Implicit function theorem #
We prove three versions of the implicit function theorem. First we define a structure
implicit_function_data
that holds arguments for the most general version of the implicit function
theorem, see implicit_function_data.implicit_function
and implicit_function_data.to_implicit_function
. This version allows a user to choose
a specific implicit function but provides only a little convenience over the inverse function
theorem.
Then we define implicit_function_of_complemented
: implicit function defined by f (g z y) = z
,
where f : E → F
is a function strictly differentiable at a
such that its derivative f'
is surjective and has a complemented
kernel.
Finally, if the codomain of f
is a finite dimensional space, then we can automatically prove
that the kernel of f'
is complemented, hence the only assumptions are has_strict_fderiv_at
and f'.range = ⊤
. This version is named implicit_function
.
TODO #
- Add a version for a function
f : E × F → G
such that $$\frac{\partial f}{\partial y}$$ is invertible. - Add a version for
f : 𝕜 × 𝕜 → 𝕜
provinghas_strict_deriv_at
andderiv φ = ...
. - Prove that in a real vector space the implicit function has the same smoothness as the original one.
- If the original function is differentiable in a neighborhood, then the implicit function is differentiable in a neighborhood as well. Current setup only proves differentiability at one point for the implicit function constructed in this file (as opposed to an unspecified implicit function). One of the ways to overcome this difficulty is to use uniqueness of the implicit function in the general version of the theorem. Another way is to prove that any implicit function satisfying some predicate is strictly differentiable.
Tags #
implicit function, inverse function
General version #
Consider two functions f : E → F
and g : E → G
and a point a
such that
- both functions are strictly differentiable at
a
; - the derivatives are surjective;
- the kernels of the derivatives are complementary subspaces of
E
.
Note that the map x ↦ (f x, g x)
has a bijective derivative, hence it is a local homeomorphism
between E
and F × G
. We use this fact to define a function φ : F → G → E
(see implicit_function_data.implicit_function
) such that for (y, z)
close enough to (f a, g a)
we have f (φ y z) = y
and g (φ y z) = z
.
We also prove a formula for $$\frac{\partial\varphi}{\partial z}.$$
Though this statement is almost symmetric with respect to F
, G
, we interpret it in the following
way. Consider a family of surfaces {x | f x = y}
, y ∈ 𝓝 (f a)
. Each of these surfaces is
parametrized by φ y
.
There are many ways to choose a (differentiable) function φ
such that f (φ y z) = y
but the
extra condition g (φ y z) = z
allows a user to select one of these functions. If we imagine
that the level surfaces f = const
form a local horizontal foliation, then the choice of
g
fixes a transverse foliation g = const
, and φ
is the inverse function of the projection
of {x | f x = y}
along this transverse foliation.
This version of the theorem is used to prove the other versions and can be used if a user needs to have a complete control over the choice of the implicit function.
- left_fun : E → F
- left_deriv : E →L[𝕜] F
- right_fun : E → G
- right_deriv : E →L[𝕜] G
- pt : E
- left_has_deriv : has_strict_fderiv_at self.left_fun self.left_deriv self.pt
- right_has_deriv : has_strict_fderiv_at self.right_fun self.right_deriv self.pt
- left_range : self.left_deriv.range = ⊤
- right_range : self.right_deriv.range = ⊤
- is_compl_ker : is_compl self.left_deriv.ker self.right_deriv.ker
Data for the general version of the implicit function theorem. It holds two functions
f : E → F
and g : E → G
(named left_fun
and right_fun
) and a point a
(named pt
)
such that
- both functions are strictly differentiable at
a
; - the derivatives are surjective;
- the kernels of the derivatives are complementary subspaces of
E
.
Instances for implicit_function_data
- implicit_function_data.has_sizeof_inst
The function given by x ↦ (left_fun x, right_fun x)
.
Implicit function theorem. If f : E → F
and g : E → G
are two maps strictly differentiable
at a
, their derivatives f'
, g'
are surjective, and the kernels of these derivatives are
complementary subspaces of E
, then x ↦ (f x, g x)
defines a local homeomorphism between
E
and F × G
. In particular, {x | f x = f a}
is locally homeomorphic to G
.
Equations
Implicit function theorem. If f : E → F
and g : E → G
are two maps strictly differentiable
at a
, their derivatives f'
, g'
are surjective, and the kernels of these derivatives are
complementary subspaces of E
, then implicit_function_of_is_compl_ker
is the unique (germ of a)
map φ : F → G → E
such that f (φ y z) = y
and g (φ y z) = z
.
Equations
Case of a complemented kernel #
In this section we prove the following version of the implicit function theorem. Consider a map
f : E → F
and a point a : E
such that f
is strictly differentiable at a
, its derivative f'
is surjective and the kernel of f'
is a complemented subspace of E
(i.e., it has a closed
complementary subspace). Then there exists a function φ : F → ker f' → E
such that for (y, z)
close to (f a, 0)
we have f (φ y z) = y
and the derivative of φ (f a)
at zero is the
embedding ker f' → E
.
Note that a map with these properties is not unique. E.g., different choices of a subspace
complementary to ker f'
lead to different maps φ
.
Data used to apply the generic implicit function theorem to the case of a strictly differentiable map such that its derivative is surjective and has a complemented kernel.
Equations
- has_strict_fderiv_at.implicit_function_data_of_complemented f f' hf hf' hker = {left_fun := f, left_deriv := f', right_fun := λ (x : E), ⇑(classical.some hker) (x - a), right_deriv := classical.some hker, pt := a, left_has_deriv := hf, right_has_deriv := _, left_range := hf', right_range := _, is_compl_ker := _}
A local homeomorphism between E
and F × f'.ker
sending level surfaces of f
to vertical subspaces.
Equations
- has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented f f' hf hf' hker = (has_strict_fderiv_at.implicit_function_data_of_complemented f f' hf hf' hker).to_local_homeomorph
Implicit function g
defined by f (g z y) = z
.
Equations
- has_strict_fderiv_at.implicit_function_of_complemented f f' hf hf' hker = (has_strict_fderiv_at.implicit_function_data_of_complemented f f' hf hf' hker).implicit_function
implicit_function_of_complemented
sends (z, y)
to a point in f ⁻¹' z
.
Any point in some neighborhood of a
can be represented as implicit_function
of some point.
Finite dimensional case #
In this section we prove the following version of the implicit function theorem. Consider a map
f : E → F
from a Banach normed space to a finite dimensional space.
Take a point a : E
such that f
is strictly differentiable at a
and its derivative f'
is surjective. Then there exists a function φ : F → ker f' → E
such that for (y, z)
close to (f a, 0)
we have f (φ y z) = y
and the derivative of φ (f a)
at zero is the
embedding ker f' → E
.
This version deduces that ker f'
is a complemented subspace from the fact that F
is a finite
dimensional space, then applies the previous version.
Note that a map with these properties is not unique. E.g., different choices of a subspace
complementary to ker f'
lead to different maps φ
.
Given a map f : E → F
to a finite dimensional space with a surjective derivative f'
,
returns a local homeomorphism between E
and F × ker f'
.
Equations
Implicit function g
defined by f (g z y) = z
.
Equations
- has_strict_fderiv_at.implicit_function f f' hf hf' = function.curry ⇑((has_strict_fderiv_at.implicit_to_local_homeomorph f f' hf hf').symm)
implicit_function
sends (z, y)
to a point in f ⁻¹' z
.
Any point in some neighborhood of a
can be represented as implicit_function
of some point.