Derivation bundle #
In this file we define the derivations at a point of a manifold on the algebra of smooth functions. Moreover, we define the differential of a function in terms of derivations.
The content of this file is not meant to be regarded as an alternative definition to the current tangent bundle but rather as a purely algebraic theory that provides a purely algebraic definition of the Lie algebra for a Lie group.
Equations
- Eq (smoothFunctionsAlgebra 𝕜 I M) inferInstance
Instances For
Type synonym, introduced to put a different SMul
action on C^n⟮I, M; 𝕜⟯
which is defined as f • r = f(x) * r
.
Denoted as C^n⟮I, M; 𝕜⟯⟨x⟩
within the Derivation
namespace.
Equations
- Eq (PointedContMDiffMap 𝕜 I M n x✝) (ContMDiffMap I (modelWithCornersSelf 𝕜 𝕜) M 𝕜 n)
Instances For
Alias of PointedContMDiffMap
.
Type synonym, introduced to put a different SMul
action on C^n⟮I, M; 𝕜⟯
which is defined as f • r = f(x) * r
.
Denoted as C^n⟮I, M; 𝕜⟯⟨x⟩
within the Derivation
namespace.
Equations
Instances For
Type synonym, introduced to put a different SMul
action on C^n⟮I, M; 𝕜⟯
which is defined as f • r = f(x) * r
.
Denoted as C^n⟮I, M; 𝕜⟯⟨x⟩
within the Derivation
namespace.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
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Equations
- Eq (PointedContMDiffMap.instInhabitedSomeENatTop I) { default := 0 }
Instances For
Equations
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ContMDiffMap.evalRingHom
gives rise to an algebra structure of C^∞⟮I, M; 𝕜⟯
on 𝕜
.
Instances For
With the evalAlgebra
algebra structure evaluation is actually an algebra morphism.
Equations
- Eq (PointedContMDiffMap.eval x) (Algebra.ofId (PointedContMDiffMap 𝕜 I M (WithTop.some Top.top) x) 𝕜)
Instances For
The derivations at a point of a manifold. Some regard this as a possible definition of the tangent space
Equations
- Eq (PointDerivation I x) (Derivation 𝕜 (PointedContMDiffMap 𝕜 I M (WithTop.some Top.top) x) 𝕜)
Instances For
Evaluation at a point gives rise to a C^∞⟮I, M; 𝕜⟯
-linear map between C^∞⟮I, M; 𝕜⟯
and 𝕜
.
Equations
Instances For
Alias of ContMDiffFunction.evalAt
.
Evaluation at a point gives rise to a C^∞⟮I, M; 𝕜⟯
-linear map between C^∞⟮I, M; 𝕜⟯
and 𝕜
.
Equations
Instances For
The evaluation at a point as a linear map.
Equations
- Eq (Derivation.evalAt x) (ContMDiffFunction.evalAt I x).compDer
Instances For
The heterogeneous differential as a linear map, denoted as 𝒅ₕ
within the Manifold
namespace.
Instead of taking a function as an argument this
differential takes h : f x = y
. It is particularly handy to deal with situations where the points
on where it has to be evaluated are equal but not definitionally equal.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The homogeneous differential as a linear map, denoted as 𝒅
within the Manifold
namespace.
Equations
- Eq (fdifferential f x) (hfdifferential ⋯)
Instances For
The homogeneous differential as a linear map, denoted as 𝒅
within the Manifold
namespace.
Equations
- Eq Manifold.«term𝒅» (Lean.ParserDescr.node `Manifold.«term𝒅» 1024 (Lean.ParserDescr.symbol "𝒅"))
Instances For
The heterogeneous differential as a linear map, denoted as 𝒅ₕ
within the Manifold
namespace.
Instead of taking a function as an argument this
differential takes h : f x = y
. It is particularly handy to deal with situations where the points
on where it has to be evaluated are equal but not definitionally equal.
Equations
- Eq Manifold.«term𝒅ₕ» (Lean.ParserDescr.node `Manifold.«term𝒅ₕ» 1024 (Lean.ParserDescr.symbol "𝒅ₕ"))
Instances For
Alias of fdifferential_apply
.
Alias of hfdifferential_apply
.