Advanced Calculus: Second Edition (Dover Books on Mathematics) 2nd Edition by David V. Widder (PDF)

Description

This classic text by a distinguished mathematician and former Professor of Mathematics at Harvard University, leads students familiar with elementary calculus into confronting and solving more theoretical problems of advanced calculus. In his preface to the first edition, Professor Widder also recommends various ways the book may be used as a text in both applied mathematics and engineering.Believing that clarity of exposition depends largely on precision of statement, the author has taken pains to state exactly what is to be proved in every case. Each section consists of definitions, theorems, proofs, examples and exercises. An effort has been made to make the statement of each theorem so concise that the student can see at a glance the essential hypotheses and conclusions.For this second edition, the author has improved the treatment of Stieltjes integrals to make it more useful to the reader less than familiar with the basic facts about theRiemann integral. In addition the material on series has been augmented by the inclusion of the method of partial summation of the Schwarz-Holder inequalities, and of additional results about power series. Carefully selected exercises, graded in difficulty, are found in abundance throughout the book; answers to many of them are contained in a final section.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Liked the chapter on line integrals. The chapter on Fourier series provides an useful introduction. Other important stuff covered.

⭐Certainly an acceptable and sufficiently thorough text for use as a reference or supplement to a course on multivarialble calculus or differential equations. (PERSONALLY) I find this book unsuitable for a formal course on advanced calculus or introductory real analysis. Useful content regardless.

⭐The book has roughly 500+ pages but contains so many topics: Differential geometry, Fourier series, Laplace transform, Stieltjes integral… Each topic can be a sole book, just for introductory. Thus, you can imagine that the book is very concise and, sometimes, “dry”. There is not so much motivation or explanation at the beginning of each chapter. I may use it as a (brief) reference rather than textbook.

⭐I bought this book when I decided to go for graduate studies in mathematics. I was away from college for 3 years and wanted to refresh some of the advanced calc material. You could not ask for any better book. The generic Calculus book I used in college was too basic and this provided a nice bridge and a solid description of the “advanced” calculus. Although the order of the chapters are a little weird, it gets the job done without any lousy explanations. One thing to note is that “brdiging” book. So not so advanced but provides a nice transition if you want to look at some advanced calc books. This neverthless is a great addition

⭐Great book for students who want to take more advanced topics in Math.The book is assuming that students should have strong foundation in math.

⭐never fails

⭐I bought this textbook as a supplementary resource book for an advanced calculus class I once took although I ended up using it for a Differential Equations II class instead (in particular the partial differential equation and fourier series sections). This book does not present proofs as one might expect from many of today’s Advanced Calculus classes. It does not present abstract theorems but rather applied Calculus and Differential Equations. You will not find logical connectives, quantifiers, techniques of proofs, set operations, induction, or completeness axioms in this book. What you will find is partial differentiation, line and surface integrals, definite integrals, fourier series, infinite series, etc. Electrical and Computer Engineers will find that they may benefit from the Vector, Fourier Series, and Laplace Transform chapters of this book. Physics majors are more likely to profit from the chapters on Partial Differentiation and Fourier Series.Here’s the textbooks chapter titles: 1) Partial Differentiation, 2) Vectors, 3) Differential Geometry’, 4) Applications of Partial Differentation, 5) Stieltjes Integral, 6) Multiple Integrals, 7) Line and Surface Integrals, 8) Limits and Indeterminate Forms, 9) Infinite Series, 10) Convergence of Improper Integrals, 11) The Gamma Function. Evaluation of Definite Integrals, 12) Fourier Series, 13) The Laplace Transform, 14) Applications of the Laplace Transform.The book may be considered as being written in the ole’ school style. It was written by a former Professor of Mathematics at Harvard and was first printed in 1947. The relatively low cost of the textbook may be attributed to it not having been `updated’ for a while, being devoid of any color, and being softbound. It has some worked out examples but focuses more on established theorems and lemmas to solve problems. The book is fairly well organized and is overall a good reference book.

⭐In 1963 I asked Lynn Loomis what I should bring to his course math 55 in abstract analysis (see Loomis and Sternberg) and he said I should know the theoremthat a continuous function on a closed bounded interval has a maximum. So I got a copy of Widder and read the proof. I recall it as clear and rigorous.Thus I presume the rest of the book is similar. It fulfilled my need for rigorous background for abstract advanced calculus so I recommend it.By the way, most of the critical comments here about it being old fashioned or criticizing the notation seem to me like the sort of remarks made by novices whodo not understand what they read and confuse terminology with content.(I got B+/A- in Loomis’ course in 1963/64 and have been a professional mathematician for the past 40 years.)

⭐Whenever I need to buy a new textbook I groan at the cost. Yes, I understand that academics need to make a living but £70 for a book is just so hard to swallow when you don’t have a lot of money! This is why I love the Dover Mathematics series – the prices are just so much more reasonable. Don’t be fooled by the cheap price – this is a proper sized textbook, very well printed.There are also minimal errata in this textbook, which is nice and again too uncommon. The first chapters of this book are absolutely key and really give the foundations for going on to enjoy the later parts. I would strongly recommend that additional time is spent here in order to appreciate the later sections of the book. It’s no short project working through this book, but once you have you will find that you are so so much better at calculus. This is often an area that lets even great mathematicians down.

⭐Nothing else to add.

⭐Book good heavily damaged

⭐There is not one word wasted in this concise disciplined text. No waffle, no pretty pictures, only useful diagrams to illustrate a point, and it never says: “the proof is omitted as it is somewhat intricate”. The style reflects the fact that the first edition was 1947 and this second edition was first published in 1961. To benefit most from this book you need to:(1) have a background in calculus with some basic linear algebra(2) be able to read mathematical writing.It actually starts right at the very beginning and sets out exactly what the author means by every term. It builds logically from there. Every proof includes a very clear list of hypotheses that are relied upon.It is probably most useful if you have done or doing a modern course and the ideas are still swimming around in your head but you would like them nailed down in a logical way. Many of the proofs will be different to ones taught today and therefore give another perspective. It is also invaluable if you are doing an “omitted proofs” type course which has given you adequate mechanical calculus skills but left you unsatisfied by lack of proof/understanding.

⭐conciseness and simplicity are the beauty of this book

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