向量值映照微分学09定理应用曲线和曲面的隐式表达

x0

˜0ˆ0,D(f1,···,f(˜0ˆD(ˆ1,ˆm−1(˜0ˆ非奇异Bλ(˜0RBµ(ˆ0Rm−1

D(f1,···,f(˜0ˆ

ˆϕ(˜)Bµ(0), ˜,ϕ(˜))X3,···,

ϕ(˜)

ϕ(˜0Bλ(˜0(ˆ0[Bλ(˜0(ˆ0

(˜01 11:(ˆ0.(˜0Bµ(ˆ

ϕ(˜)

˜(˜0

((˜)(0)˜(˜)

ϕ(˜)

⊂可确定曲线Γ˜) ˜)∆˜)(˜)(˜) x˜→0 ∆

(˜,ϕ())

Dϕ(˜) RmΣ={x∈mf(x)=0∈R}基于隐映照定理针对上述约束有以下结论x0

x0ˆ0Df(x˜,ˆ)∂f

,ˆ)

(˜ˆ ˜(ˆ0,(˜0 ˆ=ϕ˜)Bµ(ˆ0 ˜ϕ(˜))2:隐映照定理论的几何刻画,如图2所示.Bλ˜0Bµ(ˆ0Rm中,Σ

ϕ(˜)

˜(˜

⊂R.现为 中的曲 Σ(˜):Rm−1⊃

((˜(˜

)ϕ(˜)

⊂DΣ(˜)=:=Dϕ(x˜)1

·· (˜∈

TxΣ

g1,···,

(˜)⊂相应地,n(x)Rm, (Dϕ)T(˜)

n(x)=0∈1(R3中曲线（隐式表示形式）).R3{ f(x,y,z)=0∈Γ ∈Rg(x,y,z)=0∈z{ f(x,y,z)= ∂(f,

g(x0,y0,z0)=

∂(xz(x,y,z̸0(F

[x])

f(x,y,

∈ g(x,y,(F

f(x0,y0, =0∈g(x0,y0,

(x,y,z)R2×2

Bλ(y0R,µ

R2∀yB(y),ξ(y

B (

[[=[

f(x(y),y,

=0∈ g(x(y),y,亦即 Γ，表所以曲线示可以用向

Γ(y):Bλ(y0)∋y7→Γ(y)

∈Γ

ddΓ(y)=DΓ(y) ∈ λ y, =0∈R2,∀y∈Bλ

+D[x

])

)=0∈)

z

z

(x(y),y,z(y))

(x(y),y,

x = =

(x(y),y,

(x(y),y, ∂(f,= ∂(y,

(x(y),y,∂(f,g)(x(y),y,∂(x,

∂(f,∂(x,∂(f,—∂(y,z)(x(y),y,∂(f, ∂(x,

∂(f,∂(y,∂(f,dy(y)

∂(z, (x(y),y, ∀y∈∂(f, ∂(f,—∂x,—∂x, (x(y),y,

∂(x,∂(f,∂(x,

∂(f,∂(y, + ∂(f, (x(y),y,z(y))∈∂(z,

∂(f,∂(x,∂(f,

x−

=∂(f,

y−

=∂(f,

z−∂(y,

(x(y0),y0,

(x(y0),y0, ∂z,

(x(y0),y0, ∂x, 2(R4中曲线（隐式表示形式）).R4f(x,y,z,θ)=0∈x∈Γ g(x,y,z,θ)=0∈h(x,y,z,θ)=0∈θ f(x,y,z,θ)=

)=

∂(f,g,h)(x,y,z,θ̸0 h(x0,y0,z0,θ0)=

∂(x,y,

f(x,y,z, g(x,y,z,

∈ h(x,y,z, f(x0,y0,z0,

g(x0,y0,z0, =0∈h(x0,y0,z0, D[D[x z0 yθ0

(xy,z,θR3×3 ∃Bλ(z0)⊂R,∃∀z∈Bλ(z0),∃ξ(z)∈

g(x(z),y(z),z,h(x(z),y(z),z,

=0∈

Γ(z):B(z)∋z7→Γ(z) ∈ Γ

(z)=DΓ(z)

∈

=0∈ ∀z∈ +D[x =0∈

yθ

(x(z),y(z),z,θ(z))

(x(z),y(z),z, =

=

(x(z),y(z),z,

(x(z),y(z),z,

∂(f,g,∂(z,y,=− ∂(f,g,∂(x,y,θ)(x(z),y(z),z,

∂(f,g, (x(z),y(z),z,∂(x,z,∂(f,g,∂(x,y,∂(f,g,∂(z,y,—∂(f,g,h)(x(z),y(z),z,∂(x,y,∂(f,g,

∂(f,g,—∂(z,y, ∂(x,z,

—∂(f,g,∂Γ(z)∂

—(f,g,h)(x(z),y(z),z,

∂(x,z, (x(z),y(z),z, ∂(x,y,1

∂(f,g,∂(x,y,∂(f,g,∂(f,g,∂(f,g,∂(x,y,

x(z),y(z),z,

∂(x,y,∂(f,g,∂(y,z,∂(f,g,=

(x(z),y(z),z, ∀z∈∂(f,g,∂(x,y,∂(f,g,—∂(x,y,3(Rp中曲线（隐式表示形式）).Rp Γ X∈Rp|f(X)=0∈f ) ∈设有X0∈Γ亦即f(X0)

(X0)

.

∂(X1,···,Xα,···,

=f(X)∈X.XX1X0

. =f0

)=0∈0.p0X1X0.. D(f1,···,fX0D0.◦

X 000.

D(X1,···,Xα,···,

(X∈R(p−1)×(p−1)X.

pX1X0.

.⊂ ，对∀X∈B(X),∃ξ(X) p

(Xα)⊂R,∃

(Xα) p1◦.◦0 0..XpX0

F(Xα,ξ(Xα))=0∈

..

.0Γ(Xα):Bλ(Xα)∋Xα7→Γ(Xα) ∈0.Γ

dX1(X.α(Xα)=DΓ(Xα) α.

∈.◦0F(Xα,ξ(Xα))= =0∈ ∀Xα∈Bλ0.1X X DXα + X.

. (

(Xα)=0∈ . ∂f ∂f ∂f

∂f ∂f ·· ·· ∂.α

(Xα)=

.∂f

.. ∂f

.∂f

.. ∂f

∂f.

··

··

◦(−1)α−2∂(f1,···,◦∂(X1,···,. ∂(f1,···, ∂(X1,···,Xα◦1,···,−0= ∂(f1,···, ∂(f1,···,−0 ∂(X1,···,Xα,···, ∂(X1,···,Xα+1,···,.◦(−1)p−α−1∂(f1,···,◦∂(X1,···,

◦(−1)α−2∂(f1,···,◦∂(X1,···,.∂(f1,···,

(−1) ∂(X1,···,Xα−1,···, αdΓ(Xα) α

∂(f1,···,α α ∂(X1,···,Xα,···, ∂(f1,···,◦∂(X1,···,Xα+1,···,.(−1)p−α+1∂(f1,···,

∂(X1,··◦(−1)0∂(f1,···,◦∂(X1,···,

,

∂(f1,···, ◦ ∂(X1,X2,···,.

∈.曲

p−1∂(f1,···, ◦∂(X1,···, 4(R32维曲面).R32 Σ f(x,y,z)=03z

Σf(x0y0z00

∂f(x0,y0,z0)̸=0([x ,z

=f(x,y,z)∈ x0,y0

=f(x0,y0,z0)=y

,

=∂f(x0,y0,z0)̸= x

[x

x则按隐映照定理，∃

⊂R2,∃B(y)⊂R，对 ∈

,∃!ξ(x,z)Bµ(y0)⊂R

([xFz

,ξ(x,

=f(x,ξ(x,z),z)=([x ([x [x Σz

:

7→

ξ(x, ∈z

[∂Σ∂Σ

∈

∂x,∂z(x,z)= ∂x(x, ∂z(x,

[

x × (x,z)= (x, ∈

∂ξ(x,

∂ξ(x,再确定隐函数的偏导数∂ξ(x,z)和∂ξ(x,z) F

([xz

),ξ(x,z)= ([ ([ D[x x,ξ(x,z)+D x,ξ(x,z)D[x]ξ(x,z)=0∈y ∂f,

(x,ξ(x,z),z)+∂f(x,ξ(x,z),

∂ξ,

(x,z)=0∈∂x ∂x ∂x(x,z)=−(x,ξ(x,z), (x,z)=−∂z(x,ξ(x,z), n

×

(x,z)= (x,ξ(x,z),z)

(x,ξ(x,z),

ξ(x0,

(x,ξ(x,z), =

[xz

∈B

(x−x0)∂x(x0,ξ(x0,z0),z0)+(y−ξ(x0,z0))∂y(x0,ξ(x0,z0),z0)+(z−z0)∂z(x0,ξ(x0,z0),z0)=

Σ (x−x0)∂x(x0,y0,z0)+(y−y0)∂y(x0,y0,z0)+(z−z0)∂z(x0,y0,z0)=(DΣ)T(x,z)n=0∈ −∂x(x,ξ(x,z), −∂z(x,ξ(x,z),

=0∈

n10−∂x(x,ξ(x,z), n −∂z(x,ξ(x,z),

=0∈

∂x(x,ξ(x,z),n3=∂z(x,ξ(x,z),∂x 1 1n ∥

(x,ξ(x,z),z)

(x,ξ(x,z),n ∂f n∂z 5(R43维曲面).R43xΣ ∈R4f(x,y,z,θ)=0∈zθ

Σf(xy,z,θ0

∂f(x,y,z,θ̸=0 x , =f(x,y,z,θ)∈θ ,

=f(x0,y0,z0,θ0)=

z , =∂y(x0,y0,z0,θ0)̸=x⊂x⊂zθ∈,∃!ξ(x,z,θ)

θxxxΣzθ:∋zθx7→ θ

ξ(x,z,=zθ

∈

∂ξ

∂Σ∂Σ

(x,z, (x,z, (x,z, θ

(x,z,θ)

∈x =θzθ

z + ,ξ(x,z, D[x]ξ(x,z,θ)=0∈zθ ∂f,∂f,∂f(x,ξ(x,z,θ),z,θ)+∂f(x,ξ(x,z,θ),z,θ)∂ξ,∂ξ,∂ξ(x,z,θ)=0∈∂x∂z

∂x∂∂x(x,z,θ)=−∂f(x,ξ(x,z,θ),z, (x,z,θ)=−∂z(x,ξ(x,z,θ),z,∂ξ(x,z,θ)=−∂θ(x,ξ(x,z,θ),z,

(DΣ)T(x,z,θ)n=0∈1−∂x(x,ξ(x,z,θ),z,θ)00−∂z(x,ξ(x,z,θ),z,θ)1

4

0∈

0−∂x(x,ξ(x,z,θ),z,θ)0 100−∂x(x,ξ(x,z,θ),z,010−∂z(x,ξ(x,z,θ),z,

2

0∈001−∂x(x,ξ(x,z,θ),z, 1 n=∂f(x,ξ(x,z,θ),z,n3=∂z(x,ξ(x,z,θ),z,n4=∂θ(x,ξ(x,z,θ),z, nn

(x,ξ(x,z,θ),z, ξ(x0,z0, ∈Σ点的切平面方程 y ξ(x0,z0,

(x,ξ(x,z,θ),z, = 0

∈ (x−

(x0,ξ(x0,z0,θ0),z0,θ0)+(y−ξ(x0,

(x0,ξ(x0,z0,θ0),z0, +(z− (x0,ξ(x0,z0,θ0),z0,θ0)+(θ−

(x0,ξ(x0,z0,θ0),z0,θ0)= (x−

(x0,y0,z0,θ0)+(y− +(θ−

(x0,y0,z0,θ0)+(z−(x0,y0,z0,θ0)=

(x0,y0,z0,6(Rp+1p维曲面（隐式表示形式）).Rp+1p Σ X∈Rp+1f(X)=0∈

)=0Σ，亦即有f(X ∂Xα(X0)̸=0，其中α为1,···,p+1 )=0.◦ ,

=f(X1,···,

,···,

)∈.◦X1X0.. ,

=f(X1,···,Xα,···,Xp+1)=0X 0.XX0. ◦

, (X1,···,Xα,···,Xp+1)̸= X 0.XX

∃◦

X0X.◦..XX

.◦..

∈

X0X.◦ ..XX00ξ(X1···,Xα,···,Xp+1)∈Bµ(Xα)⊂R00.◦ .

,

◦1,···,Xα,···,

=

1

∈

ξ(X,···,Xα,···, .XXX XXX0