作者jimmy780331 (lucky晓筑)
看板Math
标题Re: [分析] 隐函数定理
时间Wed Feb 9 23:36:56 2011
※ 引述《Madroach (∞)》之铭言：
: (1)Show that x^2-y^2-u^3+v^2+4 = 0
: 2xy+y^2-2u^2+3v^4+8 = 0
: can be solve u, v as continuously differentiable functions of
: (x,y) for (x,y) near (2,-1) satisfying u(2,-1)=2, v(2,-1)=1.
: (2)Find the value of du dv
: ----(2,-1) + ----(2,-1)
: dx dx
: 第一小题的部分我会做
: 但是第二小题想不出办法解出来 Q Q
: 恳请各位强者能指示小弟一点方向
f_1:x^2-y^2-u^3+v^2+4
f_2:2xy+y^2-2u^2+3v^4+8
f_1(2,-1,2,1)=0
f_2(2,-1,2,1)=0
let A=f'(2,-1,2,1)=[ 4,2,-6, 2]
[-2,2,-8,12]
let Ax=[ 4,2] Ay=[-6, 2]
[-2,2] [-8,12]
detAx=/=0
thus existence of C'-mapping u v,defined in a neighborhood of (2,-1)
s.t. u(2,-1)=2 v(2,-1)=1
then -(Ax^-1)Ay= -1/12 [ 4,-10]
[-44,52]
-1/3 + 11/3 = 10/3#
--
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1F：推 Madroach :了解!Rudin的书在这部份给了很棒的找解方式呢! 02/10 08:40
2F：→ Madroach :谢谢! 02/10 08:40