ch08第八章多元函數(shù)微分法及其應(yīng)用-02第二節(jié)偏導(dǎo)數(shù)

第二節(jié)導(dǎo)分布圖★★★★★★★★★★★★★★★★8—★內(nèi)容要一、dy可看作函數(shù)的微分dy與自變量的微分dx的商.x是一個(gè)整體，在一元函數(shù)微分學(xué)中知道，如果函數(shù)在某點(diǎn)存在導(dǎo)數(shù)，則它在該點(diǎn)必定連續(xù)，二、

2

21zf(xy的兩個(gè)二階混合偏導(dǎo)數(shù)yxxyD內(nèi)連續(xù),2 2

yxxy例題選例1 求z(x,y)x23xyy2在點(diǎn)(1,2)處的偏導(dǎo)數(shù) 把y看作常數(shù),對(duì)x求導(dǎo)得fx(x,y)2xx看作常數(shù),yfy(x,y)3x2

fx(1,2)2132fy(1,2)3122例2（E02）

xz

z2z

yxy1,

xylnx

y

lnxxz1

xyxy1

xyxy原結(jié)論成立

y

lnx 3（E03）求三元函數(shù)usinxy2ez的偏導(dǎo)數(shù)uuuxy解yx看作常數(shù),xucos(xy2ezxz看作常數(shù),yu2ycos(xy2ezxy看作常數(shù),zuezcos(xy2ezx2y2例4（E04）求r 把x2y2x2y2x2y2z

xr 可ry

rz 5

f(x,y)

2y

xyxy

f

(0,0f(x,在(0,0點(diǎn)不連續(xù) f

(0,0)

f(0x,0)f

0

fy(0,0)

f(0,0y)f

0

f(0,0),f(0,0)存在.但由上節(jié)的例8知道,

不存在,f(xy) )

x0x2y6（E05）z4x33x2y3xy2xy,2x2

2

2

2y2

3x3

12x26xy3y23x26xy2

224x6y,

2

26y,

6x67

ueax

求二階偏導(dǎo)數(shù) uaeaxcosby,

beax x

cosby,

2u

2

sinby,yx

8（E06）zxlnxy的二階偏導(dǎo)數(shù)

ln(xy)

xx

,

xx2

x

xyx(x

x2y,(xy)22

x (xy)22

x

(x

(xy)22

(xy)(x

(xy)2

u(x,y)

x22ux2

x2y1ln(x2x2y2

0u

x2y

,

x2y

(x2y2)x2x(x2y2(x2y2)y2y(x2y2

y2(x2y2)2x2y(x2y2)22u2u

y2

x2

(x2y2

(x2y2 例10（E08）證明函數(shù)u1滿(mǎn) r2u2u2u 0其中r證

u

x2y2r1xx2y2 r2 r2 r

3xr

3xx

r3r4

r r

3y ,

3z2

r r

z

r r2u2u2u

3(x2y2z2

3r2

r r

r r 11

f(x,y)

x2yx2y2

(x,

fxy0,0fxy (x,y)解

(0,0)

f(x,0)f(0,0)lim0

xy0

x0f(x,y)f(0, y(x2y2fx(0,y)

x2y

所

(0,0)

fx(0,y)fx

limy0

f(0,0)yy

f(0,y)f(0,0)yf(x,y)f

x(x2y2x0fy

x2y

所

(0,0)

fy(x,0)fy(0,0)limx0

課堂練f(xyP