A Course in Advanced Calculus by Robert S. Borden
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A Course in Advanced Calculus by
Robert S. Borden .
This book has been a long time in developing, but the original goal of the author has remained unchanged: to present a course in calculus that would unify the concepts of integration in Euclidean space while at the same time grving the student a feeling for some of the other areas of mathematics which are intimately related to mathematical analysis. For several years at Knox College notes that might be called the original form of this book were used as a text for the honors section of advanced calculus, so in a sense this book has been classroom tested. It should be mentioned, however, that the students had all been through Tom Apostol
s classic two-volume text of calculus with linear algebra, and most were graduate-school bound. Thus, it is only fair to say that although the book was written with the undergraduate in mind, it might be a bit heavy going for students with limited background. On the other hand, it would be presumptuous to say that this is a reference book or a treatise; it is simply a survey of a number of topics the author feels serious mathematics students should know something about, topics that are at the core of undergraduate analysis.
Anyone who studies advanced calculus must be strongly motivated by a genuine love of mathematics. By its very nature the subject is hard, and perhaps not so glamorous. To spend a year at it requires dedication. Therefore the author has made a real effort to "break the monotony," so to speak, by shifting abruptly from one topic to another. Of course, the topics are related; topology, linear algebra, and inequalities all fit into the grand scheme of advanced calculus. To the student, however, they will probably be as distinct as night and day. If the author
s timing has been good, just when one has had all the topology one can stand, inner-product spaces present themselves in all their pristine glory, Fourier series for the most striking curves unfold, and the surprising secret of Pythagoras is revealed.
A Course in Advanced Calculus by
Robert S. Borden
cover the following topics.
1. SETS AND STRUCTURES L.L Sets
1.2 Algebraic Structures
1.3 Morphisms
1.4 Order Structures
Problems
2. LIMIT AND CONTINUITY
2.1 Limit of a Function
2.2 Sequencesin E
2.3 Limit Superior and Limit Inferior of a Function
Problems
3. INEQUALITIES
3.1 Some Basic Inequalities
Problems
4. LINEAR SPACES
4.1 Linear and Affine Mappings
4.2 Continuity of Linear Maps
4.3 Determinants
4.4 The Grassmann Algebra
Problems
5. FORMS IN E"
5.1 Orientation of Parallelotopes
5.2 l-Forms in.En
5.3 Some Applications of l-Forms
5.4 0-Forms in En
5.5 2-Forms in E"
5.6 An Application in E3
5.7 A Substantial Example
5.8 /c-Formsin En
5.9 Another Example
Problems
6. TOPOLOGY
6.1 The Open-Set ToPologY
6.2 Continuity and Limit
6.3 Metrics and Norms
6.4 Product Topologies
6.5 Compactness
6.6 Dense Sets, Connected Sets, Separability, and Category
6.7 Some Properties of Continuous Maps
6.8 Normal Spaces and the Tietze Extension Theorem
6.9 The Cantor Ternary Set
Problems
7. INNER.PRODUCT SPACES
7.1 Real Inner Products
7.2 Orthogonality and Orthonormal Sets
7.3 AnExarnple: The Space L210,2r7
7.4 Fourier Series and Convergence
7.5 The Gram-Schmidt Process
7.6 Approximation bY Projection
7.7 Complex Inner-Product SPaces
7.8 The Gram Determinant and Measures of /<-Parallelotopes
7.9 Vector Products in E3
Problems
8. MEASURE AND INTEGRATION
8.1 Measure
8.2 Measure Spaces and a Darboux Integral
8.3 The MeasureS pace( E",UlL,p) and LebesgueM easure
8.4 The Lebesgue Integral in E
8.5 Signed Measures
8.6 Affine Maps on (E
,9R
,p,)
8.? Integration by Pullbacks; the Affine Case
8.8 A Nonmeasurable Set in Er
8.9 The Riemann-Stieltjes Integral in Er
8.10 Fubini
s Theorem
8.1 I Approximate ContinuitY
Problems
9. DIFFERENTIABLE MAPPINGS
9.1 The Derivative of a Map
9.2 Taylor
s Formula
9.3 The Inverse Function Theorem
9.4 The Implicit Function Theorem
9.5 Lagrange Multipliers
9.6 Some Particular Parametric Maps
9.7 A Fixed Point Theorem
Problems
10. SEQUENCES AND SERIES
10.1 Convergenceo f Sequenceso f Functions
10.2 Series of Functions and Convergence
10.3 Power Series
10.4 Arithmetic with Series
10.5 Infinite Products
Problems
11. APPLICATIONS OF IMPROPER INTEGRALS
11.1 Improper Integrals
11.2 SomeF urther ConvergenceT heorems
11.3 Some Special Functions
11.4 Dirac Sequencesa nd Convolutions
11.5 The Fourier Transform
11.6 The Laplace Transform
11.7 Generalized Functions
Problems
12. THE GENERALIZED STOKES THEOREM
l2.l Manifolds and Partitions of Unity
12.2 ^I\e Stokes Theorem
Problems
Tips and Solutions for Selected Problems,
Bibliography
Index
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