaystyle \nabla =\sum _{j=1}^{n}{\bigg [}{\frac {\partial }{\partial x_{j}}}{\bigg ]}\mathbf {{\hat {e}}_{j}} ={\bigg [}{\frac {\partial }{\partial x_{1}}}{\bigg ]}\mathbf {{\hat {e}}_{1}} +{\bigg [}{\frac {\partial }{\partial x_{2}}}{\bigg ]}\mathbf {{\hat {e}}_{2}} +{\bigg [}{\frac {\partial }{\partial x_{3}}}{\bigg ]}\mathbf {{\hat {e}}_{3}} +\dots +{\bigg [}{\frac {\partial }{\partial x_{n}}}{\bigg ]}\mathbf {{\hat {e}}_{n}} }$

## 例子

${\displaystyle V(r,h)={\frac {\pi r^{2}h}{3}}}$

V关于r的偏导数为：

${\displaystyle {\frac {\partial V}{\partial r}}={\frac {2\pi rh}{3}}}$ ，它描述了高度固定而半径变化时，圆锥的体积的变化率。

V关于h的偏导数为：

${\displaystyle {\frac {\partial V}{\partial h}}={\frac {\pi r^{2}}{3}}}$ ，它描述了半径固定而高度变化时，圆锥的体积的变化率。

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} r}}=\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}+\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}{\frac {\partial h}{\partial r}}}$

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} h}}=\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}+\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}{\frac {\partial r}{\partial h}}}$

${\displaystyle k={\frac {h}{r}}={\frac {\partial h}{\partial r}}}$

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} r}}={\frac {2\pi rh}{3}}+k{\frac {\pi r^{2}}{3}}}$

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} r}}=k\pi r^{2}}$

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} h}}=\pi r^{2}}$

## 记法

f的一阶偏导数为：

${\displaystyle {\frac {\partial f}{\partial x}}=f_{x}=\partial _{x}f}$

${\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f_{xx}=\partial _{xx}f}$

${\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=f_{xy}=\partial _{yx}f}$

${\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\,\partial y^{j}\,\partial z^{k}}}=f^{(i,j,k)}}$

${\displaystyle \left({\frac {\partial f}{\partial x}}\right)_{y,z}}$

## 正式定义和性质

${\displaystyle {\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )=\lim _{h\rightarrow 0}{f(a_{1},\dots ,a_{i-1},a_{i}+h,a_{i+1},\dots ,a_{n})-f(a_{1},\dots ,a_{n}) \over h}}$

${\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\,\partial x_{i}}}}$

## 参考文献

• George B. Thomas & Ross L. Finney. Calculus and Analytic Geometry. Addison-Wesley Publishing Company, Inc. 1994: 833–840. ISBN 0-201-52929-7.

## 注释

1. ^ 相对于全导数，在其中所有变量都允许变化