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Mathematics --- Calculus.

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Infinite dimensional holomorphy is the study of holomorphic or analytic func­ tions over complex topological vector spaces. The terms in this description are easily stated and explained and allow the subject to project itself ini­ tially, and innocently, as a compact theory with well defined boundaries. However, a comprehensive study would include delving into, and interacting with, not only the obvious topics of topology, several complex variables theory and functional analysis but also, differential geometry, Jordan algebras, Lie groups, operator theory, logic, differential equations and fixed point theory. This diversity leads to a dynamic synthesis of ideas and to an appreciation of a remarkable feature of mathematics - its unity. Unity requires synthesis while synthesis leads to unity. It is necessary to stand back every so often, to take an overall look at one's subject and ask "How has it developed over the last ten, twenty, fifty years? Where is it going? What am I doing?" I was asking these questions during the spring of 1993 as I prepared a short course to be given at Universidade Federal do Rio de Janeiro during the following July. The abundance of suit­ able material made the selection of topics difficult. For some time I hesitated between two very different aspects of infinite dimensional holomorphy, the geometric-algebraic theory associated with bounded symmetric domains and Jordan triple systems and the topological theory which forms the subject of the present book.

Holomorphic functions --- Linear topological spaces --- Functions of complex variables --- Fonctions holomorphes --- Espaces vectoriels topologiques --- Fonctions d'une variable complexe --- Variétés (mathématiques) --- Applications holomorphes. --- Manifolds (Mathematics) --- Holomorphic mappings. --- Polynômes. --- Fonctions d'une variable complexe. --- Polynomials. --- Functions of complex variables. --- Mathematical analysis. --- Analysis (Mathematics). --- Topology. --- Analysis. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- 517.1 Mathematical analysis --- Mathematical analysis --- Analyse fonctionnelle --- Functional analysis --- Polynômes. --- Variétés (mathématiques) --- Linear topological spaces. --- Holomorphic functions. --- Topological linear spaces --- Topological vector spaces --- Vector topology --- Topology --- Vector spaces --- Functions, Holomorphic --- Functions of several complex variables --- Complex variables --- Elliptic functions --- Functions of real variables --- Functional analysis. --- Applications holomorphes --- Polynômes --- Analyse en dimension infinie

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Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

Mathematics --- wiskunde

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Mathematical analysis --- Analyse mathématique --- Fonctions de plusieurs variables réelles --- Géométrie analytique --- Analyse mathématique --- Fonctions de plusieurs variables réelles --- Géométrie analytique