Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics)
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Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.
83 reviews for Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics)
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John Robinson –
Kevin W. –
Karthik Rathinavelu –
I just want to learn Calculus in an application point of view from my school. During the time, when preparing for the college Entrance Examination, the solution suddenly contains Integral Calculus without me knowing the reason on why they have used that.
I picked up some book to equip myself with Calculus and all the book starts with limits with its formal definition, which leave me spellbound …I will ponder over the same for some days to get what they mean exactly and leave it for later, but this book touched the concept of limits only in the 800 and odd pages. Till then you will be learning Calculus and applying it INTUITIVELY …
This is THE book for person who would like to learn Calculus …great work Kline …
M. Gil –
In general I like the book. But to offend the author think he must save you by saying this part is hard, I’ll give you more details later. I found I skipped some part becuase of to much blah, blah. If you never have taken Calculus it’s just as good as any text. It won’t make learning calculus any easier. For a reference piece it’s a good addition to your collection.
Kersi Von Zerububbel –
If you have the time, patience, and the inclination to get a strong introduction to Calculus then this book will probably meet your needs. Note I say ‘probably’ because for many of the young folks today the text may seem wordy and overly pedantic. Kline forfeits the graphical techniques of learning for the verbal. So one has to carefully follow the explanations. Later in the book graphical explanations for things like differentiation are explained.
So if you expect a fast paced text like say ‘Calculus for Dummies’, ‘How to Ace Calculus’ etc you will be disappointed. What I found helps is to get going with some modern quickie texts like Dummies etc and then go with Kline’s work. This helps firm up some theory instead of just learning the mechanical stuff. Also, as others have pointed out, but this needs to be mentioned, the answers to problems can be obtained via a PDF from Dover. They answered my email promptly.
If you want a good reference for introductory Calculus without say the rigor of Apostol’s text this is a great investment that should serve you for years to come. Excellent textbook.
Schwifty Fox –
Book arrived prompt and in fine condition. No problems with the seller, all good stuff. Party on and learn math… its more than one plus one dude
Faisal Siddiqi –
I ordered this book about a month ago and have just started going through it. I really enjoy the writing style and the teaching approach. I have a math degree and know calculus but would recommend anyone who would like to understand calculus from an intuitive approach to also read this book. I am looking forward to how the material will be covered from a practical bent and what generated certain results in the calculus.
contrarian –
Kline is an excellent teacher and his philosophy of teaching mathematics is right on, in my opinion. Rigor does nothing for the understanding. How could it? Rigor makes everything neat, generalized, and the notation consistent. It also helps to make sure our intuition isn’t playing tricks on us. So, it’s important, but does nothing for learning this science. This book will enable you to understand all the important themes of calculus and proves them in a clear and natural way. No formal understanding of proof methods is necessary.
Methaya Sirichit –
I am currently working through this book. Other reviewers have mentioned all the good points and features of this text. However, given the text is 920 pages long, the reader would normally be required to invest a lot of time working with it. With that in mind, it is natural to ask the all important question: whether this book is really for you? My answer is that it depends on your mathematical backgrounds.
As we know, Kline’s approach to this book is intuitive. Kline argued that “intuition” is the way human’s mind learns things; and he is right on. Especially with calculus whose essence being a collection of “mathematical methods” fundamental to the understanding of physical world. To study these mathematical methods without understanding or appreciating the physical problems or applications which gave birth to their (the methods’) development is therefore meaningless and shallow. On this point alone, Kline’s approach is a first rate introduction to calculus. On the other hand, Kline’s text despite being quite thick never progresses beyond these intuitive functions; instead Kline spends a lot of space discussing elementary topics like analytic geometry and application in economics or examples from Newton’ Principia. Very interesting of course, but one cannot find a rigorous discussion of functions or imaginary number or convergence of infinite series, all of which are essential basics for those who would progress to higher courses in analysis. An important question that it raises is: to whom this text is actually for?
My answer is that Kline’s text works best for either those whose high-school math backgrounds is in a pretty bad shape but want to know calculus or for those who have learned calculus a long time ago and are now in a “serious need” for some brush-up. Also for certain high school students who like physics but have never been at home with math, this book is really the missing keystone. There are many other good texts out there like Spivak’s or Apostol’s or Hardy’s Pure Mathematics. These are ideal for college students whose high-school math is still fresh and strong and thus are more able to appreciate deeper/advanced topics like the foundation of number system or analytical treatment of functions. For these students (especially pure-math B.S. students), calculus texts that gear toward analysis (i.e. more rigorous) would better prepare them for future challenges. However, for people like myself, whose math education ended 14 year-ago in high school and who barely remember the cosine rule, working through texts like Spivak’s or Hardy’s simply lead to a bogged down. It should be remembered that Kline’s calculus, first written in 1960s, was introduced during the time when most students were not exposed to calculus in high school. Thus, it was quite a problem when they had to encounter calculus for the first time in college. Back then it would be quite a blunder to demand that kids have to learn both techniques and rigorous foundations of calculus, the first truly “higher math” they ever encountered, at the same time. I think Kline’s text was written especially to remedy that problem. However, as most kids of our time are all exposed to fair amount of “intuitive” calculus in their school years, it may not make much sense to require to learn intuitively again in college. Still, I would maintain that even good students would profit much from at least taking a look at Morris Kline’s text, for it develops the subject in a strong historical context and is quite broad in the materials covered.
All things discussed, this book is a truly 5-star treatment of calculus. Given the state of education and teaching in our times, no one might ever write like this again.
[Note: PDF file of solution manual (about 260 pages) can be conveniently obtained by writing to Dover Publication. Great job! Dover, for making this book available and affordable at the same time.]
Customer11 –
I purchased this book because I needed to refresh my knowledge of differential calculus, which I studied 10 years ago. After reading the first few chapters I know that I made a right choice to get this masterfully written book, which makes you think, easily grasp complex concepts as if they were very simple, and, more importantly, become fascinated with the beautiful and coherent world of mathematical symbolism. Unlike many other books in this field it is so well expounded that you will easily read through the pages without being stuck on any single section. I’m also quite enjoying being reintroduced to some forgotten concepts. A lot of value for the symbolic price!
Marc –
When my daughter was in high school and first learning Calculus I told her it really is tough if you are not applying it. And this book really applies it.
This is probably the single best Calculus book out there. This book with the Calculus Lifesaver are a great combination for the student.
What makes this book intuitive is the structure in which he lays out the material. Most Calculus texts start with the basics of functions, limits, then derivatives..and at the end you do integration. Kline takes it from the bottom up, and starts with the basics of simple derivatives and integration building a foundation. Then he builds on the foudation; chapter by chapter.
The only real issue I have is the mixing of notation from page to page. I understand the point is to get you comfortable with all of the various notations, but it really breaks up the flow of the text and the problems.
Overall this is a timeless text and if you only had one book to buy, this is it.
PersonReviewing –
This is a very good textbook in my opinion. It covers the calculus with an intuitive treatment which is sometimes easier as a first course for scientist and engineers as it is not full of the rigorous profs that are taught in classes for Mathematics majors. I like both approaches and I use this book along with the very rigorous “A Course in Pure Mathematics”.
This book, “Calculus an Intuitive and Physical Approach” is a very nice introduction to the calculus needed to start learning physics, but it doesn’t cover vector analysis or complex analysis which are needed in more advanced physics, but that’s not a problem as this book covers differentiation and integration well which are the prerequisites for basic physics.
I really like this book.
haim ben-bassat –
i am very famillyar with calculus.klein approch is very interesting.i find the explanation very clear.the examples illuminate
the subject introduce in each chapter.
Mathematical Customer –
Thank you very much for the book.I love mathematics and especially Calculus.While Calculus sums can be sometimes laborious, they make one think.For example, a circle of infinite radius is a straight line ….and…a straight line drawn on earth is a circle…and Calculus acts as a go between for physics and mathematics to fit into a single concept that does not defy gravity.I give this 5 stars.
tkc –
Very thorough and extremely well worded and as an added bonus I think it’s typeset very elegantly with beautiful diagrams. This is unquestionably the best book on Calculus I’ve ever come across, Morris Kline was a master teacher. I’d pay $200 for it and yet it’s only $16.
André Gargoura –
Kline’s worldwide and combined reputation as a mathematician, a teacher, a writer is proved again !
This book is the absolute prelude to Apostol’s trilogy of analysis (or Goursat’s “A course in Mathematical Analysis”), see my other reviews…
It’s a thrilling and unique blending of intuition and rigour, in a rarely attained clarity of exposition, i.e. an irreplaceable book in your mathematical collection.
Richard H –
Very good book to understand the basics behind calculus and its multitude applications.
Graphics were good for the time but not for today’standards. “dry” graphics presentation.
Very detailed progressive presentation to give you time to assimilate Calculus.
Cheapest book on the market and at this price it could be a very good complementary book for a detailed progressive learning experience.
Kinetic –
This is a fantastic book. Alhough I received an “A” in my college calculus class, I did not feel that I learned the material as well as I should have. It was more of a “memorize the stuff to get a good grade” instead of “learn the material”. So this summer, I decided to study calculus as a whole, both to master the concepts of last semester and to prepare myself for Calc II in the fall. This book is fantastic and challenging. I love it. I cannot stress enough the benefits of learning something intuitively versus memorizing formulas. Get this book. I paid only 15 dollars because I bought it used, and it was WAY BETTER than my textbook.
L.S. Chang –
Jack –
I have only been back into math for the last year after taking a 5 year break. Long story short, I had to build up from prealgebra (PEMDAS!) and now I am halfway through a brutal semester of calc. I purchased a few other calc help books but they are either too watered down or overly wordy. This book strikes a very nice balance that text books don’t achieve any more. Plenty of pictures, down to earth talk, very detailed explanations, and it even shows you how to derive the equations and the thought process. Used with wolfram alpha, these are a very good combination of tools
Edit: 1 Feb 2013
I am now in Calculus for Engineers which is considered to be a combination of Calc 2 and 3 in one semester (with a little bit cut out). I am still using this book in conjunction with my schools textbook and I often find myself referring back to this one when learning a new concept. I prefer to use the school textbook for problems to work through as they will most likely be similar to the one on the schools exams. I am averaging a 95% in the course so far and considering I had to build from the basics, I think I am on to something. If you are serious about learning Calc, this book is only $20 and college text books range $100-200 easy.
Noah –
Note: my 4 star review does not reflect any faults with the book.
I got this book because I sort of skipped calc II in college and wasn’t sure of my background. While reading the foreword I noticed the author explained that his goal was to give intuitive mathematical arguments which were not rigorous proofs. This is actually a good approach for someone with less of a mathematical background, but I was looking for a more rigorous approach. With that said, I still read the book, but often I skip over the sections I’m not interested in. The intuitive explanations are very good.
I’d recommend this book for anyone who didn’t have a strong foundation in math or who is learning about calculus for the first time.
John –
This is a superb text from which to learn calculus. It is ideal for self study. It reads as if one were sitting in a lecture given by a talented & friendly professor. It is well described as an intuitive & physical approach with very detailed & understandable explanations of new concepts. There is an emphasis on geometrical explanations over formal proofs. One should be able to more or less painlessly obtain proficiency in 1st year calculus from this text. It aslo has brief chapters on partial derrivatives & differential equations. This is one of the few calculus books that makes the American Association of Mathematics recommend list. Apostol & Currant are also on the list.
Adiya –
Though I have had a few courses in calculus in my undergrad degree in computer science, but the focus was always on the rigorous proof. So lately, I when I was browsing through calculus book to recommend to my niece, I came across this and was quite intrigued by its approach. Bought one for her and liked it so much that i bought one for myself as well. Absolutely must for any one who is more or less starting in calculus or even for someone like me who did calculus a long time back under duress and didn’t derive any joy whatsoever, but perhaps now can afford to take it slow and appreciate the nuances.
John –
I used this book as a review for classes I took about 16 years ago. This was a good refresher for the first year of calculus I took. It seems to lack information on vector calculus, though.
Adri –
This is by far the best book on math that I have ever come across. I must say that it is a dense read, but it contained concepts that my professors at a very prestigious university failed to include in their instruction. The link between math and physics has officially been made for me.
Davy Crockett –
This book broadens my view on Calculus. I am able to solve the problems calculus has to offer by working on those problems and noticing the structure of them, but to ‘understand’ them makes it all the more satisfying.
Mathematical Customer –
Great Book……..Kline Does His “Normal” Fantastic Job and Makes it Seem Simple…….Will Continue to Buy His Books…..A+A+A+A+A+A+A+A+A+A+
Khaled Ahmed Farouk –
I enjoye reading calculus from this book
KI –
So I’m on third chapter after downloading about two days ago. This is an amazing calc book to say the least. The things you just read in other math books are explained in plain English and in a language that is understandable.
Michael Moore –
I’m a freshman collage student who has returned to school after nearly 20 years. This books have made all the difference in my physics and calculus classes.
Eduardo –
Very clear explanations on the concepts that make calculus so important. I very much wish that I had bought this when I started calculus 1.
Whetstone Guy –
Morris Kline was a living legend and his text is proof of it. Beware–calculus is very difficult to teach oneself without outside help such as a school or a tutor.
James Herrmann –
It met my needs and in a way I can understand
COL James D. Bass, Ph.D. –
Another great book that should be used by more than just “Honors” students. Time to lecture folks – not just solve equations for student to put in their notes!
Dan C. –
Very well written and informative calculus book. This book gives more insight into the real world applications for calculus.
user_3419043 –
The difference between this older book and the newer texts is mostly in the way it explains where the concepts originated as it goes through the standard litany of calculus. For the type of person who learns more holistically (rather than by rule sets, as most things are taught these days), this is a huge improvement. This is a good secondary source to reinforce and expand on the concepts introduced by the instructor and text in your university calculus series.
Jose O. N. Souza –
The best course so far in calculus! And it comes with the student’s manual (pdf).
Ecology665 –
I’m giving this a 5 for preliminary review of the book. There are lots of examples and problems to work on in every chapter. I have yet to sit down and read the book. I have looked at the first 5 chapters and this book is indeed a helpful book. They explain a lot of formulas and problems both visually and literally. I am reading ‘Calculus Made Easy’ by Thompson & Gardner first. When I start working on this I will give a more indepth summation.
Ivan –
Exactly what I asked for.
Jay G. –
Helped me out in the beginning of DE. I only wish I knew about this book when I was in Calc 1-3.
Seong Kook Shin –
Having inferiority complexity, I always want to understand lots of mathematics better, especially on Calculus. I bought several books on this topic, but because of short of my free time to study, I couldn’t advanced in my calculus skill. In my country, basic calculus is for high-school students. It has some merits compared to other countries whose students learn the calculus on their college life, but has a serious defect; making the student lose the interest to mathematics due to the scoring policy. Thus, I know how to solve/simplify complex equation, how to solve differentiation, and so on, but never fully understood the meaning of them.
Reading this book (I admit, that I just finished to read Chapter 7), brought me the enlightenment of the differentiation and integration, especially the concept of calculus, not just how to solve them. I know other great books on calculus especially for the students. However, being a person who has limited time to study, this book shines above all. (Even for non-English speakers like me.)
There is one minor defect I think, like many mathematics books, it refers to other section and figures in a different pages a lot. Thus it prevent me from reading smoothly sometimes. Most of equations are rendered as images with reasonable size and resolution, so it does not cause problems, although they look somewhat ugly.
David Rodgers –
A good introduction to serious treatment of calculus for poets. Not a fluff text
but very approachable.
JD –
Initial review oif book looks good. However, I have not had time to go over cntent in detail.
Max K –
Very accessible. So far I prefer it very much to my other calculus text. Note that the solutions manual is not included in this tome but is available for download for free from Dover Publications. Googling “Calculus: An Intuitive and Physical Approach Solutions Manual” should do the trick.
ignacio –
Concept explained very well and a lot of exercises to practice.
Mathematical Customer –
good
Johannes H. Benink Jr. –
I took one year of calculus and didn’t grasp all the little details. This book opened my eyes, and it’s delightful to understand the long forgotten and misunderstood principles at age 66.
Martin –
A wonderful textbook for only about 20 dollars. What many people do not know about is the free to access online solutions manual that Kline published for this book specifically. Please look at it to increase your knowledge of calculus.
Kasra –
One of the best math books I have studied; thorough, engaging, and concise.
SR –
Haven’t been through it yet but excited
Nick L –
Dover is a really nice company and if you have any questions just email a representative. I had an ishue finding the answer key and got a responce from them within like 24 hours. BTW the book is good to! LOVE math LIVE math!
JF –
An excellent option for a first course in Calculus.
SaiyaJin95 –
The author does a fantastic job of leading you through the concepts. If more textbooks were written like this, then I would imagine that comprehension and interest in the topic would improve. Thanks Mr. Kline!
Jarvis –
Very comprehensive explanations and problems, the writing voice is like that of a cool professor. My Calculus class is EXTREMELY difficult and usually only the smartest of the smartest get barely an A-, so this book really helps even out the playing field. I was pleasantly surprised to see the way in which Kline breaks down even the most complex theorems and problems to simple steps. He doesn’t just show you the answer or formula, but explains how you get there and derives each definition/theorem/formula in baby steps. I came for the hard calculus problems and left with a deeper understanding and appreciation of Calculus and Physics.
Giezer55 –
I’m the only person I know that reads calculus books for fun. This book is very well written and goes into detailed explanations. It also shows lots of examples, mainly physics problems, to illustrate the application of calculus. I highly recommend this book. It is very readable.
Jorge Vega –
I started reading this book around February and I was done by the middle of July.
It certainly got me engaged in the subject to a point that, the highlight of my week was doing Calculus
I began with doing some sparse notes on the first chapters and progressively I improved and structured
my notes to even doing the graphs and summarizing the concepts on my on words with a full sequence
of the derivations and equations to solve the exercises
I did it as a way to explain to my self the concepts the author was sharing since I am doing it on my own.
and on my own pace, with no regard or consideration about tests or examinations. It was purely for myself.
To my surprise I discovered that I was enjoying this book. very much.
It was very satisfying to share on the vision, on the thoughts and on the conclusions from the author
and math took a new dimension, a dimension beyond the symbols and the equations, It became alive
as a language that was relating to me a story much clear and structured of the things around me, an of
the different phenomena and of motion and of time and of constants and initial conditions, of relations
and of rates of change, and of limits and of most of all seeing how could I interpret the equations and how to relate
theorems and concepts,
It was talking of intuition and of rigor on math, of geometric constructs and derivations, and of how could I play
with the equations and discern the meaning of the results, and how can I go from generalities and simple conditions
on to more complex problems in a progressive escalation on the restrictions, and elements at play, around a set
of conditions relating to a problem.
I could picture Fermat working in his light theory of least time, hundred years earlier and admired his accomplishment
beautiful in his derivation, Kepler in his wondering of celestial motions of planets, and of course the Newtonian achievement
of Calculus from the secant to the tangent the vision of a limit concept expressing instantaneous change and the crowning concept
of anti differentiation from a sum again approaching a limit expressed by the integral it was especially satisfying to reach to
triple integration and see how the geometry scaffold and the calculus make such great team on illuminating over the varied
physical phenomena that different sciences work on.
Now am eager to continue the discoveries on some other books on Mathematics, on Physics, on Electricity and Magnetism
on Chemistry and even my goal Quantum Mechanics, It seems like this book helped me to awaken in me
the interest on observing and analyzing a set of conditions and correlate them to mathematical terms
having the confidence to try, to err and to try again, to play with the math, to have fun with the equations and to challenge
the results, the approach, the concepts.
After all mathematical constructs are just that, an expression of our perceived reality, and of our resources
and of our limitations, It is a mind endeavor, even a mind game, if you will that comes alive in the equations and through the
effort and persistence of so many minds bent on pushing ahead the boundaries of math and science..
It certainly was fun doing this Calculus book
Thanks Mr Kline
where ever you are Thank you very much.
Garegin Ghukasyan –
High quality and delivered on time.
Jeff –
Absolutely the best book I’ve ever read on the subject!
e –
As a physicist and the owner of 10+ calculus texts I can honestly say that this is the best calculus text available today. The order of presentation is slightly different from the current texts but I believe makes the content much easier to learn.
The author gives great explanations along with detailed examples.
Rudy Ray Bowman –
Mathematical Customer –
As an engineering student i found this, if not the best, but perhaps one of the greatest books out there. The book is written in a way that you won’t get lost or confused as the subjects get more advanced; specially for a beginner or someone with little background in calculus (like me). Currently finished chapter 13, can’t wait to finish off the entire book it in the coming month. Highly recommended book.
Kathleen S. –
Best Calc Book around.
Howard L. Pearce –
I am getting too old for theory,
so, this was a good reminder of the basics.
Thanks.
lellise shewakena –
Really helpful, It elaborates the material perfectly
Dr. Terrence McGarty –
I got this for my grandson who is off to University and studying Civil Engineering. It is a classic, very reasonably priced, and presents calculus in a classic manner. Frankly why one would spend $250 for a calculus book weighing fifteen pounds and containing more pictures than the Louvre I do not know. This is a substantial book but it is superbly organized and covers calculus in a manner that meets the needs of the engineer, scientist and perhaps the budding mathematician. It is an applied approach however. The organization is perfect, introducing one element logically atop the other.
I would recommend that any engineering or science major have a copy of this to see how “it is really done” after the spend the large sums demanded by someone, often not the Professor, unless they are the author as well!
Mathematical Customer –
Very good book. It contains calc I, some II, and some calc III (not exactly sure). If u want to learn it by yourself buy this textbook. I am a sophomore in hs and can understand this book (this doesn’t mean that the material is simplified at all). It’s fantastic
Benjamin –
As a mathematics/physics student, I bought this book with hopes of finding a digestible text to read specifically when I’m dedicating a lot of time towards non-math related courses. I think the author has done a wonderful job in create just that– a text that is clear and concise, with a good amount of application/examples, easy to grasp proofs. Having taken a few different Calculus courses and using a couple different texts this book is by far the most enjoyable read I’ve found yet.
jared jeffries –
Best explanation of any calculus book ive ever seen
Sally –
Everything my husband expected and more.
Arielle Wolter –
Though old this book is still more than relevant. If you want to understand calculus and not just the equations this book may help you. I bought it for calculus 1 in college and it was a wonderful reference resource for discussions and assignments.
Lifetime Learner –
The author not only has a thorough knowledge on this topic but knows the philosophy behind important formula.
I bought this book to help my child with her calculus course, and it quickly refreshed my memories on this topic that I learned many decades ago.
“the reviewer” –
good text – but I lack concentration and don’t get much utility out of it
Aminium –
I love this book! If anyone wants to learn about calculus and broaden your scope of it. Get this book!
Katie Buckley –
Tremendous read for calculus using a physical approach. Tremendous supplement to a rigorous text on the subject.
larry ho –
Item as described.
Akmsailor –
Really was an intuitive approach!
Somehumanbeing –
This seems like a reprint. That’s probably how the Dover’s series does things, but it’s perfect.
The content itself is very clear and easy to read.
Karie Gauen –
This textbook is the most clear and beautiful textbook I have ever read in any subject. I’m not sure why anyone would ever attempt to improve upon it.
Dr. Bryan Nolan –
This book should be used in conjunction to a textbook on calculus. It helps to clarify some formulas that are not explained well in the standard texts.
It contains a fair amount of useful information for a small size book and the price is fair
It is entertaining to read this book and it gradually become a favorite book on your shelf
Antonio M. –
Much more intuitively presented—as the title suggests.
Neal W –
A classic. Haven’t found a printed error yet, unlike too many unedited mathematics books today. Take a fun ride with Morris Kline, educator and mathematician (a Rare combination) with an !! Obtuse?? Wit.
Crystal –
My son is in 8th grade and super smart. He loves math and carries this book everywhere he goes.
John Moore –
This book contains all the topics typically covered in three semesters of calculus and in that sense is similar to any number of other good books I’ve seen on the subject. The thing that makes this book unique is the intuitive approach (through geometry) taken to explain topics that would otherwise just be abstract mathematics. If you are struggling with understanding what a derivative or integral of a function really means, or especially functions of multiple variables and partial derivatives, this book may be of use to you. Highly recommended.