复旦大学高等数学教案07隐函数存在定理及应用

23xy)(xfyF0),(yxFxy0122yx)0,1()0,1(0122yxxy),1(122xyxy)0,1()0,1(x0122yxyxyyx1O01),(22yxyxF,)0,1()0,1(0),(yxFy0),(yxFyyx0),(yxFyx)(xyy),(yxF0))(,(xyxFxyyyxFxyxFxxyxFdd),(),(d))(,(d0ddxyFFyx,0),(yxFy)(xyyyxFFxydd0),(yxFy设二元函数F在点),(000yxP的某邻域),(0rPO上有定义，而且（1）0),(00yxF；（2）在rPB,0中，F的偏导数xF，yF连续；（3）0),(00yxFy，则存在0和在),(00xx中唯一确定的一元函数f，使得（1）0))(,(xfxF（),(00xxx），且)(00xfy；（2）f在),(00xx上可微，而且yxFFdxdy。f0))(,(xfxFx0)(xfFFyxyF),(00yx0yFyxFFxf)(0sinyxy10yxxyyxyxyyxFsin),(0)0,0(F1),(yxFx0cos1),(yyxFyFxFyF2RyxyFFdxdyyxcos11)(xfyF0),(yxFxnRF1n设1n元函数F在点),,,(0)0()0(10yxxPn的某邻域),(0rPO上有定义，而且0),,,(0)0()0(1yxxFn在),(0rPO中，F的各偏导数),,1(niFix，yF均连续0),,,(0)0()0(1yxxFny则存在0和在),,()0()0(10nxxx的邻域上定义的n元函数f，使得0)),,(,,,(11nnxxfxxF),(),,(01xOxxn且),,()0()0(10nxxfyf在),(0xO上可微，而且)(111nxxynFFFxyxy01322zzxyz),(yxzzxzyz)1,1,1(3221),,(zzxyzzyxF222322zzxyxzFFxzzx2232zzxyzFFyzzy1)1,1,1(xz21)1,1,1(yzF),(yxzz0),(zyzxFxzyzzyvzxu,0),(zyzxF0),(vuFx0xvFxuFvu01xzFxzFvuvuuFFFxz0),(vuFyvuvFFFyz),(),,(zyzxFzyxf7.4.2vuvuvuvuyxzFFFFFFFFfffyzxz111zzyx4222zyx,22xzyxz2zzyx4222xxzxzzx422zxxz2x222224222xzxzzxz322222)2()2(21zxzzxzxzzzyx4222yyzyzzy422zyyz2xyxzyxzzyzxz2242232)2(2zxyzyzxzyxz0),,(zyxF),,(0000zyxPF0P0),,(000zyxFz7.4.2),(00yx),(yxfz0Pkjin),(),(0000yxfyxfyx000)(1PzyxzPzyPzxFFFFFFFFkjikji0)(PzyxFFFkjiNN0),,(zyxF0P0),,(zyxFzyx,,zyxFFF,,0N0),,(zyxF0P0P0))(,,())(,,())(,,(000000000000zzzyxFyyzyxFxxzyxFzyx1222222czbyax3,3,3cbaP1),,(222222czbyaxzyxF),,(zyxF=0PPczbyaxkjiN222222cbakji32P0313131czcbybaxa3czbyax)2(2zyfezxf),,(zyxF=0)2(2zyfezx),,(zyxn)2(2),2(,2),,(22zyfezyfeFFFzxzxzyx),,(nmlana0an0)2(2)2(222zyfnnezyfmlezxzxl22m2n)2,22,(a0an)2,22,(a12