Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick
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About this book :-
Advanced Calculus (Second Edition) written by
Patrick M. Fitzpatrick .
Book Detail :-
Title: Advanced Calculus
Edition: 2nd
Author(s): Patrick M. Fitzpatrick
Publisher: Brooks/Cole
Series:
Year: 2006
Pages: 609
Type: PDF
Language: English
ISBN: 0534376037,9780534376031
Country: US
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About Author :-
The author Patrick M. Fitzpatrick was born in March 1946, Youghal, Republic of Ireland. He complete his B.A. in Mathematics, Rutgers University (1966) and Ph.D. in Mathematics, Rutgers University (1966)
He done his B.S. in Mathematics, 1966, Rutgers University and Ph.D. in Mathematics, 1971, Rutgers University
He start his career as Chair, Department of Mathematics, UMd (1996–2007) then Associate Chair for Undergraduate Studies, UMd (1994–1996). He start as Assistant Professor, UMd (1975–1984) and Professor, UMd (1984)
He also work as Visiting Assistant Professor, Rutgers University (1972–1973) and Visiting Member, Courant Institute of Mathematical Sciences, N.Y.U (1971–1972) and Visiting Member, Institute for Physical Sciences and Technology, UMd (1977–1986)
My present research interests center on the study of topological methods in nonlinear operator theory, with particular interest in the study of bifurcation of solutions of parametrized families of nonlinear partial differential equations. One aspect of this has been the development of a topological degree for nonlinear Fredholm mappings. The essential novelty of this degree is that it presents a new, precise description of the homotopy property of degree that is needed to establish global bofurcation results for one parameter families of such mappings.
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Book Contents :-
Advanced Calculus (Second Edition) written by
Patrick M. Fitzpatrick
cover the following topics.
Preface
Preliminaries
1. TOOLS FOR ANALYSIS
1.1 The Completeness Axiom and Some of Its Consequences
1.2 The Distribution of the Integers and the Rational Numbers
1.3 Inequalities and Identities
2. CONVERGENT SEQUENCES
2.1 The Convergence of Sequences
2.2 Sequences and Sets
2.3 The Monotone Convergence Theorem
2.4 The Sequential Compactness Theorem
2.5 Covering Properties of Sets*
3. CONTINUOUS FUNCTIONS
3.1 Continuity
3.2 The Extreme Value Theorem
3.3 The Intermediate Value Theorem
3.4 Uniform Continuity
3.5 The E-o Criterion for Continuity
3.6 Images and Inverses; Monotone Functions
3.7 Limits
4. DIFFERENTIATION
4.1 The Algebra of Derivatives
4.2 Differentiating Inverses and Compositions
4.3 The Mean Value Theorem and Its Geometric Consequences
4.4 The Cauchy Mean Value Theorem and Its Analytic Consequences
4.5 The Notation of Leibnitz
5. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS
5.1 Solutions of Differential Equations
5.2 The Natural Logarithm and Exponential Functions
5.3 The Trigonometric Functions
5.4 The Inverse Trigonometric Functions
6. INTEGRATION: TWO FUNDAMENTAL THEOREMS
6.1 Darboux Sums; Upper and Lower Integrals
6.2 The Archimedes-Riemann Theorem
6.3 Additivity, Monotonicity, and Linearity
6.4 Continuity and Integrability
6.5 The First Fundamental Theorem: Integrating Derivatives
6.6 The Second Fundamental Theorem: Differentiating Integrals
7.* INTEGRATION: FURTHER TOPICS
7.1 Solutions of Differential Equations
7.2 Integration by Parts and by Substitution
7.3 The Convergence of Darboux and Riemann Sums
7.4 The Approximation of Integrals
8. APPROXIMATION BY TAYLOR POLYNOMIALS
8.1 Taylor Polynomials
8.2 The Lagrange Remainder Theorem
8.3 The Convergence of Taylor Polynomials
8.4 A Power Series for the Logarithm
8.5 The Cauchy Integral Remainder Theorem
8.6 A Nonanalytic, Infinitely Differentiable Function
8.7 The Weierstrass Approximation Theorem
9. SEQUENCES AND SERIES OF FUNCTIONS
9.1 Sequences and Series of Numbers
9.2 Pointwise Convergence of Sequences of Functions
9.3 Uniform Convergence of Sequences of Functions
9.4 The Uniform Limit of Functions
9.5 Power Series
9.6 A Continuous Nowhere Differentiable Function
10. THE EUCLIDEAN SPACE IR.n
10.1 The Linear Structure of IR.n and the Scalar Product
10.2 Convergence of Sequences in IR.n
10.3 Open Sets and Closed Sets in IR.n
11. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS
11.1 Continuous Functions and Mappings
11.2 Sequential Compactness, Extreme Values, and Uniform Continuity
11.3 Pathwise Connectedness and the Intermediate Value Theorem*
11.4 Connectedness and the Intermediate Value Property*
12.* METRIC SPACES
12.1 Open Sets, Closed Sets, and Sequential Convergence
12.2 Completeness and the Contraction Mapping Principle
12.3 The Existence Theorem for Nonlinear Differential Equations
12.4 Continuous Mappings between Metric Spaces
12.5 Sequential Compactness and Connectedness
13. DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES
13.1 Limits
13.2 Partial Derivatives
13.3 The Mean Value Theorem and Directional Derivatives
14. LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS
14.1 First-Order Approximation, Tangent Planes, and Affine Functions
14.2 Quadratic Functions, Hessian Matrices, and Second Derivatives*
14.3 Second-Order Approximation and the Second-Derivative Test*
15. APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS
15.1 Linear Mappings and Matrices
15.2 The Derivative Matrix and the Differential
15.3 The Chain Rule
16. IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM
16.1 Functions of a Single Variable and Maps in the Plane
16.2 Stability of Nonlinear Mappings
16.3 A Minimization Principle and the General Inverse Function Theorem
17. THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS
17.1 A Scalar Equation in Two Unknowns: Dini
s Theorem
17.2 The General Implicit Function Theorem
17.3 Equations of Surfaces and Paths in IR3
17.4 Constrained Extrema Problems and Lagrange Multipliers
18. INTEGRATING FUNCTIONS OF SEVERAl VARIABLES
18.1 Integration of Functions on Generalized Rectangles
18.2 Continuity and Integrability
18.3 Integration of Functions on jordan Domains
19. ITERATED INTEGRATION AND CHANGES OF VARIABLES
19.1 Fubini
s Theorem
19.2 The Change of Variables Theorem: Statements and Examples
19.3 Proof of the Change of Variables Theorem
20. LINE AND SURFACE INTEGRALS
20.1 Arclength and Line Integrals
20.2 Surface Area and Surface Integrals
20.3 The Integral Formulas of Green and Stokes
Appendix
A. CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS
A.1 The Field Axioms and Their Consequences
A.2 The Positivity Axioms and Their Consequences
B.LINEAR ALGEBRA
Index
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