在数学分析 中，隐函数定理 是一個(數學上的)工具用來回答下面的問題：以隐函数 表示的多變量函數，這函數的變量在局部上是否存在显式的关系？隐函数定理说明，对于一个由关系 f (x , y )=0微分 满足某些条件，则在这点有鄰域 使得在該鄰域內 y x
y
=
h
(
x
)
{\displaystyle y=h(x)}
这样就把隐函数关系变成了常见的函数 关系。
舉一個簡單例子：假設兩個變數 x , y x 2 + y 2 − 1 = 0鄰域 跟定義在鄰域裡的顯函數 y =h (x )
答案是：除了(-1,0) (1,0 ) y =h (x )
讓函数
f
(
x
,
y
)
=
x
2
+
y
2
{\displaystyle f(x,y)=x^{2}+y^{2}}
，則单位圆就可以写成满足方程式
f
(
x
,
y
)
−
1
=
0
{\displaystyle f(x,y)-1=0}
的点的集合。在圆上的点A附近，
y 可以表示成
x 的函数：
y
(
x
)
=
1
−
x
2
{\displaystyle y(x)={\sqrt {1-x^{2}}}}
，但點B就不行（因為在點B附近，一個
x 會對應到兩個
y 的值）。
讓函數
f
(
x
,
y
)
=
x
2
+
y
2
{\displaystyle f(x,y)=x^{2}+y^{2}}
f
(
x
,
y
)
−
1
=
0
{\displaystyle f(x,y)-1=0}
单位圆 。圆上的点整體上是无法表示成單變數函數
y
=
h
(
x
)
{\displaystyle y=h(x)}
x
∈
(
−
1
,
1
)
,
{\displaystyle x\in (-1,1),}
y
{\displaystyle y}
±
1
−
x
2
{\displaystyle \pm {\sqrt {1-x^{2}}}}
然而在某些點附近，局部 地用
x
{\displaystyle x}
y
{\displaystyle y}
(
x
,
y
)
{\displaystyle (x,y)}
y
>
0
{\displaystyle y>0}
y
{\displaystyle y}
x
{\displaystyle x}
y
=
1
−
x
2
{\displaystyle y={\sqrt {1-x^{2}}}}
y
<
0
{\displaystyle y<0}
y
{\displaystyle y}
x
{\displaystyle x}
y
=
−
1
−
x
2
{\displaystyle y=-{\sqrt {1-x^{2}}}}
但是，在点
(
±
1
,
0
)
{\displaystyle (\pm 1,0)}
y
{\displaystyle y}
x
{\displaystyle x}
x
{\displaystyle x}
y
{\displaystyle y}
y
{\displaystyle y}
x
{\displaystyle x}
定理的叙述：欧几里得空间的情况 [ 编辑 ] 设 f : R n+m R m 连续可微 函数。这里R n+m 直积 ： R n R m R n+m (x ,y ) = (x1 , ..., xn , y1 , ..., ym ) h : R n R m (x , h (x )) (x , y ) | f (x ,y ) = 0 h
固定一点(a ,b ) = (a1 , ..., an , b1 , ..., bm ) f (a , b ) = 0 (a ,b ) y x h a U ⊆ R n b 邻域 V ⊆ R m h : U → V h (x , h (x )) U × V f (x ,y ) = 0
{
(
x
,
h
(
x
)
)
∣
x
∈
U
}
=
{
(
x
,
y
)
∈
U
×
V
∣
f
(
x
,
y
)
=
0
}
{\displaystyle \{(\mathbf {x} ,h(\mathbf {x} ))\mid \mathbf {x} \in U\}=\{(\mathbf {x} ,\mathbf {y} )\in U\times V\mid f(\mathbf {x} ,\mathbf {y} )=\mathbf {0} \}}
要保證这样的函数 h f 雅可比矩阵 要满足某些性質。对于给定的一点 (a ,b ) f 雅可比矩阵 写作：
(
D
f
)
(
a
,
b
)
=
[
∂
f
1
∂
x
1
(
a
,
b
)
⋯
∂
f
1
∂
x
n
(
a
,
b
)
⋮
⋱
⋮
∂
f
m
∂
x
1
(
a
,
b
)
⋯
∂
f
m
∂
x
n
(
a
,
b
)
|
∂
f
1
∂
y
1
(
a
,
b
)
⋯
∂
f
1
∂
y
m
(
a
,
b
)
⋮
⋱
⋮
∂
f
m
∂
y
1
(
a
,
b
)
⋯
∂
f
m
∂
y
m
(
a
,
b
)
]
=
[
X
|
Y
]
{\displaystyle (Df)(\mathbf {a} ,\mathbf {b} )=\left[{\begin{matrix}{\frac {\partial f_{1}}{\partial x_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}(\mathbf {a} ,\mathbf {b} )\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{m}}{\partial x_{n}}}(\mathbf {a} ,\mathbf {b} )\end{matrix}}\right|\left.{\begin{matrix}{\frac {\partial f_{1}}{\partial y_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{1}}{\partial y_{m}}}(\mathbf {a} ,\mathbf {b} )\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial y_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{m}}{\partial y_{m}}}(\mathbf {a} ,\mathbf {b} )\\\end{matrix}}\right]=[X|Y]}
其中的矩阵
X
{\displaystyle X}
f x
Y
{\displaystyle Y}
f y
Y
{\displaystyle Y}
可逆 矩阵的话，那么满足前面性质的鄰域 U V h (x )
设 f : R n+m → R m 连续可微 函数，讓 R n+m (x , y ) (x , y ) = (x1 , ..., xn , y1 , ..., ym ) (a1 , ..., an , b1 , ..., bm ) = (a ,b ) f (a ,b )=0 0 ∈ R m m ×m [(∂fi / ∂yj )(a , b )
Y
{\displaystyle Y}
a U ⊆ R n b V ⊆ R m h :U → V
h
(
a
)
=
b
{\displaystyle h(\mathbf {a} )=\mathbf {b} }
且
f
(
x
,
h
(
x
)
)
=
0
,
{\displaystyle f(\mathbf {x} ,h(\mathbf {x} ))=\mathbf {0} ,\,\,}
x
∈
U
{\displaystyle \mathbf {x} \in U}
一般情形 [ 编辑 ] 设
E
1
{\displaystyle E_{1}}
E
2
{\displaystyle E_{2}}
F
{\displaystyle F}
U
{\displaystyle U}
V
{\displaystyle V}
E
1
{\displaystyle E_{1}}
E
2
{\displaystyle E_{2}}
开集 。设函数：
f
:
U
×
V
→
F
{\displaystyle f:U\times V\rightarrow F}
是一个
C
k
{\displaystyle C^{k}}
光滑函数 ），其中
k
≥
1
{\displaystyle k\geq 1}
E
1
×
E
2
{\displaystyle E_{1}\times E_{2}}
(
x
0
,
y
0
)
{\displaystyle (x_{0},y_{0})}
f
(
x
0
,
y
0
)
=
0
{\displaystyle f(x_{0},y_{0})=0}
D
y
f
(
x
0
,
y
0
)
≠
0
{\displaystyle D_{y}f(x_{0},y_{0})\neq 0}
那么有如下结论：
存在
x
0
{\displaystyle x_{0}}
邻域
U
0
⊂
U
{\displaystyle U_{0}\subset U}
y
0
{\displaystyle y_{0}}
邻域
V
0
⊂
V
{\displaystyle V_{0}\subset V}
存在一个
C
k
{\displaystyle C^{k}}
φ
:
U
0
→
V
0
{\displaystyle \varphi :U_{0}\rightarrow V_{0}}
(
x
,
y
)
∈
U
0
×
V
0
{\displaystyle (x,y)\in U_{0}\times V_{0}}
f
(
x
,
y
)
=
0
{\displaystyle f(x,y)=0}
y
=
φ
(
x
)
{\displaystyle y=\varphi (x)}
参考来源 [ 编辑 ] （英文） Arne Hallam. The implicite function theorem (PDF) . Iowa State University. Chiang, Alpha C. Fundamental Methods of Mathematical Economics 3rd. McGraw-Hill. 1984. Edwards, Charles Henry. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. 1994 [1973]. ISBN 978-0-486-68336-2 Jittorntrum, K. An Implicit Function Theorem. Journal of Optimization Theory and Applications. 1978, 25 (4). doi:10.1007/BF00933522 Kumagai, S. An implicit function theorem: Comment. Journal of Optimization Theory and Applications. 1980, 31 (2). doi:10.1007/BF00934117 
![(Df)({\mathbf {a}},{\mathbf {b}})=\left[{\begin{matrix}{\frac {\partial f_{1}}{\partial x_{1}}}({\mathbf {a}},{\mathbf {b}})&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}({\mathbf {a}},{\mathbf {b}})\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}({\mathbf {a}},{\mathbf {b}})&\cdots &{\frac {\partial f_{m}}{\partial x_{n}}}({\mathbf {a}},{\mathbf {b}})\end{matrix}}\right|\left.{\begin{matrix}{\frac {\partial f_{1}}{\partial y_{1}}}({\mathbf {a}},{\mathbf {b}})&\cdots &{\frac {\partial f_{1}}{\partial y_{m}}}({\mathbf {a}},{\mathbf {b}})\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial y_{1}}}({\mathbf {a}},{\mathbf {b}})&\cdots &{\frac {\partial f_{m}}{\partial y_{m}}}({\mathbf {a}},{\mathbf {b}})\\\end{matrix}}\right]=[X|Y]](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcfcdca6f88f8a6d7d0264d4bfab73bd2ff795f)
