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Introduction To Mathematical Analysis written by John Hutchinson .
Pure mathematics have one peculiar advantage, that they occasion no disputes among wrangling disputants, as in other branches of knowledge; and the reason is, because the de¯nitions of the terms are premised, and everybody that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies by shewing, either that our adversary has not stuck to his de¯nitions, or has not laid down true premises, or else that he has drawn false conclusions from true principles; and in case we are able to do neither of these, we must acknowledge the truth of what he has proved
The mathematics, he [Isaac Barrow] observes, e®ectually exercise, not vainly delude, nor vexatiously torment, studious minds with obscure subtilties; but plainly demonstrate everything within their reach, draw certain conclusions, instruct by pro¯table rules, and unfold pleasant questions.
Book Detail :-
Title: Introduction To Mathematical Analysis
Author(s): John E. Hutchinson
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About Author :- The author John E. Hutchinson , Professor of Mathematics, Department of Mathematics, School of Mathematical Sciences, Australian National University.
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Book Contents :-
Introduction To Mathematical Analysis written by John Hutchinson cover the following topics. '
Preliminary Remarks, History of Calculus, Why “Prove” Theorems?, “Summary and Problems” Book, The approach to be used, Acknowledgments
2. Some Elementary Logic
Mathematical Statements, Quantifiers, Order of Quantifiers, Connectives (Not, And, Or, Implies, Iff), Truth Tables, Proofs (Proofs of Statements Involving Connectives, Proofs of Statements Involving “There Exists”, Proofs of Statements Involving “For Every”, Proof by Cases)
3. The Real Number System
Introduction, Algebraic Axioms (Consequences of the Algebraic Axioms, Important Sets of Real Numbers, The Order Axioms, Ordered Fields, Completeness Axiom, Upper and Lower Bounds, *Existence and Uniqueness of the Real Number System, The Archimedean Property)
4. Set Theory
Introduction, Russell’s Paradox, Union, Intersection and Difference of Sets, Functions (Functions as Sets, Notation Associated with Functions, Elementary Properties of Functions), Equivalence of Sets, Denumerable Sets, Uncountable Sets, Cardinal Numbers, More Properties of Sets of Cardinality c and d, *Further Remarks (The Axiom of choice, Other Cardinal Numbers, The Continuum Hypothesis, Cardinal Arithmetic,Ordinal numbers)
5. Vector Space Properties of Rn
Vector Spaces, Normed Vector Spaces, Inner Product Spaces
6. Metric Spaces
Basic Metric Notions in Rn, General Metric Spaces, Interior, Exterior, Boundary and Closure, Open and Closed Sets, Metric Subspaces
7. Sequences and Convergence
Notation, Convergence of Sequences, Elementary Properties, Sequences in R, Sequences and Components in Rk, Sequences and the Closure of a Set, Algebraic Properties of Limits
8. Cauchy Sequences
Cauchy Sequences, Complete Metric Spaces, Contraction Mapping Theorem
9. Sequences and Compactness
Subsequences, Existence of Convergent Subsequences, Compact Sets, Nearest Points
10. Limits of Functions
Diagrammatic Representation of Functions, Definition of Limit, Equivalent Definition, Elementary Properties of Limits
Continuity at a Point, Basic Consequences of Continuity, Lipschitz and H¨older Functions, Another Definition of Continuity, Continuous Functions on Compact Sets, Uniform Continuity
12. Uniform Convergence of Functions
Discussion and Definitions, The Uniform Metric, Uniform Convergence and Continuity, Uniform Convergence and Integration, Uniform Convergence and Differentiation
13. First Order Systems of Differential Equations
Examples (Predator-Prey Problem, A Simple Spring System), Reduction to a First Order System, Initial Value Problems, Heuristic Justification for the Existence of Solutions, Phase Space Diagrams, Examples of Non-Uniqueness and Non-Existence, A Lipschitz Condition, Reduction to an Integral Equation, Local Existence, Global Existence, Extension of Results to Systems
Examples (Koch Curve, Cantor Set, Sierpinski Sponge), Fractals and Similitudes, Dimension of Fractals, Fractals as Fixed Points, *The Metric Space of Compact Subsets of Rn, *Random Fractals
Definitions, Compactness and Sequential Compactness, *Lebesgue covering theorem, Consequences of Compactness, A Criterion for Compactness, Equicontinuous Families of Functions, Arzela-Ascoli Theorem, Peano’s Existence Theorem
Introduction, Connected Sets, Connectedness in R, Path Connected Sets, Basic Results
17. Differentiation of Real-Valued Functions
Introduction, Algebraic Preliminaries, Partial Derivatives, Directional Derivatives, The Differential (or Derivative), The Gradient, Geometric Interpretation of the Gradient, Level Sets and the Gradient, Some Interesting Examples, Differentiability Implies Continuity, Mean Value Theorem and Consequences, Continuously Differentiable Functions, Higher-Order Partial Derivatives, Taylor’s Theorem
18. Differentiation of Vector-Valued Functions
Introduction, Paths in Rm, Arc length, Partial and Directional Derivatives, The Differential, The Chain Rule
19. The Inverse Function Theorem and its Applications
Inverse Function Theorem, Implicit Function Theorem, Manifolds, Tangent and Normal vectors, Maximum, Minimum, and Critical Points, Lagrange Multipliers
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