Differentiability in Banach spaces, differential forms and applications / / Celso Melchiades Doria

• Record Nr.: UNINA9910495154603321
• Autore: Doria Celso Melchiades
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-77834-7
• Descrizione fisica: 1 online resource (369 pages)
• Disciplina: 515.732
• Soggetti: Banach spaces; Espais de Banach; Stokes' theorem; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: 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 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.; 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.; 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.

Molecular aggregation [[electronic resource] ] : structure analysis and molecular simulation of crystals and liquids / / Angelo Gavezzotti

• Record Nr.: UNINA9910792246503321
• Autore: Gavezzotti Angelo
• Pubbl/distr/stampa: Oxford ; ; New York, : Oxford University Press, 2007
• ISBN: 0-19-967365-9; 0-19-152477-8; 1-280-75410-9; 1-4294-9265-1
• Descrizione fisica: 1 online resource (442 p.)
• Collana: IUCr Monographs on crystallography ; ; 19
• Disciplina: 548
• Soggetti: Crystallography; Intermolecular forces - Computer simulation; Molecular dynamics - Computer simulation; Quantum chemistry - Computer simulation; Crystals; Liquids
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Contents; PART I: FUNDAMENTALS; 1 The molecule: structure, size and shape; 1.1 Atoms and bonds; 1.2 Classification concepts in many particle systems; 1.3 Must a molecule have a size?; 1.4 Must a molecule have a shape?; 1.5 Historical portraits: a chemistry course in the early 1960's; 2 Molecular vibrations and molecular force fields; 2.1 Vibrational modes and force constants; 2.2 Molecular mechanics; 2.3 Evolution of molecular force fields; 2.4 Appendix: an example of coordinate transformation; 2.5 Historical portraits: Got a force constant?; 3 Quantum chemistry; 3.1 Some fundamentals of quantum mechanics 3.2 The hydrogen atom and atomic orbitals; 3.3 Spin; 3.4 Many-electron systems; 3.5 Molecular orbitals: The Fock and Roothaan equations; 3.6 Approximate quantum chemical methods: NDO and EHT; 3.7 Evolution of quantum chemical calculations: Beyond Hartree-Fock; 3.8 Dimerization energies and basis set superposition error; 3.9 Historical portraits: early ; experiences in quantum chemistry; 4 The physical nature and the computer simulation of the intermolecular potential; 4.1 Experimental facts and conceptual framework; 4.2 The representation of the molecular charge distribution and of the electric potential 4.3 Coulombic potential energy; 4.4 Polarization (electrostatic induction) energy; 4.5 Dispersion energy; 4.6 Pauli (exchange) repulsion energy; 4.7 Total energies versus partitioned energies; 4.8 Intermolecular hydrogen bonding; 4.9 Simulation methods; 4.10 Ad hoc or transferable? Force field fitting from ab initio calculations; 5 Crystal symmetry and X-ray diffraction; 5.1 A structural view of crystal symmetry: bottom-up crystallography; 5.2 Space group symmetry and its mathematical representation; 5.3 von Laue's idea, 1912 5.4 The structure factor; 5.5 Miller indices and Bragg's law; 5.6 The electron density in a crystal; 5.7 The atomic prejudice; 5.8 Structure and X-ray diffraction: Some examples; 5.9 Historical portraits: Training of a crystallographer in the 1960's; 6 Periodic systems: Crystal orbitals and lattice dynamics; 6.1 The mathematical description of crystal periodicity; 6.2 The electronic structure of solids; 6.3 Lattice dynamics and lattice vibrations; 7 Molecular structure and macroscopic properties: Calorimetry and thermodynamics; 7.1 Molecules and macroscopic bodies; 7.2 Energy 7.3 Heat capacity; 7.4 Entropy; 7.5 Free energy and chemical equilibrium; 7.6 Thermodynamic measurements; 7.7 Derivatives; 8 Correlation studies in organic solids; 8.1 The Cambridge Structural Database (CSD) of organic crystals; 8.2 Structure correlation; 8.3 Retrieval of molecular and crystal structures from the CSD; 8.4 The SubHeat database; 8.5 The geometrical categorization of intermolecular bonding; 8.6 Space analysis of molecular packing modes; 8.7 The calculation of intermolecular energies in crystals; 8.8 General-purpose force fields for organic crystals; 8.9 Accuracy and reproducibility
• Sommario/riassunto: The book is divided in two parts, to supply first the basic elements of the language, with short but complete explanations of terms, methods and theories; and then to describe the present status of studies on the processes by which organic molecules aggregate to form observable bodies and to determine their physical and chemical properties. - ;This book is divided in two parts. Part I provides a brief but accurate summary of all the basic ideas, theories, methods, and conspicuous results of structure analysis and molecular modelling of the condensed phases of organic compounds: quantum chemist