UID:
almafu_9960119371502883
Umfang:
1 online resource (vii, 359 pages) :
,
digital, PDF file(s).
Ausgabe:
1st ed.
ISBN:
0-511-89719-7
Inhalt:
T. M. Flett was a Professor of Pure Mathematics at the University of Sheffield from 1967 until his death in 1976. This book, which he had almost finished, has been edited for publication by Professor J. S. Pym. This text is a treatise on the differential calculus of functions taking values in normed spaces. The exposition is essentially elementary, though on are occasions appeal is made to deeper results. The theory of vector-valued functions of one real variable is particularly straightforward, and this forms the substance of the initial chapter. A large part of the book is devoted to applications. An extensive study is made of ordinary differential equations. Extremum problems for functions of a vector variable lead to the calculus of variations and general optimisation problems. Other applications include the geometry of tangents and the Newton-Kantorovich method in normed spaces. The three historical notes show how the masters of the past (Cauchy, Peano...) created the subject by examining in depth the evolution of certain theories and proofs.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover -- Frontmatter -- Contents -- Preface -- Introduction -- Differentiation of functions of one real variable -- 1.1 The derivative of a real- or vector-valued function of a real variable -- 1.2 Tangents to paths -- 1.3 The mean value theorems of Rolle, Lagrange, and Cauchy -- 1.4 Monotonicity theorems and an increment inequality -- 1.5 Applications of the increment inequality to differential equations and to a differential inequality -- 1.6 Increment and mean value inequalities for vector-valued functions -- 1.7 Applications of the increment and mean value theorems -- 1.8 Derivatives of second and higher orders -- Taylor's theorem -- 1.9 Regulated functions and integration -- 1.10 Further monotonicity theorems and increment inequalities -- 1.11 Historical notes on the classical mean value theorems, monotonicity theorems and increment inequalities -- Ordinary differential equations -- 2.1 Definitions -- 2.2 Preliminary results -- 2.3 Approximate solutions -- 2.4 Existence theorems for y′ = f(t, y) when Y is finite-dimensional -- 2.5 Some global existence theorems and other comparison theorems -- 2.6 Peano's linear differential inequality and the integral inequalities of Gronwall and Bellman -- 2.7 Lipschitz conditions -- 2.8 Linear equations -- 2.9 Linear equations with constant coefficients -- 2.10 Dependence on initial conditions and parameters -- 2.11 Further existence and uniqueness theorems -- 2.12 Successive approximations -- 2.13 An existence theorem for a discontinuous function -- 2.14 Historical notes on existence and uniqueness theorems for differential equations and on differential and integral inequalities -- The Fréchet differential -- 3.1 The Fréchet differential of a function -- 3.2 Mean value inequalities for Fréchet differentiable functions -- 3.3 The partial Fréchet differentials of a function with domain in a product space.
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3.4 The partial derivatives of a function with domain in Rn -- 3.5 Fréchet differentials of higher order -- 3.6 Taylor's theorem for Fréchet differentiable functions -- 3.7 The inverse function theorem -- 3.8 The implicit function theorem -- 3.9 Examples of Fréchet differentiable functions -- 3.10 Higher order differentiability of solutions of differential equations -- differentiability with respect to the initial conditions and parameters -- 3.11 Applications of Fréchet differentiation to the calculus of variations -- 3.12 Newton's method for the solution of the equation f(x) = 0 -- The Gâteaux and Hadamard variations and differentials -- 4.1 The Gâteaux variation and the Gâteaux differential -- 4.2 The Hadamard variation and the Hadamard differential -- 4.3 The tangent cones to the graph and the level surfaces of a function -- 4.4 Constrained maxima and minima (equality constraints) -- 4.5 Constrained maxima and minima (inequality constraints) -- 4.6 Theorems of Lyapunov type for differential equations -- 4.7 Historical note on differentials -- Appendix -- A.1 Metric and topological spaces -- A.2 Normed spaces -- A.3 Convex sets and functions -- A.4 The Hahn-Banach theorem -- A.5 Cones and their duality -- A.6 Measurable vector-valued functions -- Notes -- Bibliography -- Mathematical notation -- Author Index -- Subject Index.
,
English
Weitere Ausg.:
ISBN 0-521-09030-X
Weitere Ausg.:
ISBN 0-521-22420-9
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9780511897191
