高数笔记-第九章 多元函数微分法及其应用-2

第九章 多元函数微分法及其应用

第二节 偏导数

偏导数的定义及其计算法

二元函数 z = f ( x , y ) z = f(x, y) z=f(x,y)对于 x x x的偏导数有如下定义:
定义 设函数 z = f ( x , y ) z = f(x, y) z=f(x,y) 在点 ( x 0 , y 0 ) (x_0, y_0) (x0,y0)的某一邻域内有定义,当 y y y固定在 y 0 y_0 y0 x x x x 0 x_0 x0处有增量 Δ x \Delta x Δx时,相应地函数有增量
f ( x 0 + Δ x , y 0 ) − f ( x 0 , y 0 ) f(x_0 + \Delta x, y_0) - f(x_0, y_0) f(x0+Δx,y0)f(x0,y0)
如果
lim ⁡ Δ x → 0 f ( x 0 + Δ x , y 0 ) − f ( x 0 , y 0 ) Δ x (2-1) \lim\limits_{\Delta x \to 0}{\dfrac{f(x_0 + \Delta x, y_0) -f(x_0, y_0)}{\Delta x}} \tag {2-1} Δx0limΔxf(x0+Δx,y0)f(x0,y0)(2-1)
存在,那么称此极限为函数 z = f ( x , y ) z = f(x, y) z=f(x,y)在点 ( x 0 , y 0 ) (x_0, y_0) (x0,y0)处对 x x x的偏导数,记作
∂ z ∂ x ∣ x = x 0 y = y 0 \left. \dfrac{\partial z}{\partial x} \right|_{x = x_0\atop y = y_0} xzy=y0x=x0, ∂ f ∂ x ∣ x = x 0 y = y 0 \left. \dfrac{\partial f}{\partial x}\right |_{x = x_0 \atop y = y_0} xfy=y0x=x0, z x ∣ x = x 0 y = y 0 \left .z_x\right|_{x = x_0 \atop y = y_0} zxy=y0x=x0 f x ( x 0 , y 0 ) f_x(x_0, y_0) fx(x0,y0)

极限(2-1)可以表示为
f x ( x 0 , y 0 ) = lim ⁡ Δ x → 0 f ( x 0 + Δ x ) − f ( x 0 , y 0 ) Δ x f_x(x_0, y_0) = \lim\limits_{\Delta x \to 0}{\dfrac{f(x_0 + \Delta x) - f(x_0, y_0)}{\Delta x}} fx(x0,y0)=Δx0limΔxf(x0+Δx)f(x0,y0)

类似地,函数 z = f ( x , y ) z = f(x, y) z=f(x,y)在点 ( x 0 , y 0 ) (x_0, y_0) (x0,y0)处对 y y y的偏导数定义为
lim ⁡ Δ y → 0 f ( x 0 , y 0 + Δ y ) − f ( x 0 , y 0 ) Δ y \lim\limits_{\Delta y \to 0}{\dfrac{f(x_0, y_0 + \Delta y) - f(x_0, y_0)}{\Delta y}} Δy0limΔyf(x0,y0+Δy)f(x0,y0)
记作 ∂ z ∂ y ∣ x = x 0 y = y 0 \left. \dfrac{\partial z}{\partial y}\right |_{x = x_0 \atop y = y_0} yzy=y0x=x0 , ∂ f ∂ y ∣ x = x 0 y = y 0 \left. \dfrac{\partial f}{\partial y}\right |_{x = x_0 \atop y = y_0} yfy=y0x=x0, z y ∣ x = x 0 y = y 0 \left . z_y \right|_{x = x_0 \atop y = y_0} zyy=y0x=x0 f y ( x 0 , y 0 ) f_y(x_0, y_0) fy(x0,y0)

三元函数的偏导数

f x ( x , y , z ) = lim ⁡ Δ x → 0 f ( x + Δ x , y , z ) − f ( x , y , z ) Δ x f_x(x, y, z) = \lim\limits_{\Delta x \to 0}{\dfrac{f(x + \Delta x, y, z) - f(x, y, z)}{\Delta x}} fx(x,y,z)=Δx0limΔxf(x+Δx,y,z)f(x,y,z)

例题

例 1 求 z = x 2 + 3 x y + y 2 z = x^2 + 3xy + y^2 z=x2+3xy+y2在点 ( 1 , 2 ) (1, 2) (1,2)处的偏导数.
例 2 求 z = x 2 sin ⁡ 2 y z = x^2\sin2y z=x2sin2y的偏导数.
例 3 设 z = x y ( x > 0 , x ≠ 1 ) z = x ^y (x > 0, x \neq1) z=xy(x>0,x=1),求证:
x y ∂ z ∂ x + 1 ln ⁡ x ∂ z ∂ y = 2 z \dfrac{x}{y}\dfrac{\partial z}{\partial x} + \dfrac{1}{\ln x}\dfrac{\partial z}{\partial y} = 2z yxxz+lnx1yz=2z.
例 4 求 r = x 2 + y 2 + z 2 r = \sqrt{x^2 + y^2 + z^2} r=x2+y2+z2 的偏导数.
例 5 已知理想气体的状态方程 p V = R T pV = RT pV=RT R R R为常量),求证:
∂ p ∂ V ⋅ ∂ V ∂ T ⋅ ∂ T ∂ p = − 1. \dfrac{\partial p}{\partial V} \cdot \dfrac{\partial V}{\partial T} \cdot \dfrac{\partial T}{\partial p} = -1. VpTVpT=1.

二元函数偏导数的几何意义.
高阶偏导数

未完待续 ⋯ \cdots

你可能感兴趣的:(高数)