Vector Calculus

新的版本，除了符合應用工程學上，新的需求之外，筆者還特別精挑細選，補充了一些例題、習題以及應用題。重新編訂後的版本，筆者將其命名為《Vector Calculus with its Applications (向量演算學及其應用)》，無論是在字體或是符號上而言，都有新的面貌。

This concise text was created as a workbook for learning to use vector calculus in practical calculations and derivations. Its only prerequisite is a familiarity with one-dimensional differential and integral calculus. Though it often makes use of physical examples, knowledge of physics itself is not required to study the mathematics of vector calculus. The approach is suitable for advanced undergraduates and graduate students in mathematics, physics, and other areas of science.The two-part treatment opens with a brief text that develops vector calculus from the very beginning and then addresses some more detailed applications. Topics include vector differential operators, vector identities, integral theorems, Dirac delta function, Green's functions, general coordinate systems, and dyadics. The second part consists of answered problems, all closely related to the development of vector calculus in the text. Those who study this book and work out the problems will find that rather than m

Designed for undergraduates in mathematics, engineering, the physical sciences and for practicing engineers, the book focuses on practical applications of engineering and science used in industry. It

This is the second volume of "A Course in Analysis" and it is devoted to the study of mappings between subsets of Euclidean spaces. The metric, hence the topological structure is discussed as well as

This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus. It is a sequel to the textLinear and Geometric Algebra by the same author. Tha

This book serves as an introduction to calculus on normed vector spaces at a higher undergraduate or beginning graduate level. The prerequisites include basic calculus and linear algebra, as well as a

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The aim of this book is to facilitate the use of Stokes' Theorem in applications. The text takes a differential geometric point of view and provides for the student a bridge between pure and applied

This text helps students foster computational skills and intuitive understanding with a careful balance of theory, applications and optional materials. This edition offers revised coverage in several

This introduction to the differential and integral calculus of functions of several variables offers a rigorous and comprehensive treatment. The classical theorems of vector calculus are amply illustr

This brief book presents an accessible treatment of multivariable calculus with an early emphasis on linear algebra as a tool. Its organization draws strong analogies with the basic ideas of elementar

Vector calculus is the foundation stone on which a vast amount of applied mathematics is based. Topics such as fluid dynamics, solid mechanics and electromagnetism depend heavily on the calculus of ve

Building on previous texts in the Modular Mathematics series, in particular 'Vectors in Two or Three Dimensions' and 'Calculus and ODEs', this book introduces the student to the concept of vector calc

Table of Contents:Chapter 0 PrecalculusChapter 1 Functions and LimitsChapter 2 DerivativesChapter 3 Inverse FunctionsChapter 4 Applications of DerivativesChapter 5 IntegralsChapter 6 Techniques of IntegrationChapter 7 Applications of IntegrationChapter 8 Parametric Equations and Polar CoordinatesChapter 9 Vector SpacesChapter 10 Partial DerivativesChapter 11 Multiple IntegralsChapter 12 Sequences and SeriesAppendix A The Precise Definition of LimitsAppendix B Change of VariablesAppendix C Vector CalculusExercise AnswersIndex

This text for undergraduates was designed as a short introductory course to give students the tools of vector algebra and calculus, as well as a brief glimpse into the subjects' manifold applications.

Part I of this classic work offers a rigorous presentation of tensor calculus as a development of vector analysis. Part II discusses the most important applications of tensor calculus, including the use of tensors in classical analytical dynamics and the role of tensors in special relativity. A concluding section examines field equations of general relativity theory.

"Classroom-tested in a Princeton University honors course, this text offers a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the

"Assuming only a knowledge of basic calculus, this text presents an elementary and gradual development of tensor theory. From this treatment, the traditional material of courses on vector analysis is

This book is concerned with the bifurcation theory, the study of the changes in the structures of the solution of ordinary differential equations as parameters of the model vary. The theory has developed rapidly over the past two decades. Chapters 1 and 2 of the book introduce two systematic methods of simplifying equations: centre manifold theory and normal form theory, by which the dimension of equations may be reduced and the forms changed so that they are as simple as possible. Chapters 3–5 of the book study in considerable detail the bifurcation of those one- or two-dimensional equations with one, two or several parameters. This book is aimed at mathematicians and graduate students interested in dynamical systems, ordinary differential equations and/or bifurcation theory. The basic knowledge required by this book is advanced calculus, functional analysis and qualitative theory of ordinary differential equations.

This 2002 book presents the reader with mathematical tools taken from matrix calculus and zero-one matrices and demonstrates how these tools greatly facilitate the application of classical statistical procedures to econometric models. The matrix calculus results are derived from a few basic rules that are generalizations of the rules of ordinary calculus. These results are summarized in a useful table. Well-known zero-one matrices, together with some newer ones, are defined, their mathematical roles explained, and their useful properties presented. The basic building blocks of classical statistics, namely the score vector, the information matrix, and the Cramer-Rao lower bound, are obtained for a sequence of linear econometric models of increasing statistical complexity. From these are obtained interactive interpretations of maximum likelihood estimators, linking them with efficient econometric estimators. Classical test statistics are also derived and compared for hypotheses of intere

Designed to familiarize undergraduates with the methods of vector algebra and vector calculus, this text offers both a clear view of the abstract theory as well as a concise survey of the theory's app

De Rham cohomology is the cohomology of differential forms. This book offers a self-contained exposition to this subject and to the theory of characteristic classes from the curvature point of view. It requires no prior knowledge of the concepts of algebraic topology or cohomology. The first ten chapters study cohomology of open sets in Euclidean space, treat smooth manifolds and their cohomology and end with integration on manifolds. The last eleven chapters include Morse theory, index of vector fields, Poincaré duality, vector bundles, connections and curvature, and the book ends with the general Gauss-Bonnet theorem. The text includes well over 150 exercises, and gives the background to the modern developments in gauge theory and geometry in four dimensions, but it also serves as an introductory course in algebraic topology. It will be invaluable to anyone studying cohomology, curvature, and their applications.

Master the practical aspects of implementing deep learning solutions with PyTorch, using a hands-on approach to understanding both theory and practice. This updated edition will prepare you for applying deep learning to real world problems with a sound theoretical foundation and practical know-how with PyTorch, a platform developed by Facebook's Artificial Intelligence Research Group.You'll start with a perspective on how and why deep learning with PyTorch has emerged as an path-breaking framework with a set of tools and techniques to solve real-world problems. Next, the book will ground you with the mathematical fundamentals of linear algebra, vector calculus, probability and optimization. Having established this foundation, you'll move on to key components and functionality of PyTorch including layers, loss functions and optimization algorithms. You'll also gain an understanding of Graphical Processing Unit (GPU) based computation, which is essential for training deep learning models

The fundamental mathematical tools needed to understand machine learning include linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. These topics are traditionally taught in disparate courses, making it hard for data science or computer science students, or professionals, to efficiently learn the mathematics. This self-contained textbook bridges the gap between mathematical and machine learning texts, introducing the mathematical concepts with a minimum of prerequisites. It uses these concepts to derive four central machine learning methods: linear regression, principal component analysis, Gaussian mixture models and support vector machines. For students and others with a mathematical background, these derivations provide a starting point to machine learning texts. For those learning the mathematics for the first time, the methods help build intuition and practical experience with applying mathematical concepts. Every cha

Designed for engineers, mathematicians, computer scientists, financial analysts, and anyone interested in using numerical linear algebra, matrix theory, and vector calculus concepts to maximize effici

This book is designed as a calculus-based introduction to the concepts of physics. Numerous examples, applications, and figures provide readers with simple explanations. Standard topics include vector

Magnetohydrodynamics (MHD) plays a crucial role in astrophysics, planetary magnetism, engineering and controlled nuclear fusion. This comprehensive textbook emphasizes physical ideas, rather than mathematical detail, making it accessible to a broad audience. Starting from elementary chapters on fluid mechanics and electromagnetism, it takes the reader all the way through to the latest ideas in more advanced topics, including planetary dynamos, stellar magnetism, fusion plasmas and engineering applications. With the new edition, readers will benefit from additional material on MHD instabilities, planetary dynamos and applications in astrophysics, as well as a whole new chapter on fusion plasma MHD. The development of the material from first principles and its pedagogical style makes this an ideal companion for both undergraduate students and postgraduate students in physics, applied mathematics and engineering. Elementary knowledge of vector calculus is the only prerequisite.