第六章多元函数微分学答案习题68

1习题6.833232332233232.1.(,):(1)20.3320,3220.32,.3232(2)ln.1fFzzxyxzzxyzzzzxzxzxzyxxxzzzxzxzyyyyxzzzzxxzyxxxzyyzzxyzzyxz在本节习题中所涉及的函数或都是有连续一阶偏导数的函数求由下列方程确定的隐函数的所有一阶偏导数211111,1.11,.1111(3)sin(01).1(1cos)0,(1cos)1.11,.1cos1cos(4).lnln,lxzxxzxzzzzzyxyzyzzzzzzxyzyyzyyzzxzzyzzzzxyzzxzyzzyzzzzzxzyyxzzyyxxy1112111n.lnlnln,lnlnln.lnlnlnln(5)coscoscos1.cossincossin(sincos)0,sinxxxzxxzzxxzxzxxzyyzzzzzzzxxzyyxzzyxzyzzyzyzzzyxzyyxyzyyyxyzzyyxyzyyxyyzzxzzyzxyzxyzxxxy22221222212212.cossincossincos(sincos)0,.cossin2.(,)0(),.(,)(2)(,)(1)0,(,)(,)2(,)(xxzzxyzxyzyzxyyxyzdyfxyxyyyxdxfxyxyyxyyfxyxyyyfxyxyfxyxydydxxyfxyxyf设由方程确定隐函数为求解2.,)xyxy22222222222233.cos,.sincos0.sin(1)0,.1(1)cossin()(1)sin(1)cossin()1(1)(1)cossin.(1)4.zzzzzzzzzzzzzzzzzxyexxzzyxyzxyeyxyexxezyexyyxyezxxeyxyyexyyxyeeeexyexyyeF设求及设解1231233232331231232333(,,)0,,.(1)0,.(1)0,.(,,)0,()()0,,zzxxyxyzxyFFFzzFFFxxFFFzzFFyyFFxxyxyzFdxFdxdyFdxdydzFFFFFzdzdxdyFFx求解另解d12323332222,.5.(,)(,,)0,,(0),(,)0(,)8.(,,)0,(0).(yxzzzyxxyzzzzFFFFFzFyFzzxyFxyzFFdzdxdyFFFFxyzxyzzzxydzFFdFxyzFdxFdyFdzdzdxdyFFFdFx设是方程确定的隐函数利用一阶微分形式的不变型证明并求确定的隐函数的微分记证2222121212121212212,)0.(222)(2)0,(2)(2)(22)0,(2)(2).2()6.Jacobisin.sincossinsinyzxyzFxdxydyzdzFydxxdyzdzxFyFdxyFxFdyzFzFdzxFyFdxyFxFdydzzFFJrxryr证明球坐标变换的行列式证coszr322232sincoscoscossinsin,sinsincossinsincos,cossin.sincoscoscossinsinsinsincossinsincoscossin0cossinsinsin.7.,dxdrrdrddydrrdrddzdrrdrrJrrrrrrxuvy设由证22332(,)110,,223.22,.222222110,,22,,2231313()()42424uvzuvzzxyzzxyuvxyduvdvdudvdxvdxdydyudxdudvuduvdvdyvuvuxyuvdxdydydxdudvdzdxdydxdy，确定函数，求当时，与的值.2解dz=3u,3,0.4dxzzxy422222220,8.5.1,1,2.0,0.,.()()()()xuyvuvxyuuxyuvxxyudxxduvdyydvvduudvydxxdyxduydvudxvdyvduudvydxxdyuudxvdyyydxxdyuydxuvxydyduxuyvxuyvuuyx设求当时与的值=-解2222222222,.()()()(),().2()().()uuvxyxuyvyxuyvxydxxdyvudxvdyxyuvdxxvdydvxuyvxuyvvxyuvxxuyvuuvuxuyvuyuxyuuyxxxxxuyvxuyvuuuvxyxyxyxx=-=-x22222222()().()1,1,25()3,442()()()4(5)5(uyvuvuvvuyxuyvuxyuvxyxxxxxuyvxyuvuuyvxyuvxxuyvxxuyvuuvuxuyvuyuxyuxxxxxuyv当时,=-=2253)5544.1632()()25.()32uvuvvuyxuyvuxyuvxyuxxxxxyxuyv522219.,2,1,1,2.22202222,,222222,.22221,1,22020,224dxdyxyzxyzxyzdzdzxdxydyzdzdxdydzxdxydyzdzdxdydzzyxzdxdzdydzyydxzydyxzdzydzyxyzdxzydyxdzydz设求当时与的值当时解41.22410.coscos,cossin,sin,.cossinsinsincoscoscoscossincoscoscoscoscocossinsinsincoscoszyzxyzxdddxdddyddzdxdydxd设求由前两个方程解出-sincos解-sincos22222222ssinsincoscossin,sinsincoscoscossincossinsincoscos.sin1,220,.1,.1dydxdydzddxdyzxxzxyzxdxzdzzxxzzxyzzxxxzxyz2另解再解611.(,)(,)(,)(,)((,),(,))((,),(,))(,)(,)(,).(,)(,)(,),,uuxyvvxyxxyyuuxyvvxyDuvDuvDxyDDxyDuuxuyuuxuyxyxyvv设及有连续一阶偏导数,又设及也有连续一阶偏导数,且使复合函数及有定义.证明证,,(,)(,)(,)(,).(,)(,)xvyvvxvyxyxyuxuyuxuyuuxyxyDuvvvvxvyvxvyDxyxyuuxxxyDuvDxyvvyyDxyDxy