by parts and weak vanishing condition for a function
6.7. Change of variables in an integral
6.8. Some particular changes of variables 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
List of Illustrations
1 Chapter 4Figure 4.1 Paving ω by the cubes Δs,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 ΩD of the weighted function f ⋄ μFigure 7.2 Domain ΩK extending up to part of the boundary. The contour of ΩKFigure 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
7 Chapter 10Figure 10.1 Decomposition of the crown Cr,r+ϵ. A is dark gray and L i...
Guide
1 Cover
2 Table of Contents
Pages
1 ii
2 iii
3 iv
