Theorems · Definition · global analysis

HasMFDerivAt

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_2} →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : NormedSpace 𝕜 E] →
          {H : Type u_3} →
            [inst_3 : TopologicalSpace H] →
              (I : ModelWithCorners 𝕜 E H) →
                {M : Type u_4} →
                  [inst_4 : TopologicalSpace M] →
                    [inst_5 : ChartedSpace H M] →
                      {E' : Type u_5} →
                        [inst_6 : NormedAddCommGroup E'] →
                          [inst_7 : NormedSpace 𝕜 E'] →
                            {H' : Type u_6} →
                              [inst_8 : TopologicalSpace H'] →
                                (I' : ModelWithCorners 𝕜 E' H') →
                                  {M' : Type u_7} →
                                    [inst_9 : TopologicalSpace M'] →
                                      [inst_10 : ChartedSpace H' M'] →
                                        (f : M → M') → (x : M) → (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 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.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Defs

Cited by

46 results in Mathlib

Foundations

Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace