Page 64 - Vector Analysis

60 CHAPTER 2. Differentiation of Functions of Several Variables

2.7 The Implicit Function Theorem（隱函數定理）

Theorem 2.68 (Implicit Function Theorem). Let D Ď Rn ˆ Rm be open, and F : D Ñ Rm

be a function of class C 1. Suppose that for some (x0, y0) P D, where x0 P Rn and y0 P Rm,

F (x0, y0) = 0 and  B F1 B F1 

[ )(x0, ] =  B y1 ¨¨¨ B ym  (x0, y0)
(Dy F y0) ...
... ¨¨¨ ...

B Fm B Fm

B y1 B ym

is invertible. Then there exists an open neighborhood U Ď Rn of x0, an open neighborhood

V Ď Rm of y0, and f : U Ñ V such that

()
1. F x, f (x) = 0 for all x P U;

2. y0 = f (x0);

3. (Df )(x) = ( )(x, f (x)))´1(DxF ( ) for al l x P U;
´ (DyF ) x, f (x)

4. f is of class C 1;

5. If F is of class C r for some r ą 1, so is f .

Proof. Let z = (x, y) and w =( (u, v), where x, u P Rn and y, v P Rm. Define w = G(z),
)
where G is given by G(x, y) = x, F (x, y) . Then G : D Ñ Rn+m, and

[]
[ ] In 0
(DG)(x, y) = ,
(DxF )(x, y) (DyF )(x, y)

where In is the n ˆ n identity matrix and (DxF )(x, y) P B(Rn, Rm) whose matrix represen-

tation is given by 

B F1 ¨¨¨ B F1
...
[ )(x, ] =  B x1 B xn  (x, y) .
(DxF y) ¨¨¨
... ...

B Fm B Fm

B x1 ( B xn )
We note that the Jacobian of G at (x0, y0) is det [(DyF )(x0, y0)] which does not vanish

since (DyF )(x0, y0) is invertible, so the inverse function theorem implies that there exists
( )
open neighborhoods O of (x0, y0) and W of x0, F (x0, y0) = (x0, 0) such that