Continuous Functions. Jacques Simon

in an integral

      13  Chapter 7: Weighting and Regularization of Functions 7.1. Weighting 7.2. Properties of weighting 7.3. Weighting of differentiable functions 7.4. Local regularization 7.5. Global regularization 7.6. Partition of unity 7.7. Separability of K^∞(Ω)

      14  Chapter 8: Line Integral of a Vector Field Along a Path 8.1. Paths 8.2. Line integral of a field along a path 8.3. Line integral along a concatenation of paths 8.4. Tubular flow and the concentration theorem 8.5. Invariance under homotopy of the line integral of a local gradient

      15  Chapter 9: Primitives of Continuous Functions 9.1. Explicit primitive of a field with line integral zero 9.2. Primitive of a field orthogonal to the divergence-free test fields 9.3. Gluing of local primitives on a simply connected open set 9.4. Explicit primitive on a star-shaped set: Poincaré’s theorem 9.5. Explicit primitive under the weak Poincaré condition 9.6. Primitives on a simply connected open set 9.7. Comparison of the existence conditions for a primitive 9.8. Fields with local primitives but no global primitive 9.9. Uniqueness of primitives 9.10. Continuous primitive mapping

      16  Chapter 10: Additional Results: Integration on a Sphere 10.1. Surface integration on a sphere 10.2. Properties of the integral on a sphere 10.3. Radial calculation of integrals 10.4. Surface integral as an integral of dimension d − 1 10.5. A Stokes formula

      17  Appendix: Reminders A.1. Notation and numbering A.2. Semi-normed spaces A.3. Continuous mappings and duality A.4. Differentiable mappings and differentiable functions

      18  Bibliography

      19  Index

      20  Other titles from ISTE in Mathematics and Statistics

      21  End User License Agreement

      List of Illustrations

      1 Chapter 4Figure 4.1 Paving ω by the cubes Δ_s,n

      2 Chapter 5Figure 5.1 Equivalent decompositions for “measuring” a parallelepiped...

      3 Chapter 6Figure 6.1. Domain delimited by the graphs of ϒ₁ and ϒ₂Figure 6.2. Change of variables.

      4 Chapter 7Figure 7.1 Domain Ω_D of the weighted function f ⋄ μFigure 7.2 Domain Ω_K extending up to part of the boundary. The contour of Ω_KFigure 7.3 Crown-shaped set κn and potato-shaped set

....

      5 Chapter 8Figure 8.1 Divergence-free tubular flow in the tube T of axis [Γ]Figure 8.2 Intermediate closed paths

      6 Chapter 9Figure 9.1 Subset

that is star shaped with respect to a.

is dark gray an...Figure 9.2 Field q with local primitive θ but no global primitive. The set...

      7 Chapter 10Figure 10.1 Decomposition of the crown C_r,r+ϵ. A is dark gray and L i...

      Guide

      1  Cover

      2 Table of Contents

      3  Begin Reading

      Pages

      1  ii

      2  iii

      3  iv

      4  ix

      5  x

      6  xi

      7  xii

      8  xiii

      9  xiv