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This textbook offers a high-level introduction to multi-variable differential calculus. Differential forms are introduced incrementally in the narrative, eventually leading to a unified treatment of Green's, Stokes' and Gauss' theorems. Furthermore, the presentation offers a natural route to differential geometry. Contents:Calculus of Vector FunctionsTangent Spaces and 1-formsLine IntegralsDifferential Calculus of MappingsApplications of Differential CalculusDouble and Triple IntegralsWedge Products and Exterior DerivativesIntegration of FormsStokes' Theorem and Applications

Differential calculus. --- Mathematical analysis. --- Stokes' theorem. --- Integrals --- Vector valued functions --- 517.1 Mathematical analysis --- Mathematical analysis --- Calculus, Differential --- Calculus

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The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and teachers. Volume 1 focuses on the analysis of real-valued functions of a real variable. This second volume goes on to consider metric and topological spaces. Topics such as completeness, compactness and connectedness are developed, with emphasis on their applications to analysis. This leads to the theory of functions of several variables. Differential manifolds in Euclidean space are introduced in a final chapter, which includes an account of Lagrange multipliers and a detailed proof of the divergence theorem. Volume 3 covers complex analysis and the theory of measure and integration.

Metric spaces. --- Topological spaces. --- Vector valued functions. --- Functions, Vector --- Functions, Vector valued --- Functional analysis --- Functions of real variables --- Spaces, Topological --- Spaces, Metric --- Generalized spaces --- Set theory --- Topology

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The subject of complex vector functional equations is a new area in the theory of functional equations. This monograph provides a systematic overview of the authors' recently obtained results concerning both linear and nonlinear complex vector functional equations, in all aspects of their utilization. It is intended for mathematicians, physicists and engineers who use functional equations in their investigations. Contents: Linear Complex Vector Functional Equations: General Classes of Cyclic Functional Equations; Functional Equations with Operations Between Arguments; Functional Equations with

Functional equations. --- Vector fields. --- Vector valued functions. --- Functions, Vector --- Functions, Vector valued --- Functional analysis --- Functions of real variables --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Vector analysis --- Equations, Functional

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Approximation of vector valued functions

Functional analysis --- Function spaces --- Vector valued functions --- Approximation polynomiale --- Espaces fonctionnels --- Fonctions vectorielles --- Approximation theory --- 517.518.8 --- 519.6 --- 681.3*G12 --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations --- Functions, Vector --- Functions, Vector valued --- Functions of real variables --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Vector valued functions. --- Approximation theory.

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Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integration theory, existence and uniqueness results, and stability theory. The results will be of particular interest to natural scientists and the finance community. Readers should ideally be familiar with stochastic processes and probability theory in general, as well as functional analysis, and in particular the theory of operator semigroups.

Mathematics. --- Probability Theory and Stochastic Processes. --- Quantitative Finance. --- Partial Differential Equations. --- Differential equations, partial. --- Finance. --- Distribution (Probability theory). --- Mathématiques --- Finances --- Distribution (Théorie des probabilités) --- Banach spaces. --- Stochastic processes. --- Vector valued functions. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Stochastic integrals. --- Integrals, Stochastic --- Partial differential equations. --- Economics, Mathematical. --- Probabilities. --- Functions of complex variables --- Generalized spaces --- Topology --- Stochastic analysis --- Distribution (Probability theory. --- Partial differential equations --- Funding --- Funds --- Economics --- Currency question --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Economics, Mathematical . --- Mathematical economics --- Econometrics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Methodology --- Social sciences --- Differential equations. --- Probability Theory. --- Mathematics in Business, Economics and Finance. --- Differential Equations. --- 517.91 Differential equations --- Differential equations