The derivative of functions between smooth manifolds

Let M and M' be two smooth manifolds with corners over a field 𝕜 (with respective models with corners I on (E, H) and I' on (E', H')), and let f : M → M'. We define the derivative of the function at a point, within a set or along the whole space, mimicking the API for (Fréchet) derivatives. It is denoted by mfderiv I I' f x, where "m" stands for "manifold" and "f" for "Fréchet" (as in the usual derivative fderiv 𝕜 f x).

Main definitions

• UniqueMDiffOn I s : predicate saying that, at each point of the set s, a function can have at most one derivative. This technical condition is important when we define mfderivWithin below, as otherwise there is an arbitrary choice in the derivative, and many properties will fail (for instance the chain rule). This is analogous to UniqueDiffOn 𝕜 s in a vector space.

Let f be a map between smooth manifolds. The following definitions follow the fderiv API.

• mfderiv I I' f x : the derivative of f at x, as a continuous linear map from the tangent space at x to the tangent space at f x. If the map is not differentiable, this is 0.
• mfderivWithin I I' f s x : the derivative of f at x within s, as a continuous linear map from the tangent space at x to the tangent space at f x. If the map is not differentiable within s, this is 0.
• MDifferentiableAt I I' f x : Prop expressing whether f is differentiable at x.
• MDifferentiableWithinAt 𝕜 f s x : Prop expressing whether f is differentiable within s at x.
• HasMFDerivAt I I' f s x f' : Prop expressing whether f has f' as a derivative at x.
• HasMFDerivWithinAt I I' f s x f' : Prop expressing whether f has f' as a derivative within s at x.
• MDifferentiableOn I I' f s : Prop expressing that f is differentiable on the set s.
• MDifferentiable I I' f : Prop expressing that f is differentiable everywhere.
• tangentMap I I' f : the derivative of f, as a map from the tangent bundle of M to the tangent bundle of M'.

Various related results are proven in separate files: see

• Basic.lean for basic properties of the mfderiv, mimicking the API of the Fréchet derivative,
• FDeriv.lean for the equivalence of the manifold notions with the usual Fréchet derivative for functions between vector spaces,
• SpecificFunctions.lean for results on the differential of the identity, constant functions, products and arithmetic operators (like addition or scalar multiplication),
• Atlas.lean for differentiability of charts, models with corners and extended charts,
• UniqueDifferential.lean for various properties of unique differentiability sets in manifolds.

Implementation notes

The tangent bundle is constructed using the machinery of topological fiber bundles, for which one can define bundled morphisms and construct canonically maps from the total space of one bundle to the total space of another one. One could use this mechanism to construct directly the derivative of a smooth map. However, we want to define the derivative of any map (and let it be zero if the map is not differentiable) to avoid proof arguments everywhere. This means we have to go back to the details of the definition of the total space of a fiber bundle constructed from core, to cook up a suitable definition of the derivative. It is the following: at each point, we have a preferred chart (used to identify the fiber above the point with the model vector space in fiber bundles). Then one should read the function using these preferred charts at x and f x, and take the derivative of f in these charts.

Due to the fact that we are working in a model with corners, with an additional embedding I of the model space H in the model vector space E, the charts taking values in E are not the original charts of the manifold, but those ones composed with I, called extended charts. We define writtenInExtChartAt I I' x f for the function f written in the preferred extended charts. Then the manifold derivative of f, at x, is just the usual derivative of writtenInExtChartAt I I' x f, at the point (extChartAt I x) x.

There is a subtlety with respect to continuity: if the function is not continuous, then the image of a small open set around x will not be contained in the source of the preferred chart around f x, which means that when reading f in the chart one is losing some information. To avoid this, we include continuity in the definition of differentiablity (which is reasonable since with any definition, differentiability implies continuity).

Warning: the derivative (even within a subset) is a linear map on the whole tangent space. Suppose that one is given a smooth submanifold N, and a function which is smooth on N (i.e., its restriction to the subtype N is smooth). Then, in the whole manifold M, the property MDifferentiableOn I I' f N holds. However, mfderivWithin I I' f N is not uniquely defined (what values would one choose for vectors that are transverse to N?), which can create issues down the road. The problem here is that knowing the value of f along N does not determine the differential of f in all directions. This is in contrast to the case where N would be an open subset, or a submanifold with boundary of maximal dimension, where this issue does not appear. The predicate UniqueMDiffOn I N indicates that the derivative along N is unique if it exists, and is an assumption in most statements requiring a form of uniqueness.

On a vector space, the manifold derivative and the usual derivative are equal. This means in particular that they live on the same space, i.e., the tangent space is defeq to the original vector space. To get this property is a motivation for our definition of the tangent space as a single copy of the vector space, instead of more usual definitions such as the space of derivations, or the space of equivalence classes of smooth curves in the manifold.

Tags

derivative, manifold

Derivative of maps between manifolds

The derivative of a smooth map f between smooth manifold M and M' at x is a bounded linear map from the tangent space to M at x, to the tangent space to M' at f x. Since we defined the tangent space using one specific chart, the formula for the derivative is written in terms of this specific chart.

We use the names MDifferentiable and mfderiv, where the prefix letter m means "manifold".

def DifferentiableWithinAtProp {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') (f : H → H') (s : Set H) (x : H) : Prop

Property in the model space of a model with corners of being differentiable within at set at a point, when read in the model vector space. This property will be lifted to manifolds to define differentiable functions between manifolds.

Equations

• DifferentiableWithinAtProp I I' f s x = DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ Set.range ↑I) (↑I x)

theorem differentiable_within_at_localInvariantProp {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') : StructureGroupoid.LocalInvariantProp (contDiffGroupoid ⊤ I) (contDiffGroupoid ⊤ I') (DifferentiableWithinAtProp I I')

Being differentiable in the model space is a local property, invariant under smooth maps. Therefore, it will lift nicely to manifolds.

def UniqueMDiffWithinAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (s : Set M) (x : M) : Prop

Predicate ensuring that, at a point and within a set, a function can have at most one derivative. This is expressed using the preferred chart at the considered point.

Equations

• UniqueMDiffWithinAt I s x = UniqueDiffWithinAt 𝕜 (↑(PartialEquiv.symm (extChartAt I x)) ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x) x)

def UniqueMDiffOn {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (s : Set M) : Prop

Predicate ensuring that, at all points of a set, a function can have at most one derivative.

Equations

• UniqueMDiffOn I s = ∀ x ∈ s, UniqueMDiffWithinAt I s x

def MDifferentiableWithinAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (s : Set M) (x : M) : Prop

MDifferentiableWithinAt I I' f s x indicates that the function f between manifolds has a derivative at the point x within the set s. This is a generalization of DifferentiableWithinAt to manifolds.

We require continuity in the definition, as otherwise points close to x in s could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence writtenInExtChartAt I I' x f could be differentiable, while this would not mean anything relevant.

Equations

• One or more equations did not get rendered due to their size.

theorem mdifferentiableWithinAt_iff_liftPropWithinAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (s : Set M) (x : M) : MDifferentiableWithinAt I I' f s x ↔ ChartedSpace.LiftPropWithinAt (DifferentiableWithinAtProp I I') f s x

def MDifferentiableAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (x : M) : Prop

MDifferentiableAt I I' f x indicates that the function f between manifolds has a derivative at the point x. This is a generalization of DifferentiableAt to manifolds.

We require continuity in the definition, as otherwise points close to x could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence writtenInExtChartAt I I' x f could be differentiable, while this would not mean anything relevant.

Equations

• MDifferentiableAt I I' f x = (ContinuousAt f x ∧ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (Set.range ↑I) (↑(extChartAt I x) x))

theorem mdifferentiableAt_iff_liftPropAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (x : M) : MDifferentiableAt I I' f x ↔ ChartedSpace.LiftPropAt (DifferentiableWithinAtProp I I') f x

def MDifferentiableOn {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') (s : Set M) : Prop

MDifferentiableOn I I' f s indicates that the function f between manifolds has a derivative within s at all points of s. This is a generalization of DifferentiableOn to manifolds.

Equations

• MDifferentiableOn I I' f s = ∀ x ∈ s, MDifferentiableWithinAt I I' f s x

def MDifferentiable {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : M → M') : Prop

MDifferentiable I I' f indicates that the function f between manifolds has a derivative everywhere. This is a generalization of Differentiable to manifolds.

Equations

• MDifferentiable I I' f = ∀ (x : M), MDifferentiableAt I I' f x

def PartialHomeomorph.MDifferentiable {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] (f : PartialHomeomorph M M') : Prop

Prop registering if a partial homeomorphism is a local diffeomorphism on its source

Equations

• PartialHomeomorph.MDifferentiable I I' f = (MDifferentiableOn I I' (↑f) f.source ∧ MDifferentiableOn I' I (↑(PartialHomeomorph.symm f)) f.target)

def HasMFDerivWithinAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] (f : M → M') (s : Set M) (x : M) (f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)) : Prop

HasMFDerivWithinAt I I' f s x f' indicates that the function f between manifolds has, at the point x and within the set s, the derivative f'. Here, f' is a continuous linear map from the tangent space at x to the tangent space at f x.

This is a generalization of HasFDerivWithinAt to manifolds (as indicated by the prefix m). The order of arguments is changed as the type of the derivative f' depends on the choice of x.

We require continuity in the definition, as otherwise points close to x in s could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence writtenInExtChartAt I I' x f could be differentiable, while this would not mean anything relevant.

Equations

• One or more equations did not get rendered due to their size.

def HasMFDerivAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] (f : M → M') (x : M) (f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)) : Prop

HasMFDerivAt I I' f x f' indicates that the function f between manifolds has, at the point x, the derivative f'. Here, f' is a continuous linear map from the tangent space at x to the tangent space at f x.

We require continuity in the definition, as otherwise points close to x s could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence writtenInExtChartAt I I' x f could be differentiable, while this would not mean anything relevant.

Equations

• HasMFDerivAt I I' f x f' = (ContinuousAt f x ∧ HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (Set.range ↑I) (↑(extChartAt I x) x))

def mfderivWithin {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] (f : M → M') (s : Set M) (x : M) : TangentSpace I x →L[𝕜] TangentSpace I' (f x)

Let f be a function between two smooth manifolds. Then mfderivWithin I I' f s x is the derivative of f at x within s, as a continuous linear map from the tangent space at x to the tangent space at f x.

Equations

• One or more equations did not get rendered due to their size.

def mfderiv {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] (f : M → M') (x : M) : TangentSpace I x →L[𝕜] TangentSpace I' (f x)

Let f be a function between two smooth manifolds. Then mfderiv I I' f x is the derivative of f at x, as a continuous linear map from the tangent space at x to the tangent space at f x.

Equations

• mfderiv I I' f x = if MDifferentiableAt I I' f x then fderivWithin 𝕜 (writtenInExtChartAt I I' x f) (Set.range ↑I) (↑(extChartAt I x) x) else 0

def tangentMapWithin {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] (f : M → M') (s : Set M) : TangentBundle I M → TangentBundle I' M'

The derivative within a set, as a map between the tangent bundles

Equations

• tangentMapWithin I I' f s p = { proj := f p.proj, snd := (mfderivWithin I I' f s p.proj) p.snd }

def tangentMap {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] (f : M → M') : TangentBundle I M → TangentBundle I' M'

The derivative, as a map between the tangent bundles

Equations

• tangentMap I I' f p = { proj := f p.proj, snd := (mfderiv I I' f p.proj) p.snd }