# 偏導數

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 簡介

$z = f(x, y) = x^2 + xy + y^2$
f = x2 + xy + y2的圖像。我們希望求出函數在點（1, 1, 3）的對x的偏導數；對應的切線與xOz平面平行。

$\frac{\partial z}{\partial x} = 2x+y$

$\frac{\part f}{\part x} = 3$

## 定義

$f(x,y) = f_x(y) = \,\! x^2 + xy + y^2$

$f_x(y) = x^2 + xy + y^2$

$f_a(y) = a^2 + ay + y^2$

$f_a'(y)= a + 2y$

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

$\frac{\part f}{\part x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots,a_n)}{h}$

$\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(a_1,\ldots,a_n) = \frac{\part f}{\part x_i}(a_1,\ldots,a_n)$

$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right)$

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

$\nabla = \sum_{j=1}^n \bigg[{\frac{\partial}{\partial x_j}}\bigg] \mathbf{\hat{e}_j} = \bigg[{\frac{\partial}{\partial x_1}}\bigg] \mathbf{\hat{e}_1} + \bigg[{\frac{\partial}{\partial x_2}}\bigg] \mathbf{\hat{e}_2} + \bigg[{\frac{\partial}{\partial x_3}}\bigg] \mathbf{\hat{e}_3} + \dots + \bigg[{\frac{\partial}{\partial x_n}}\bigg] \mathbf{\hat{e}_n}$

## 例子

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

V關於r的偏導數為：

$\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3}$

$\frac{ \partial V}{\partial h} = \frac{\pi r^2}{3}$

$\frac{\operatorname dV}{\operatorname dr} = \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{\partial h}{\partial r}$

$\frac{\operatorname dV}{\operatorname dh} = \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{\partial r}{\partial h}$

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

$\frac{\operatorname dV}{\operatorname dr} = \frac{2 \pi r h}{3} + k\frac{\pi r^2}{3}$

$\frac{\operatorname dV}{\operatorname dr} = k\pi r^2$

$\frac{\operatorname dV}{\operatorname dh} = k\pi r^2$

## 記法

f的一階偏導數為：

$\frac{ \partial f}{ \partial x} = f_x = \partial_x f$

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

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

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

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

## 正式定義和性質

$\frac{ \partial }{\partial x_i }f(\mathbf{a}) = \lim_{h \rightarrow 0}{ f(a_1, \dots , a_{i-1}, a_i+h, a_{i+1}, \dots ,a_n) - f(a_1, \dots ,a_n) \over h }$

$\frac{\partial^2f}{\partial x_i\, \partial x_j} = \frac{\partial^2f} {\partial x_j\, \partial x_i}$

## 參考文獻

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