Theorems · Definition · global analysis

MDifferentiableWithinAt

{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] →
              ModelWithCorners 𝕜 E H →
                {M : Type u_4} →
                  [inst_4 : TopologicalSpace M] →
                    [ChartedSpace H M] →
                      {E' : Type u_5} →
                        [inst_6 : NormedAddCommGroup E'] →
                          [inst_7 : NormedSpace 𝕜 E'] →
                            {H' : Type u_6} →
                              [inst_8 : TopologicalSpace H'] →
                                ModelWithCorners 𝕜 E' H' →
                                  {M' : Type u_7} →
                                    [inst : TopologicalSpace M'] → [ChartedSpace H' M'] → (M → M') → Set M → 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.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Defs

Cited by

190 results in Mathlib

Foundations

Depth 63 from the axioms, rests on 928 definitions · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace