Advanced Calculus for Applications (2nd Edition)

Rating:
Summary: Good supplement to Mathematical Methods Courses
Review: I've been using this book to supplement an intense course on mathematical methods of physics that I'm currently taking correspondence in Germany, and it's been a real help to me thus far. It doesn't seem to be super mathematical (in the pure theoretical sense), but it is most definitely useful for applications, particularly for physical problems. Anyone who has had the calculus I-III sequence at a mediocre American university, along with differential equations and linear algebra, should be able to make great use of this book. It's been 17 years since I finished my undergrad math degree, and I feel that this book is just my speed!

Rating:
Summary: Good supplement to Mathematical Methods Courses
Review: This book helped me get a High Pass in Applied Mathematics here at the Thayer School of Engineering of Dartmouth College. It's not a great intro text to this material and it skips alot of details but it's a fantastic additional reference book.

For example there is a solved problem of a wave function PDE, which involves Bessel functions. Earlier sections of the text refer to how the ODE's are solved by Bessel, which many books leave out.

Later sections of the text contain an intriguing treatment of the Laplace transform.

This book is not for beginners but is a good stepping stone to more advanced concepts such as the Calculus of Variations. Beyond this you are wise to consider additional texts such as Hildebrand's Methods of Applied Mathematics which contains a more rigorous section on C of V, Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers, Wunsch's Complex Variables, and even a good review on PDE's such as Haberman's Elementary Applied PDE's is not a bad idea if you are in a review situation like I was (having spent 8 yrs in Dot Com's and then back to grad school in engineering).

Rating:
Summary: Great additional reference
Review: This book helped me get a High Pass in Applied Mathematics here at the Thayer School of Engineering of Dartmouth College. It's not a great intro text to this material and it skips alot of details but it's a fantastic additional reference book.

For example there is a solved problem of a wave function PDE, which involves Bessel functions. Earlier sections of the text refer to how the ODE's are solved by Bessel, which many books leave out.

Later sections of the text contain an intriguing treatment of the Laplace transform.

This book is not for beginners but is a good stepping stone to more advanced concepts such as the Calculus of Variations. Beyond this you are wise to consider additional texts such as Hildebrand's Methods of Applied Mathematics which contains a more rigorous section on C of V, Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers, Wunsch's Complex Variables, and even a good review on PDE's such as Haberman's Elementary Applied PDE's is not a bad idea if you are in a review situation like I was (having spent 8 yrs in Dot Com's and then back to grad school in engineering).

Rating:
Summary: Solid Advanced Calculus text
Review: This is a one of the best introductions to advanced calculus out there, in my opinion. Once you've read it, you'll have enough background to start work on serious PDEs and analysis. Further, the book has an rather extensive set of problems for the reader to work, with solutions provided.

There are several flaws, however, which is why I gave it 4 stars. In particular, the first couple chapters are an excrutiating read. It was so boring, in fact, that I quit reading the book and shelved it for three years (math was just a hobby at the time). Once you get past those initial few series-choked chapters, the book picks up the pace quite a bit; the book actually becomes quite the page turner once you get to chapter 6 and for the rest of the text --- took only three weeks to soak up the last half of the book, and that without rushing (versus three years for the first fourth ;) ).

This book is, however, an introductory text --- I'd say probably Junior level. Thus, many points are not explicitly proven. Calculus of variations, for example, only gets a couple pages, and as I recall, they only prove the necessity of d/dx(df/du') - df/du = 0, omitting any discussion of sufficiency: many other similar examples occur throughout the text. This shouldn't be a problem, however, as such issues are often discussed in the literature or in more advanced (graduate-level or advanced undergraduate) texts.

Another potential problem is that the text is fairly old. As such, it doesn't address the use of numerical methods on computers, although it does discuss numerical methods (somewhat outdated ones at times, however, as reflected in the strong emphasis on series solutions thoughout). Personally, I didn't find this to be a problem; there are plenty of mathematical methods texts out there that address these issues, so let them handle it.

So, overall, I'd say Hildebrand is a pretty good book if you're looking for a way to extend your knowledge of elementary calculus. After Hildebrand, you should do pretty well reading graduate-level texts, monographs and journal/conference proceedings, although the going might be pretty rough at first.

Rating:
Summary: Solid Advanced Calculus text
Review: This is a one of the best introductions to advanced calculus out there, in my opinion. Once you've read it, you'll have enough background to start work on serious PDEs and analysis. Further, the book has an rather extensive set of problems for the reader to work, with solutions provided.

There are several flaws, however, which is why I gave it 4 stars. In particular, the first couple chapters are an excrutiating read. It was so boring, in fact, that I quit reading the book and shelved it for three years (math was just a hobby at the time). Once you get past those initial few series-choked chapters, the book picks up the pace quite a bit; the book actually becomes quite the page turner once you get to chapter 6 and for the rest of the text --- took only three weeks to soak up the last half of the book, and that without rushing (versus three years for the first fourth ;) ).

This book is, however, an introductory text --- I'd say probably Junior level. Thus, many points are not explicitly proven. Calculus of variations, for example, only gets a couple pages, and as I recall, they only prove the necessity of d/dx(df/du') - df/du = 0, omitting any discussion of sufficiency: many other similar examples occur throughout the text. This shouldn't be a problem, however, as such issues are often discussed in the literature or in more advanced (graduate-level or advanced undergraduate) texts.

Another potential problem is that the text is fairly old. As such, it doesn't address the use of numerical methods on computers, although it does discuss numerical methods (somewhat outdated ones at times, however, as reflected in the strong emphasis on series solutions thoughout). Personally, I didn't find this to be a problem; there are plenty of mathematical methods texts out there that address these issues, so let them handle it.

So, overall, I'd say Hildebrand is a pretty good book if you're looking for a way to extend your knowledge of elementary calculus. After Hildebrand, you should do pretty well reading graduate-level texts, monographs and journal/conference proceedings, although the going might be pretty rough at first.