Continuous Functions
1st Edition
Published on 4. September 2020
272 pages
978-1-119-77723-6 (ISBN)
Description
This book is the second of a set dedicated to the mathematical tools used in partial differential equations derived from physics. It presents the properties of continuous functions, which are useful for solving partial differential equations, and, more particularly, for constructing distributions valued in a Neumann space. The author examines partial derivatives, the construction of primitives, integration and the weighting of value functions in a Neumann space. Many of them are new generalizations of classical properties for values in a Banach space. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students - doctoral students, postgraduate students - engineers and researchers, without restricting or generalizing the results.
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Jacques Simon
Continuous Functions
Book
11/2020
1st Edition
Wiley-ISTE
€164.50
Shipment within 3-4 weeks
Person
Jacques Simon is Honorary Research Director at CNRS. His research focuses on Navier-Stokes equations, particularly in shape optimization and in the functional spaces they use.
Content
- Intro
- Title Page
- Copyright Page
- Contents
- Introduction
- Familiarization with Semi-normed Spaces
- Notations
- Chapter 1 Spaces of Continuous Functions
- 1.1. Notions of continuity
- 1.2. Spaces C(O
- E), Cb(O
- E), CK(O
- E), C(O
- E) and Cb(O
- E)
- 1.3. Comparison of spaces of continuous functions
- 1.4. Sequential completeness of spaces of continuous functions
- 1.5. Metrizability of spaces of continuous functions
- 1.6. The space K(O
- E)
- 1.7. Continuous mappings
- 1.8. Continuous extension and restriction
- 1.9. Separation and permutation of variables
- 1.10. Sequential compactness in Cb(O
- E)
- Chapter 2 Differentiable Functions
- 2.1. Differentiability
- 2.2. Finite increment theorem
- 2.3. Partial derivatives
- 2.4. Higher order partial derivatives
- 2.5. Spaces
- 2.5. Spaces Cm(O
- E), Cmb (O
- E), CmK(O
- E), Cmb(O
- E) and Km(O
- E)
- 2.6. Comparison and metrizability of spaces of differentiable functions
- 2.7. Filtering properties of spaces of differentiable functions
- 2.8. Sequential completeness of spaces of differentiable functions
- 2.9. The space Cm(O
- E) and the set Cm(O
- U)
- Chapter 3 Differentiating Composite Functions and Others
- 3.1. Image under a linear mapping
- 3.2. Image under a multilinear mapping: Leibniz rule
- 3.3. Dual formula of the Leibniz rule
- 3.4. Continuity of the image under a multilinear mapping
- 3.5. Change of variables in a derivative
- 3.6. Differentiation with respect to a separated variable
- 3.7. Image under a differentiable mapping
- 3.8. Differentiation and translation
- 3.9. Localizing functions
- Chapter 4 Integrating Uniformly Continuous Functions
- 4.1. Measure of an open subset of Rd
- 4.2. Integral of a uniformly continuous function
- 4.3. Case where E is not a Neumann space
- 4.4. Properties of the integral
- 4.5. Dependence of the integral on the domain of integration
- 4.6. Additivity with respect to the domain of integration
- 4.7. Continuity of the integral
- 4.8. Differentiating under the integral sign
- Chapter 5 Properties of the Measure of an Open Set
- 5.1. Additivity of the measure
- 5.2. Negligible sets
- 5.3. Determinant of d vectors
- 5.4. Measure of a parallelepiped
- Chapter 6 Additional Properties of the Integral
- 6.1. Contribution of a negligible set to the integral
- 6.2. Integration and differentiation in one dimension
- 6.3. Integration of a function of functions
- 6.4. Integrating a function of multiple variables
- 6.5. Integration between graphs
- 6.6. Integration 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
- 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 K8(O)
- 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
- 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
- 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
- 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
- Bibliography
- Index
- Other titles from in Mathematics and Statistics
- EULA
