Re: [分析] 隱函數定理

[jimmy780331 (lucky曉筑)]

※ 引述《Madroach (∞)》之銘言：
: (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
: 第一小題的部分我會做
: 但是第二小題想不出辦法解出來 Q Q
: 懇請各位強者能指示小弟一點方向


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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