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[null,"1.","Record Nr.",null,"UNISA996466394503316"]
[null,null,"Autore",null,"Doria Celso Melchiades"]
[null,null,"Titolo",null,"Differentiability in Banach spaces, differential forms and applications / / Celso Melchiades Doria"]
[null,null,"Pubbl/distr/stampa",null,"Cham, Switzerland : , : Springer, , [2021]"]
[{"$nulls":4},"©2021"]
[null,null,"ISBN",null,"3-030-77834-7"]
[null,null,"Descrizione fisica",null,"1 online resource (369 pages)"]
[null,null,"Disciplina",null,"515.732"]
[null,null,"Soggetti",null,"Banach spaces"]
[{"$nulls":4},"Espais de Banach"]
[{"$nulls":4},"Stokes' theorem"]
[{"$nulls":4},"Llibres electrònics"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,"Nota di contenuto",null,"Intro -- Preface -- Introduction -- Contents -- 1 Differentiation in mathbbRn -- 1 Differentiability of Functions f:mathbbRnrightarrowmathbbR -- 1.1 Directional Derivatives -- 1.2 Differentiable Functions -- 1.3 Differentials -- 1.4 Multiple Derivatives -- 1.5 Higher Order Differentials -- 2 Taylor's Formula -- 3 Critical Points and Local Extremes -- 3.1 Morse Functions -- 4 The Implicit Function Theorem and Applications -- 5 Lagrange Multipliers -- 5.1 The Ultraviolet Catastrophe: The Dawn of Quantum Mechanics -- 6 Differentiable Maps I -- 6.1 Basics Concepts -- 6.2 Coordinate Systems -- 6.3 The Local Form of an Immersion -- 6.4 The Local Form of Submersions -- 6.5 Generalization of the Implicit Function Theorem -- 7 Fundamental Theorem of Algebra -- 8 Jacobian Conjecture -- 8.1 Case n=1 -- 8.2 Case nge2 -- 8.3 Covering Spaces -- 8.4 Degree Reduction -- 2 Linear Operators in Banach Spaces -- 1 Bounded Linear Operators on Normed Spaces -- 2 Closed Operators and Closed Range Operators -- 3 Dual Spaces -- 4 The Spectrum of a Bounded Linear Operator -- 5 Compact Linear Operators -- 6 Fredholm Operators -- 6.1 The Spectral Theory of Compact Operators -- 7 Linear Operators on Hilbert Spaces -- 7.1 Characterization of Compact Operators on Hilbert Spaces -- 7.2 Self-adjoint Compact Operators on Hilbert Spaces -- 7.3 Fredholm Alternative -- 7.4 Hilbert-Schmidt Integral Operators -- 8 Closed Unbounded Linear Operators on Hilbert Spaces -- 3 "]
[{"$nulls":4},"Differentiation in Banach Spaces -- 1 Maps on Banach Spaces -- 1.1 Extension by Continuity -- 2 Derivation and Integration of Functions f:[a,b]rightarrowE -- 2.1 Derivation of a Single Variable Function -- 2.2 Integration of a Single Variable Function -- 3 Differentiable Maps II -- 4 Inverse Function Theorem (InFT) -- 4.1 Prelude for the Inverse Function Theorem -- 4.2 InFT for Functions of a Single Real Variable."]
[{"$nulls":4},"4.3 Proof of the Inverse Function Theorem (InFT) -- 4.4 Applications of InFT -- 5 Classical Examples in Variational Calculus -- 5.1 Euler-Lagrange Equations -- 5.2 Examples -- 6 Fredholm Maps -- 6.1 Final Comments and Examples -- 7 An Application of the Inverse Function Theorem to Geometry -- 4 Vector Fields -- 1 Vector Fields in mathbbRn -- 2 Conservative Vector Fields -- 3 Existence and Uniqueness Theorem for ODE -- 4 Flow of a Vector Field -- 5 Vector Fields as Differential Operators -- 6 Integrability, Frobenius Theorem -- 7 Lie Groups and Lie Algebras -- 8 Variations over a Flow, Lie Derivative -- 9 Gradient, Curl and Divergent Differential Operators -- 5 Vector Integration, Potential Theory -- 1 Vector Calculus -- 1.1 Line Integral -- 1.2 Surface Integral -- 2 Classical Theorems of Integration -- 2.1 Interpretation of the Curl and Div Operators -- 3 Elementary Aspects of the Theory of Potential -- 6 Differential Forms, Stokes Theorem -- 1 Exterior Algebra -- 2 Orientation on V and on the Inner Product on Λ(V) -- 2.1 Orientation -- 2.2 Inner Product in Λ(V) -- 2.3 Pseudo-Inner Product, the Lorentz Form -- 3 Differential Forms -- 3.1 Exterior Derivative -- 4 De Rham Cohomology -- 4.1 Short Exact Sequence -- 5 De Rham Cohomology of Spheres and Surfaces -- 6 Stokes Theorem -- 7 Orientation, Hodge Star-Operator and Exterior Co-derivative -- 8 Differential Forms on Manifolds, Stokes Theorem -- 8.1 Orientation -- 8.2 Integration on Manifolds -- 8.3 Exterior Derivative -- 8.4 Stokes Theorem on Manifolds -- 7 Applications to the Stokes Theorem -- 1 Volumes of the (n+1)-Disk and of the n-Sphere -- 2 Harmonic Functions -- 2.1 Laplacian Operator -- 2.2 Properties of Harmonic Functions -- 3 Poisson Kernel for the n-Disk DnR -- 4 Harmonic Differential Forms -- 4.1 Hodge Theorem on Manifolds -- 5 Geometric Formulation of the Electromagnetic Theory."]
[{"$nulls":4},"5.1 Electromagnetic Potentials -- 5.2 Geometric Formulation -- 5.3 Variational Formulation -- 6 Helmholtz's Decomposition Theorem -- Appendix A Basics of Analysis -- 1 Sets -- 2 Finite-dimensional Linear Algebra: V=mathbbRn -- 2.1 Matrix Spaces -- 2.2 Linear Transformations -- 2.3 Primary Decomposition Theorem -- 2.4 Inner Product and Sesquilinear Forms -- 2.5 The Sylvester Theorem -- 2.6 Dual Vector Spaces -- 3 Metric and Banach Spaces -- 4 Calculus Theorems -- 4.1 One Real Variable Functions -- 4.2 Functions of Several Real Variables -- 5 Proper Maps -- 6 Equicontinuity and the Ascoli-Arzelà Theorem -- 7 Functional Analysis Theorems -- 7.1 Riesz and Hahn-Banach Theorems -- 7.2 Topological Complementary Subspace -- 8 The Contraction Lemma -- Appendix B Differentiable Manifolds, Lie Groups -- 1 Differentiable Manifolds -- 2 Bundles: Tangent and Cotangent -- 3 Lie Groups -- Appendix C Tensor Algebra -- 1 Tensor Product -- 2 Tensor Algebra -- Appendix  References --  -- Index."]