2010, XIX, 234 p. 5 illus., Softcover
ISBN: 978-0-8176-4379-9
Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. Modern differential geometry from the author’s perspective is used in this work to describe physical theories of a geometric character without using any notion of calculus (smoothness). Instead, an axiomatic treatment of differential geometry is presented via sheaf theory (geometry) and sheaf cohomology (analysis). Using vector sheaves, in place of bundles, based on arbitrary topological spaces, this unique approach in general furthers new perspectives and calculations that generate unexpected potential applications.
Modern Differential Geometry in Gauge Theories is a two-volume research monograph that systematically applies a sheaf-theoretic approach to such physical theories as gauge theory. Beginning with Volume 1, the focus is on Maxwell fields. All the basic concepts of this mathematical approach are formulated and used thereafter to describe elementary particles, electromagnetism, and geometric prequantization. Maxwell fields are fully examined and classified in the language of sheaf theory and sheaf cohomology. Continuing in Volume 2, this sheaf-theoretic approach is applied to Yang?Mills fields in general.
The text contains a wealth of detailed and rigorous computations and will appeal to mathematicians and physicists, along with advanced undergraduate and graduate students, interested in applications of differential geometry to physical theories such as general relativity, elementary particle physics and quantum gravity.
Physicists, mathematicians; graduate and advanced undergraduate students specializing in applications of differential geometry to physical theories such as general relativity, elementary particle physics and quantum gravity
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Series: Applied Mathematical Sciences , Vol. 158
2010, XII, 796 p., Hardcover
ISBN: 978-1-4419-1563-4
Due: December 2009
Ludwig Prandtl has been called the "father" of modern fluid mechanics
This is an expanded version of Prandtl’s original text, which is a classic in the field of fluid mechanics
Ludwig Prandtl, with his fundamental contributions to hydrodynamics, aerodynamics and gasdynamics, greatly influenced the development of fluid mechanics as a whole and it was his pioneering research in the first half of the 20th century that founded modern fluid mechanics. His book `Fuhrer durch die Stromungslehre’ or `Essentials of Fluid Mechanics’ appeared in 1942. Even today it is considered one of the most important books in the area.
It is based on the 12th German edition with additional material included. All Chapters have been revised and extended, and there are new chapters on fluid mechanical instabilities and turbulence, microflows and biofluid mechanics.
Essentials of Fluid Mechanics is aimed at science and engineering students and researchers wishing to obtain an overview of the different branches of fluid mechanics. The book is extensively illustrated throughout and includes problems to aid learning in many chapters.
Preface.- Introduction.- Properties of Liquids and Gases.- Kinematics of Fluid Flow.- Dynamics of Fluid Flow.- Fundamental Equations of Fluid Mechanics.- Instabilities and Turbulent Flows.- Convective Heat and Mass Transfer.- Multiphase Flows.- Reactive Flows.- Flows in the Atmosphere and in the Ocean.- Microflows.- Biofluid Mechanics.- Selected Bibliography.- Index.
Series: Lecture Notes in Mathematics , Vol. 1986
2009, XII, 332 p., Softcover
ISBN: 978-3-642-05135-7
Due: November 2009
Partial Inner Product (PIP) Spaces are ubiquitous, e.g. Rigged Hilbert spaces, chains of Hilbert or Banach spaces (such as the Lebesgue spaces Lp over the real line), etc. In fact, most functional spaces used in (quantum) physics and in signal processing are of this type. The book contains a systematic analysis of PIP spaces and operators defined on them. Numerous examples are described in detail and a large bibliography is provided. Finally, the last chapters cover the many applications of PIP spaces in physics and in signal/image processing, respectively.
As such, the book will be useful both for researchers in mathematics and practitioners of these disciplines.
Researchers and graduate students in mathematics, physics and signal and image processing
1 General Theory : Algebraic Point of View.- 2 General Theory : Topological Aspects.- 3 Operators on PIP-spaces and Indexed PIP-spaces.- 4 Examples of Indexed PIP-spaces.- 5 Refinements of PIP-spaces.- 6 Partial *-algebras of Operators in a PIP-space.- 7 Applications in Mathematical Physics.- 8 PIP-spaces and Signal Processing.
Series: Undergraduate Texts in Mathematics
2010, XII, 475 p. 79 illus., 76 in color., Hardcover
ISBN: 978-1-4419-1620-4
Due: January 2010
Self-contained, rigorous book of reasonable size
Neatly ties up multivariable calculus with its relics in one variable calculus
Caters to theoretical as well as practical aspects of multivariable calculus
Follows a highly organized structure, where each chapter includes sections, subsections, notes and comments, and exercises
Includes high quality exercises split into two parts: Part A consists of relatively routine problems and Part B contains those that are either more theoretical or challenging
Contains extensive material on topics not typically covered in multivariable
calculus textbooks, such as: monotonicity and bimonotonicity of functions
of two variables and their relationship with partial differentiation; higher
order directional derivatives and their use in Taylor's Theorem; and rectangular
Rolle's and Mean Value Theorem
This self-contained textbook gives a thorough exposition of multivariable calculus. It can be viewed as a sequel to the one-variable calculus text, A Course in Calculus and Real Analysis, published in the same series. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in one-variable calculus. For example, when the general definition of the volume of a solid is given using triple integrals, the authors explain why the shell and washer methods of one-variable calculus for computing the volume of a solid of revolution must give the same answer. Further, the book includes genuine analogues of basic results in one-variable calculus, such as the mean value theorem and the fundamental theorem of calculus.
This book is distinguished from others on the subject: it examines topics not typically covered, such as monotonicity, bimonotonicity, and convexity, together with their relation to partial differentiation, cubature rules for approximate evaluation of double integrals, and conditional as well as unconditional convergence of double series and improper double integrals. Moreover, the emphasis is on a geometric approach to such basic notions as local extremum and saddle point.
Each chapter contains detailed proofs of relevant results, along with numerous examples and a wide collection of exercises of varying degrees of difficulty, making the book useful to undergraduate and graduate students alike. There is also an informative section of "Notes and Comments’’ indicating some novel features of the treatment of topics in that chapter as well as references to relevant literature. The only prerequisite for this text is a course in one-variable calculus.