A friend of me who is not studying mathematics now needs to deal with integrals, double integrals and triple integrals within his study of chemistry. He asked me to give him a suggestion for a basic book that explains basic facts, rules etc., about integrals. I think this means Riemann integral and that it should not be too complicated.
I do not know such a book, do you know a book that deal with basic things concerning integrals for a non-mathematician?
Thanks for your tips!
I don't know exactly what kind of book you want. Perhaps not a Bourbaki-like book, but more friendly ones. They may be classified into two types (This is only my classification):
Too many books (both types of books) are published in Japan. Then, if you live in a big city in Japan, there are several bookstores that sell such books. Unfortunately, however, these are written in Japanese.
I regret that particularly Type 1 books are not translated into English.
The concept of Type-1 books seems to be close to the concept of The Feynman Lectures on Physics.These books tend to be written by famous Japanese mathematicians. No advanced techniques and strict proofs are written. The authors of this type of book seem to be trying to show their sharpened sensibility to the next generation.
The main target include students who belong to top-level universities in Japan. I think, most Type-1's readers actually can understand advanced techniques (such as the partition of unity, the ε-δ theory, and.so on). However, they desire deeper understanding than that of such an "autopilot-like" understanding.
Textbook (1-1):(Written in Japanese)
Hideki Omori et.al "直観世界からの微・積分入門 (Calculus from the Intuitive World)" Yuseisha, Japan (1998/04) .
The textbook (1-1) is based on lectures given at the Tokyo University of Science by Prof. Dr.Hideki Omori, an Japanese top-level expert in differential geometry. Tokyo University of Science is one of the top-level universities in Japan, and famous graduates include Satoshi Omura (2015 Nobel Laureate).
By developing and nurturing a sense that even elementary school children can understand, you can arrive at an intuitive understanding of the multivariate differentiation and the multivariate integration.- That is the concept of this book. The goal of this book is Stokes' theorem.
The level of content is that all science students must understand in their first year of university. But, the book has a very strong personality. Therefore, in addition to a high mathematical sense, a literary sense will be necessary for the translation.
Textbook (1-2):(Written in Japanese)
Shoushich Kobayashi:"続 微分積分読本 (Textbook on differentiation and integration 2)", Shoka-bo, Japan (2001/8/25)
This book was written by Prof. Dr.Shoshichi Kobayashi, was a famous mathematician. However, this book was written in Japanese. It was written after he retired from UC Berkeley, unfortunately not translated into English. This is a continuation of the author's ”Textbook on differentiation and integration.” This book also aims at Stokes' theorem. It is written in a classic style and does not require much prior knowledge. In addition, the application of physics is considered.
Type 2 books are intended to help students who can't keep up with their class. Many authors of this type of book are Yobikō's teachers. Recently these kind of books sell well even to top-level university students in Japan. Style that emphasizes "How to" rather than "Why" seems to be popular. It seems to be very useful as the first book read by modern students who have to learn a lot.
I'm sorry, but I haven't read these books. However, using Amazon's trial reading function or Book reviews, I selected something that looks good.
Prof. Dr.Sonoko Ishimura, the author of Text Book (2-1), is a famous woman among science students in Japan. She and her husband have written many books to help college/university students who cannot keep up with their classes. Some of them are bestsellers. Surprisingly, her sons publishing same type of textbooks. The Ishimura family has at least two such sons. Such a family would be rare in the world. She seems to be in charge of mathematics in her family and her husband,Prof. Dr Sadao Ishimura is in charge of statistics. You can buy this book on Amazon.com instead of Amazon.co.jp (But written in Japanese).
Text Book (2-1):(Written in Japanease)
Sonoko Ishimura;"やさしく学べる微積分学(Take it easy for calucules)" Tokyo-Tosho,Japan 1999/12/1
Textbook (2-2):(Written in Japanese)
石井 俊全(Ishi Shun-zen?): "1冊でマスター 大学の微分積分 (With this book, you can understand the university level calculus)", Gijyutu-Hyouron-Shha, Japan (2014/7/9)
Textbook (2-3):(Written in Japanese)
寺田文行 (Terada Bun-Gyo?),et. al.;"基本演習微分積分(Basic Exercises of Calculus )" Science-sha (1993/4/1)
P.S.
I'm not very good at English, so I'm sorry if I have some impolite or unclear expressions. I welcome any corrections and English review. (You can edit my question and description to improve them)
Surprised to see many people endorsing Stewart. Usually it is considered incomparable with the likes of Spivak but given the context of a non-mathematical student I see the point. But I still don't think it is the best suggestion.
I think the best book to learn multi-dimensional integration from is Div, Grad, Curl, and All That - An Informal Text on Vector Calculus by H.M.Schey. It is not a chemical approach. But an electro-magnetic approach which I think the Chemistry student can relate to quite easily. It is a story-like exposition done in a beautiful sequence. It provides incentive or motivation for the Gauss and Stokes Theorems and hence makes the results easier to remember. Really is an exception to those who think these subjects cannot be understood but only memorised unless done rigorously.
But you might want to accompany it with a standard Calculus textbook too. But if you are reading Schey then you could go with something like the third volume of Marsden, Weinstein. Or even the one by Gilbert Strang which is free I think. Note these two books aren't considered friendly to non-mathematical students. But the one by Schey surely is and use these two to look up some stuff you would want to see a rigorous exposition of.
Finally there is another free MIT book called Street Fighting Mathematics which has a couple of chapters on quick integration methods. I haven't read it properly but judging by the preface I think it is made for students who want to bypass rigour.
Hope I helped.
I am one to vouch for the calculus books by James Stewart that are alluded to above by cjferes. I teach my calc courses out of these books (7E). The books are easy to understand and have quality examples. They are in color and really explain how to do the integration methods. The book has proofs, but is light on proofs. This is most likely the preferred method for a student who is not math heavy.
Another thing that is great about Stewart is that it has a large volume of diverse problems for each topic. I think this is great for self study. The odd answers are in the back of the book.
Also, there is a way to get two volumes of the single variable calc book for Stewart. The first volume is more focused on derivatives and fundamental integrals (polynomials, basic $\cos, \sin$ integrals). The second volume has a lot more about integration (by parts, trig sub, partial fractions) and engineering applications. Thus if your friend only needs some basics, volume 1 might be good. If he needs more, volume 2 can be of great use.
The multivariable book is only a single volume.
