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E-raamat: Basic Analysis II: A Modern Calculus in Many Variables

(Clemson University, USA)
  • Formaat: 529 pages
  • Ilmumisaeg: 19-Jul-2020
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351679329
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Basic Analysis II: A Modern Calculus in Many VariablesSuurem pilt
  • Formaat: 529 pages
  • Ilmumisaeg: 19-Jul-2020
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351679329
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Basic Analysis II: A Modern Calculus in Many Variables focuses on differentiation in Rn and important concepts about mappings from Rn to Rm, such as the inverse and implicit function theorem and change of variable formulae for multidimensional integration. These topics converge nicely with many other important applied and theoretical areas which are no longer covered in mathematical science curricula. Although it follows on from the preceding volume, this is a self-contained book, accessible to undergraduates with a minimal grounding in analysis.

    Features

    • Can be used as a traditional textbook as well as for self-study

    • Suitable for undergraduates in mathematics and associated disciplines

        • Emphasises learning how to understand the consequences of assumptions using a variety of tools to provide the proofs of propositions

    Arvustused

    "Mathematics is fortunate to be populated by bright practitioners. Nonetheless, amongst these we are fortunate to have rare individuals who are wise. Professor Peterson is a member of this distinguished group. His works clearly demonstrate the importance of a long career of research and teaching where he combines the two perspectives of: (1) clearly understanding the needs of diverse readers for clear exposition that scaffolds their exposure to complex material with a transparency about both where they are going and what the utility is of what they are currently reading; and, (2) the benefits of having used the mathematics under consideration in so many diverse applications. The masterly synthesis of so much complex material by a single individual is a superb achievement which will reward serious readers with insight, surprise, and breadth as well as depth." Professor John R. Jungck, University of Delaware

    "Analysis is the bedrock of rigorous mathematical thinking and abstraction. Prof. Peterson's book does a fascinating job by taking a critical approach - highly recommended."

    Professor Nithin Nagaraj, National Institute of Advanced Studies "Mathematics is fortunate to be populated by bright practitioners. Nonetheless, amongst these we are fortunate to have rare individuals who are wise. Professor Peterson is a member of this distinguished group. His works clearly demonstrate the importance of a long career of research and teaching where he combines the two perspectives of: (1) clearly understanding the needs of diverse readers for clear exposition that scaffolds their exposure to complex material with a transparency about both where they are going and what the utility is of what they are currently reading; and, (2) the benefits of having used the mathematics under consideration in so many diverse applications. The masterly synthesis of so much complex material by a single individual is a superb achievement which will reward serious readers with insight, surprise, and breadth as well as depth." Professor John R. Jungck, University of Delaware

    "Analysis is the bedrock of rigorous mathematical thinking and abstraction. Prof. Peterson's book does a fascinating job by taking a critical approach - highly recommended."

    Professor Nithin Nagaraj, National Institute of Advanced Studies

    "Dr. Peterson's thoughtful and detailed explanations reflect his insights to a very fundamental but complex subject in Mathematics. The treatment in the book does justice to recent trends in Mathematical Analysis while staying true to the classical spirit of the subject. A thoroughly enjoyable read."

    Professor Snehanshu Saha, BITS PIlani (K K Birla Goa Campus)

    I Introduction
    		1(6)
    1 Beginning Remarks
    		3(4)
    1.1 Table of Contents
    		4(2)
    1.2 Acknowledgments
    		6(1)
    II Linear Mappings
    		7(202)
    2 Preliminaries
    		9(16)
    2.1 The Basics
    		9(9)
    2.2 Some Topology in R2
    		18(2)
    2.2.1 Homework
    		20(1)
    2.3 Bolzano - Weierstrass in R2
    		20(5)
    2.3.1 Homework
    		22(3)
    3 Vector Spaces
    		25(36)
    3.1 Vector Spaces over a Field
    		25(5)
    3.1.1 The Span of a Set of Vectors
    		28(2)
    3.2 Inner Products
    		30(3)
    3.2.1 Homework
    		32(1)
    3.3 Examples
    		33(23)
    3.3.1 Two Dimensional Vectors in the Plane
    		33(3)
    3.3.2 The Connection between Two Orthonormal Bases
    		36(2)
    3.3.3 The Invariance of the Inner Product
    		38(1)
    3.3.4 Two Dimensional Vectors as Functions
    		39(9)
    3.3.5 Three Dimensional Vectors in Space
    		48(4)
    3.3.6 The Solution Space of Higher Dimensional ODE Systems
    		52(4)
    3.4 Best Approximation in a Vector Space with Inner Product
    		56(5)
    3.4.1 Homework
    		59(2)
    4 Linear Transformations
    		61(24)
    4.1 Organizing Point Cloud Data
    		61(1)
    4.1.1 Homework
    		62(1)
    4.2 Linear Transformations
    		62(1)
    4.2.1 Homework
    		63(1)
    4.3 Sequence Spaces Revisited
    		63(8)
    4.3.1 Homework
    		70(1)
    4.4 Linear Transformations between Normed Linear Spaces
    		71(11)
    4.4.1 Basic Properties
    		73(1)
    4.4.2 Mappings between Finite Dimensional Vector Spaces
    		74(8)
    4.5 Magnitudes of Linear Transformations
    		82(3)
    5 Symmetric Matrices
    		85(24)
    5.1 The General Two by Two Symmetric Matrix
    		85(7)
    5.1.1 Examples
    		86(1)
    5.1.2 A Canonical Form
    		87(3)
    5.1.3 Two Dimensional Rotations
    		90(1)
    5.1.4 Homework
    		91(1)
    5.2 Rotating Surfaces
    		92(3)
    5.2.1 Homework
    		94(1)
    5.3 A Complex ODE System Example
    		95(7)
    5.3.1 The General Real and Complex Solution
    		95(2)
    5.3.2 Rewriting the Real Solution
    		97(3)
    5.3.3 Signed Definite Matrices
    		100(1)
    5.3.4 Summarizing
    		101(1)
    5.4 Symmetric Systems of ODEs
    		102(7)
    5.4.1 Writing the Solution Another Way
    		104(2)
    5.4.2 Homework
    		106(3)
    6 Continuity and Topology
    		109(18)
    6.1 Topology in n Dimensions
    		109(3)
    6.1.1 Homework
    		112(1)
    6.2 Cauchy Sequences
    		112(1)
    6.3 Compactness
    		113(5)
    6.3.1 Homework
    		118(1)
    6.4 Functions of Many Variables
    		118(9)
    6.4.1 Limits and Continuity for Functions of Many Variables
    		119(8)
    7 Abstract Symmetric Matrices
    		127(18)
    7.1 Input-Output Ratios for Matrices
    		127(1)
    7.1.1 Homework
    		128(1)
    7.2 The Norm of a Symmetric Matrix
    		128(13)
    7.2.1 Constructing Eigenvalues
    		133(8)
    7.3 What Does This Mean?
    		141(1)
    7.4 Signed Definite Matrices Again
    		142(3)
    7.4.1 Homework
    		143(2)
    8 Rotations and Orbital Mechanics
    		145(38)
    8.1 Introduction
    		145(3)
    8.1.1 Homework
    		147(1)
    8.2 Orbital Planes
    		148(9)
    8.2.1 Orbital Constants
    		149(2)
    8.2.2 The Orbital Motion
    		151(1)
    8.2.3 The Constant B Vector
    		151(2)
    8.2.4 The Orbital Conic
    		153(4)
    8.3 Three Dimensional Rotations
    		157(9)
    8.3.1 Homework
    		160(1)
    8.3.2 Drawing Rotations
    		160(5)
    8.3.3 Rotated Ellipses
    		165(1)
    8.4 Drawing the Orbital Plane
    		166(7)
    8.4.1 The Perifocal Coordinate System
    		169(2)
    8.4.2 Orbital Elements
    		171(2)
    8.5 Drawing Orbital Planes Given Radius and Velocity Vectors
    		173(10)
    8.5.1 Homework
    		180(3)
    9 Determinants and Matrix Manipulations
    		183(26)
    9.1 Determinants
    		183(10)
    9.1.1 Consequences One
    		184(2)
    9.1.2 Homework
    		186(1)
    9.1.3 Consequences Two
    		187(5)
    9.1.4 Homework
    		192(1)
    9.2 Matrix Manipulation
    		193(15)
    9.2.1 Elementary Row Operations and Determinants
    		193(3)
    9.2.2 Code Implementations
    		196(5)
    9.2.3 Matrix Inverse Calculations
    		201(7)
    9.3 Back to Definite Matrices
    		208(1)
    III Calculus of Many Variables
    		209(108)
    10 Differentiability
    		211(40)
    10.1 Partial Derivatives
    		211(2)
    10.1.1 Homework
    		212(1)
    10.2 Tangent Planes
    		213(4)
    10.3 Derivatives for Scalar Functions of n Variables
    		217(7)
    10.3.1 The Chain Rule for Scalar Functions of n Variables
    		219(5)
    10.4 Partials and Differentiability
    		224(11)
    10.4.1 Partials Can Exist but Not be Continuous
    		224(5)
    10.4.2 Higher Order Partials
    		229(2)
    10.4.3 When Do Mixed Partials Match?
    		231(4)
    10.5 Derivatives for Vector Functions of n Variables
    		235(5)
    10.5.1 The Chain Rule for Vector-Valued Functions
    		238(2)
    10.6 Tangent Plane Error
    		240(6)
    10.6.1 The Mean Value Theorem
    		241(1)
    10.6.2 Hessian Approximations
    		242(4)
    10.7 A Specific Coordinate Transformation
    		246(5)
    10.7.1 Homework
    		249(2)
    11 Multivariate Extremal Theory
    		251(12)
    11.1 Differentiability and Extremals
    		251(1)
    11.2 Second Order Extremal Conditions
    		252(11)
    11.2.1 Positive and Negative Definite Hessians
    		253(4)
    11.2.2 Expressing Conditions in Terms of Partials
    		257(6)
    12 The Inverse and Implicit Function Theorems
    		263(28)
    12.1 Mappings
    		263(6)
    12.2 Invertibility Results
    		269(7)
    12.2.1 Homework
    		274(2)
    12.3 Implicit Function Results
    		276(8)
    12.3.1 Homework
    		282(2)
    12.4 Constrained Optimization
    		284(7)
    12.4.1 What Does the Lagrange Multiplier Mean?
    		287(2)
    12.4.2 Homework
    		289(2)
    13 Linear Approximation Applications
    		291(26)
    13.1 Linear Approximations to Nonlinear ODE
    		291(5)
    13.1.1 An Insulin Model
    		291(5)
    13.2 Finite Difference Approximations in PDE
    		296(6)
    13.2.1 First Order Approximations
    		297(3)
    13.2.2 Second Order Approximations
    		300(1)
    13.2.3 Homework
    		301(1)
    13.3 FD Approximations
    		302(9)
    13.3.1 Error Analysis
    		306(4)
    13.3.2 Homework
    		310(1)
    13.4 FD Diffusion Code
    		311(6)
    13.4.1 Homework
    		314(3)
    IV Integration
    		317(114)
    14 Integration in Multiple Dimensions
    		319(32)
    14.1 The Darboux Integral
    		319(12)
    14.1.1 Homework
    		331(1)
    14.2 The Riemann Integral in n Dimensions
    		331(4)
    14.2.1 Homework
    		334(1)
    14.3 Volume Zero and Measure Zero
    		335(5)
    14.3.1 Measure Zero
    		335(4)
    14.3.2 Volume Zero
    		339(1)
    14.4 When is a Function Riemann Integrate?
    		340(5)
    14.4.1 Homework
    		345(1)
    14.5 Integration and Sets of Measure Zero
    		345(6)
    14.5.1 Homework
    		347(4)
    15 Change of Variables and Fubini's Theorem
    		351(20)
    15.1 Linear Maps
    		351(5)
    15.1.1 Homework
    		355(1)
    15.2 The Change of Variable Theorem
    		356(5)
    15.2.1 Homework
    		360(1)
    15.3 Fubini Type Results
    		361(10)
    15.3.1 Fubini on a Rectangle
    		362(7)
    15.3.2 Homework
    		369(2)
    16 Line Integrals
    		371(34)
    16.1 Paths
    		371(6)
    16.1.1 Homework
    		375(2)
    16.2 Conservative Force Fields
    		377(3)
    16.2.1 Homework
    		379(1)
    16.3 Potential Functions
    		380(5)
    16.3.1 Homework
    		383(2)
    16.4 Green's Theorem
    		385(8)
    16.4.1 Homework
    		390(3)
    16.5 Green's Theorem for Images of the Unit Square
    		393(6)
    16.5.1 Homework
    		396(3)
    16.6 Motivational Notation
    		399(6)
    16.6.1 Homework
    		400(5)
    17 Differential Forms
    		405(26)
    17.1 One Forms
    		405(12)
    17.1.1 Smooth Paths
    		410(3)
    17.1.2 What is a Winding Number?
    		413(4)
    17.2 Exact and Closed 1-Forms
    		417(8)
    17.2.1 Smooth Segmented Paths
    		421(4)
    17.3 Two and Three Forms
    		425(6)
    V Applications
    		431(56)
    18 The Exponential Matrix
    		433(32)
    18.1 The Exponential Matrix
    		434(1)
    18.2 The Jordan Canonical Form
    		435(6)
    18.2.1 Homework
    		440(1)
    18.3 Exponential Matrix Calculations
    		441(9)
    18.3.1 Jordan Block Matrices
    		444(2)
    18.3.2 General Matrices
    		446(2)
    18.3.3 Homework
    		448(2)
    18.4 Applications to Linear ODE
    		450(8)
    18.4.1 The Homogeneous Solution
    		452(4)
    18.4.2 Homework
    		456(2)
    18.5 The Non-Homogeneous Solution
    		458(3)
    18.5.1 Homework
    		460(1)
    18.6 A Diagonalizable Test Problem
    		461(2)
    18.6.1 Homework
    		462(1)
    18.7 Simple Jordan Blocks
    		463(2)
    18.7.1 Homework
    		464(1)
    19 Nonlinear Parametric Optimization Theory
    		465(22)
    19.1 The More Precise and Careful Way
    		466(3)
    19.2 Unconstrained Parametric Optimization
    		469(3)
    19.3 Constrained Parametric Optimization
    		472(12)
    19.3.1 Hessian Error Estimates
    		473(3)
    19.3.2 A First Look at Constraint Satisfaction
    		476(1)
    19.3.3 Constraint Satisfaction and the Implicit Function Theorem
    		477(7)
    19.4 Lagrange Multipliers
    		484(3)
    VI Summing It All Up
    		487(16)
    20 Summing It All Up
    		489(14)
    VII References
    		503(4)
    VIII Detailed Index
    		507
    James Peterson has been an associate professor in the School of Mathematical and Statistical Sciences since 1990. He tries hard to build interesting models of complex phenomena using a blend of mathematics, computation and science. To this end, he has written four books on how to teach such things to biologists and cognitive scientists. These books grew out of his Calculus for Biologists courses offered to the biology majors from 2007 to 2016.

    He has taught the analysis courses since he started teaching both at Clemson and at his previous post at Michigan Technological University. In between, he spent time as a senior engineer in various aerospace firms and even did a short stint in a software development company. The problems he was exposed to were very hard and not amenable to solution using just one approach. Using tools from many branches of mathematics, from many types of computational languages and from first principles analysis of natural phenomena was absolutely essential to make progress.

    In both mathematical and applied areas, students often need to use advanced mathematics tools they have not learned properly. So recently, he has written a series of books on analysis to help researchers with the problem of learning new things after their degrees are done and they are practicing scientists. Along the way, he has also written papers in immunology, cognitive science and neural network technology in addition to having grants from NSF, NASA and the Army.

    He also likes to paint, build furniture and write stories.
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