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Comprehensive and state-of-the art study of the basic concepts and principles of variational analysis and generalized differentiation in both finite-dimensional and infinite-dimensional spaces

Presents numerous applications to problems in the optimization, equilibria, stability and sensitivity, control theory, economics, mechanics, etc.

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Topics related to the differentiation of real functions have received considerable attention during the last few decades. This book provides an efficient account of the present state of the subject. Bruckner addresses in detail the problems that arise when dealing with the class $\Delta '$ of derivatives, a class that is difficult to handle for a number of reasons. Several generalized forms of differentiation have assumed importance in the solution of various problems. Some generalized derivatives are excellent substitutes for the ordinary derivative when the latter is not known to exist; others are not. Bruckner studies generalized derivatives and indicates ``geometric'' conditions that determine whether or not a generalized derivative will be a good substitute for the ordinary derivative. There are a number of classes of functions closely linked to differentiation theory, and these are examined in some detail. The book unifies many important results from the literature as well as some results not previously published. The first edition of this book, which was current through 1976, has been referenced by most researchers in this subject. This second edition contains a new chapter dealing with most of the important advances between 1976 and 1993. ... Read more

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Automatic Differentiation (AD) is a maturing computational technology and has become a mainstream tool used by practicing scientists and computer engineers. The rapid advance of hardware computing power and AD toolshas enabled practitioners to quickly generate derivative-enhanced versions of their code for a broad range of applications in applied research and development.

"Automatic Differentiation of Algorithms" provides a comprehensive and authoritative survey of all recent developments, new techniques, and tools for AD use. The book covers all aspects of the subject: mathematics, scientific programming ( i.e., use of adjoints in optimization) and implementation (i.e., memory management problems). A strong theme of the book is the relationships between AD tools and other software tools, such as compilers and parallelizers. A rich variety of significant applications are presented as well, including optimum-shape design problems, for which AD offers more efficient tools and techniques.

Topics and features:

* helpful introductory AD survey chapter for brief overview of the field

*extensive applications chapters, i.e., for circuit simulation, optimization and optimal-control shape design, structural mechanics, and multibody dynamical systems modeling

*comprehensive bibliography for all current literature and results for the field

*performance issues

*optimal control sensitivity analysis

*AD use with object oriented software tool kits

The book is an ideal and accessible survey of recent developments and applications of AD tools and techniques for a broad scientific computing and computer engineering readership. Practitioners, professionals, and advanced graduates working in AD development will find the book auseful reference and essential resource for their work. ... Read more

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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. ... Read more

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*An inexpensive and accessible supplement to class work, it provides essential practice for specific areas of mathematics and helps reinforce and consolidate the techniques learnt in class. *The series can be used alongside any published text, and is suitable for the new AS and A Level specifications. *The Help Yourself series supports the algebra and calculus content os the Scottish Higher Grade course. ... Read more

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must read if you are a computer major
Preferred by most IITs top brains!
good foundation for Computer science... ... Read more

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This book presents a new basis for differential calculus. Classical differentiation in linear spaces of arbitrary dimension uses Banach spaces--but most function spaces are not Banach spaces. Any attempts to develop a theory of differentiation covering non-normable linear spaces have always involved arbitrary conditions. This book bases the theory of differentiability of linear spaces on the fundamental idea of reducing the differentiability of general maps to that of functions on the real numbers. And the property ``continuously differentiable'' is replaced by that of ``Lipschitz differentiable.'' The result is a more natural theory, of conceptual simplicity that leads to the the same categories of linear spaces, but in a more general setting. ... Read more

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Theory of differentiation includes all aspects of various kinds of derivates and derivatives, and the theory of various Perron and Denjoy-Perron integrals. Derivative theorems covered are theorems on unilateral (or Dini) derivates. Through a cohesive format, outstanding problems are resolved, new ones are presented, and developments in this field, both past and present, are covered. ... Read more

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This book is concerned with the principles of differentiation and integration. The principles are then applied to solve engineering problems. A familiarity with basic algebra and a basic knowledge of common functions, such as polynomials, trigonometric, exponential, logarithmic and hyperbolic is assumed but reference material on these is included in an appendix. ... Read more

Customer Reviews (1)

Suitable for BTEC NC/D or HNC/D students
The book assumes previous knowledge of Trig , complex numbers, and manipulation of equations. I bought the book for the intention on relearning all the things lost from the College days, to further my career.Clear step by step examples and problems soon brought the past learningexperience back in focus. The information I gained has enabled me tofurther understand and appreciate Power System Engineering design andcommissioning. A good buy. ... Read more

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This textbook is written for everyone who has experienced difficulties understanding Calculus. This book really teaches you, helps you understand and master Calculus through clear and meaningful explanations of all the ideas, concepts, problems and procedures of Calculus; effective problem solving skills and strategies; fully worked problems with complete, step-by-step explanations. The material in this textbook is prepared and sequenced in a particular way to allow for an easy, logical progression from the simple to the more elaborate concepts and exercises. By carefully reading all material, working through all examples and solving all problems, the student will easily master the fundamentals of Calculus in one semester. ... Read more

Customer Reviews (2)

Do It Yourself Calculus
I was interested in teaching myself calculus but didn't want a boring textbook. I came across this book and it satisfied my goal without being boring. I found the text to be very understandable and I can say with confidence that this book has taught me calculus well in place of an undergraduate college course. I recommend the book to anyone who wants to teach themselves calculus.

Disappointing
The beginning of this book laments that many calculus textbooks are needlessly complicated and promises to deliver a simple and clear textbook.The beginning of the book is promising; the author states the major problems of calculus:to find the slope of the line tangent to a curve, to find to find the area under a curve, and to find velocity and acceleration given a formula for distance as a function of time.The implication is that this is the forest; the rest is trees.

But then he describes the trees:i.e. formulas for differentiation such as the chain rule.I doubt that anyone who has not studied calculus before could follow his brief explanations.Calculus may be a few concepts accompanied by hundreds of examples; but he examples require more explanation than is provided here.

A final point.The Kindle format does not seem well suited to a book with diagrams and graphs.I have Kindle 1.Perhaps the latest iteration is better for a books like this one. ... Read more

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Algorithmic, or automatic, differentiation (AD) is a growing area of theoretical research and software development concerned with the accurate and efficient evaluation of derivatives for function evaluations given as computer programs. The resulting derivative values are useful for all scientific computations that are based on linear, quadratic, or higher order approximations to nonlinear scalar or vector functions.

AD has been applied in particular to optimization, parameter identification, nonlinear equation solving, the numerical integration of differential equations, and combinations of these. Apart from quantifying sensitivities numerically, AD also yields structural dependence information, such as the sparsity pattern and generic rank of Jacobian matrices. The field opens up an exciting opportunity to develop new algorithms that reflect the true cost of accurate derivatives and to use them for improvements in speed and reliability.

This second edition has been updated and expanded to cover recent developments in applications and theory, including an elegant NP completeness argument by Uwe Naumann and a brief introduction toscarcity, a generalization of sparsity. There is also added material on checkpointing and iterative differentiation. To improve readability the more detailed analysis of memory and complexity bounds has been relegated to separate, optional chapters.The book consists of three parts: a stand-alone introduction to the fundamentals of AD and its software; a thorough treatment of methods for sparse problems; and final chapters on program-reversal schedules, higher derivatives, nonsmooth problems and iterative processes. Each of the 15 chapters concludes with examples and exercises.

Audience: This volume will be valuable to designers of algorithms and software for nonlinear computational problems. Current numerical software users should gain the insight necessary to choose and deploy existing AD software tools to the best advantage.

Contents: Rules; Preface; Prologue; Mathematical Symbols; Chapter 1: Introduction; Chapter 2: A Framework for Evaluating Functions; Chapter 3: Fundamentals of Forward and Reverse; Chapter 4: Memory Issues and Complexity Bounds; Chapter 5: Repeating and Extending Reverse; Chapter 6: Implementation and Software; Chapter 7: Sparse Forward and Reverse; Chapter 8: Exploiting Sparsity by Compression; Chapter 9: Going beyond Forward and Reverse; Chapter 10: Jacobian and Hessian Accumulation; Chapter 11: Observations on Efficiency; Chapter 12: Reversal Schedules and Checkpointing; Chapter 13: Taylor and Tensor Coefficients; Chapter 14: Differentiation without Differentiability; Chapter 15: Implicit and Iterative Differentiation; Epilogue; List of Figures; List of Tables; Assumptions and Definitions; Propositions, Corollaries, and Lemmas; Bibliography; Index ... Read more

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Variational analysis is a rapidly growing field within pure and applied mathematics, with numerous applications to optimization, control theory, economics, engineering, and other disciplines. This volume brings together state-of-the-art results in variational analysis and its applications, with an emphasis on optimization and control.
 
The included chapters, written by international experts in the field of variational analysis and related topics, are dedicated to Boris S. Mordukhovich, a renowned mathematician, and aim to celebrate his fundamental contributions to variational analysis, generalized differentiation
and their applications.
 
This volume is intended for mathematicians studying variational analysis as well as other researchers interested in applying the principles of variational analysis to their area of study. ... Read more

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In this book the author presents the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. Results for the above are derived using three different types of fractional derivatives, namely by Canavati, Riemann-Liouville and Caputo. The univariate and multivariate cases are both examined. Each chapter is self-contained. The theory is presented systematically along with the applications. The application to information theory is also examined.

This monograph is suitable for researchers and graduate students in pure mathematics. Applied mathematicians, engineers, and other applied scientists will also find this book useful.

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Chapters: Chain Rule, Inverse Functions and Differentiation, Sum Rule in Differentiation, Constant Factor Rule in Differentiation, Product Rule, Quotient Rule, Reciprocal Rule. Source: Wikipedia. Pages: 43. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Power rule, Product rule, Quotient rule, Chain rule In calculus, the chain rule is a formula for the derivative of the composition of two functions. In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, that is y = y(u(x)) , then the rate of change of y with respect to x can be computed as the rate of change of y with respect to u multiplied by the rate of change of u with respect to x. Schematically, For an explanation of notation used in this section, see Function composition.The chain rule states that, under appropriate conditions, which in short form is written as .Alternatively, in the Leibniz notation, the chain rule is The chain rule can be applied to as many composed functions as needed: .In integration, the counterpart to the chain rule is the substitution rule. The chain rule in one variable may be stated more completely as follows. Let g be a real-valued function on which is differentiable at c ; and suppose that f is a real-valued function defined on an interval I containing the range of g and suppose further that g(c) is an interior point of I. If f is differentiable at g(c), then Suppose that a mountain climber ascends at a rate of 0.5 kilometers per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometer. To calculate the decrease in air temperature per hour that the climber experiences, one multiplies 6 °C per kilometer by 0.5 kilometer per hour, to obtain 3 °C per hour. This calculation is a typical chain ...More: http://booksllc.net/?id=6113 ... Read more

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This text provides clear information about differentiation. The text is supported by worked examples and includes past examination questions and solutions. This title is from the "Maths Made Simple" series. ... Read more