多元函数微分学及应用(隐函数反函数)_23405397.pdf

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1、习题课：多元函数求偏导，多元函数微分的应用多元复合函数、隐函数的求导法(1)多元复合函数设二元函数),(vufz在点),(00vu处偏导数连续，二元函数),(),(yxvvyxuu在点),(00yx处 偏 导 数 连 续,并 且),(),(000000yxvvyxuu,则 复 合 函 数),(),(yxvyxufz在点),(00yx处可微，且xyxvvvufxyxuuvufxzyx00000000),(,00yyxvvvufyyxuuvufyzyx00000000),(,00多元函数微分形式的不变性：设),(),(),(yxvvyxuuvufz，均为连续可微，则将z看成yx,的函数，有dyyz

2、dxxzdz计算yvvfyuufyzxvvfxuufxz,，代人，dvvfduufdyyvdxxvvfdyyudxxuufdyyvvfyuufdxxvvfxuufdyyzdxxzdz我们将dvvfduufdyyzdxxzdz叫做微分形式不变性。例1 设xyxyfxz,3，求yzxz,。解：xydfxydfxfdxxdfxdxxfdz213232)(3322132(3xydxxdyfydxxdyfxfdxxdyfxfxdxxyfyfxfx221421323由微分形式不变性，dyfxfxdxxyfyfxfxdyyzdxxzdz221421323故22142132,3fxfxyzxyfyfxfxxz

3、。例2 已知)1(1xyx，求dydx.解 考虑二元函数vuy,uxvx11,，应用推论得.dxdvvydxduuydxdy).ln1(11)(ln112221xxxuuxvuxvv(2)隐函数若函数xyy,由方程0,yxF确定，求导之函数？按隐函数定义有恒等式：0,xyxF0,xyxFdxd，0,xyxyxFxyxFyxxyxFxyxFxyyx,。从这是可见：函数xyy可导有一个必要条件是，0,yxFy.例3 已知函数 yfx()由方程,22bayxfbyax是常数，求导函数。解：方程22yxfbyax两边对 x 求导，dxdyyxyxfdxdyba22)(22)(2)(22222yxfyb

4、ayxfxdxdy一般来说，若函数xyy,由方程0,yxF确定，求导之函数？将y看作是nxx,.,1的函数),.,(1nxxyxyy,对于方程0),.,(,.,(11nnxxyxxF两端分别关于ix求偏导数得到，并解ixf,可得到公式 :yxFyxFxyyxii,例4 设函数y(z)yzxx),(由方程组01201222222zyxzyx确定，求文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编

5、码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I1

6、0 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7

7、 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文

8、档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8

9、I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4

10、J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N

11、9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9dzdydzdx,.解121222222zyxzyxzdydzydxdzxzdydzydxdzx242222解方程得：dzdydzdx=xzyzxyzzxxyyxy8124122222441由此得到yzdzdyxzdzdx2,3.例5 已知函数yxzz,由参数方程:uvzvuyvuxsincos,给定，试求yzxz,.解这个问题涉及到复合函数微分法与隐函数微分法.yx,是自变量,vu,是中间变量(vu,是yx,的函数),先由zuv得到xvuxuvxvvzxuuzxzyvuyuvyvvzyuuzyzu v,是由方

12、程),(),(yxvvyxuu的 x y,的隐函数，在这两个等式两端分别关于x y,求偏导数，得xvvuxuvxvvuxuvcossin0sincos1，yvvuyuvyvvuyuvcossin1sincos0得到uvxvvyuuuxvvxucos,sin,sin,cos将这个结果代入前面的式子,得到vvvxvuxuvxzsincos与vvvyvuyuvyzcossin(3)隐函数 函数),(yxuu由方程0),(0),(),(tzhtzygtzyxfu确定，求yuxu,解：函数关系分析:5(变量)3(方程)=2(自变量);一函 (u),二自(x,y),二中(z,t)文档编码:CG3N9K9B

13、8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R

14、4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7

15、N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K

16、9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U

17、9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8

18、P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N

19、9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9xfxu,yttfyzzfyfyu0),(),(1tgzgzhtgthtzhgytyz,zhtgthzgygthzfzhtfyfyu.二阶偏导数：一阶导函数的偏导数例6),(yxzz由2222azyx决定，求yxz2解：022xzzx，022yzzyzyyzzxxz,xzzyyxz223z

20、xy例7 设22,xxxfxg，其中函数f 于的二阶偏导数连续，求22dxxgd例8 设 zfxyxy(,)，f 二阶连续可微，求22xz.解记yxvxyu,;vffuff21,22222211,vffuff,uvffvuff221212,则211fyfyxvvfxuufxz,xfyxfyxzxxz21221因为vffuff21,都是以u v,为中间变量，以yx,为自变量的函数，所以xvfxufxf1211112111fyfyxvfxufxf2221222211fyfy将以上两式代入前式得:fyffyxz222121122212.文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN

21、8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码

22、:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10

23、 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7

24、ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档

25、编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I

26、10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J

27、7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9例9 设),(yxzz二阶连续可微，并且满足方程0222222yzCyxzBxzA若令,yxvyxu试确定,为何值时能变原方程为02vuz.解将yx,看成自变量，vu,看成中间变量，利用链式法则得zvuvzuzxvvzxuuzxzzvuvzuzyvvzyuuzyzzvuvzvuzuzvzuzxxz222222222222222

28、2222vzvuzuzvzuzyyzzvu2222222vzvuzuzvzuzxyxz=zvuvu由此可得,2222220yzCyxzBxzA=vuzCBAuzCBA2222222222vzCBA=0 只要选取,使得020222CBACBA,可得02vuz.问题成为方程022tCtBA有两不同实根，即要求：02CAB.令ACBB2,ACBB2,即可。此时，02vuz02vuz0vzuvvzufdvvz.yxgyxfvgufz.例10设2),(Cyxu,又02222yuxu,xxxu)2,(,2)2,(xxxux,求)2,(xxuxx,)2,(xxuxy)2,(xxuyy解:2)2,(xxxxu

29、,两边对 x 求导,文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编

30、码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I1

31、0 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7

32、 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文

33、档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8

34、I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4

35、J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9xxxyxuxxxu22)2,()2,(222.(1)xxxu)2,(,两边对 x 求导,122,2,xxyuxxxu,212,2xxxyu.两再边对 x 求导,xxxyuxxyxu2)2,()2,(222.(2)由已知02,2,2222xxy

36、uxxxu,(3)(1),(2),(3)联立可解得:xxxyxuxxxyuxxxu352,342,2,22222多元微分的应用:几何应用，物理应用极值与条件极值问题条件极值无条件极值极值切平面法线曲面法平面切线曲线几何多元微分的应用空间曲面(1)空间曲面的表达式显函数表示：yxfz,隐函数表示:0,zyxF参数表示：2),(),(),(),(RDvuvuzzvuyyvuxxuv(2)空间曲面的切平面与法线文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8

37、I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4

38、J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N

39、9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9

40、B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9

41、R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P

42、7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9

43、K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9空间曲面 S 由显函数表示yxfz,，设000,yxfz，空间曲面 S 过000,yxP切平面方程为0,000000zzyyyyxfxxxyxf法线方程是1,0000000zzyyxfyyxyxfxx法向量为1,0000yyxfxyxfn空间曲面 S 存在切平面的条件：若曲面 S 由显函数表示yxfz,在点00,yxp可微,则曲面 S 在点00,yxp有不平行z轴的切平面.若曲面 S 由隐函数0,zyxF表示,曲面 S 过0,00,zyx切平面方程为0

44、0000000,yyyzyxFxxxzyxF0,0000zzzzyxF法线方程为zzyxFzzyzyxFyyxzyxFxx000000000000,法向量zzyxFyzyxFxzyxFn,若曲面 S 由参数表示：2),(),(),(),(RDvuvuzzvuyyvuxxuv，其切平面为)(,()(,(),()(,()(,(),()(,()(,(),(000000000000000000000000vvvuvzuuvuuzvuzzvvvuvyuuvuuyvuyyvvvuvxuuvuuxvuxx或0),(),(),(00),(00),(00),(000000vuzzvyuyvxuxvuyyvxux

45、vzuzvuxxvzuzvyuyvuvuvu文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN

46、8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码

47、:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10

48、 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7

49、ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档

50、编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I10 HM7A7U9R4J7 ZN8L1K8P7N9文档编码:CG3N9K9B8I