A First Course in Mathematical Analysis

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Cambridge University Press, Aug 17, 2006 - Mathematics
2 Reviews
Mathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject.Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a 'continuous' function? And how exactly can one give a careful definition of 'integral'? The latter question is often one of the mysterious points in a Calculus course - and it is quite difficult to give a rigorous treatment of integration! The text has a large number of diagrams and helpful margin notes; and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self-study or use in parallel with a standard university course on the subject.
 

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This is a fantastic textbook. I graduated from a BSc maths degree in 07 and I always quite liked analysis. I'm a teacher now and recently started missing the 'buzz' of doing proper, degree-type maths again. I saw this in a bookstore and immediately saw that it is very well written. I've used in side by side with my previous uni analysis lecture notes and it's great. It really is an excellent textbook, far greater than any year 1, 2 analysis book that i have seen. I hope that the author proceeds to make a follow up text, with uniform continuity, M-test, functional sequences/series etc, i.e, like the further year 2 university material.
In summery: Excellent text. The reason why my review gives 4 stars and not 5:
Throughout the chapters, there are 'problem' excercises, of which full, good solutions are provided at the book's end. However, at the close of each chapeter there are the usual, overall, 'end of chapter' excercises, of which NO SOLUTIONS are provided! Okay, the autor stipulates that these solutions are not provided during his introduction, but why not? Indeed, if the author chooses to take this route, I feel that he should provide solutions via a webpage etc. Either way, I feel that this approach is an error, within what is otherwise an excellent text book.
 

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The book emphasizes basic details that are probably assumed known in many analysis courses. Someone at the graduate level (who understands the material at that level) would find this book to be too simple, but I found it to be very helpful while taking an undergraduate analysis course.

Contents

Preface page ix
1
Sequences
37
Series
83
Continuity
130
Limits and continuity
167
Differentiation
205
Integration
255
Power series
313
Sets functions and proofs 354
354
Standard derivatives and primitives 359
x

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Page 21 - R2, it is simply the fact that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. We will not give the solution to Example 2.2 as we generalize this example in Example 2.3. As F...
Page 20 - - d, then a + c <- b + d and ab -
Page 354 - B, is the set of elements that belong to both A and B. In the previous example, An B = (3, 5J.
Page 354 - We say that a set A is a subset of a set B if each element of A is also an element of B.

About the author (2006)

David Brannan is Professor, Dean and Director of Studies of the Maths & Computing Faculty at the Open University

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