Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics)

***On the Book I already have taken basic calculus courses not that long ago, so I used this book more for reviewing and "honing" purposes, and I couldn't be more pleased with it. The author details many derivations that were skipped in my calculus courses, for example the derivatives and integrals of inverse trig functions, the formulas of which us clueless students were forced to memorize. After reading this book, never again would I have to futilely batter my brain cells in the frantic search for a formula I was not able to retain - this is the best quality about the book. Ordinarily, students would be taught math (or any other subject) in a way that would induce them to score well on tests. This technique is shallow and often cuts corners, as some steps are left out for fear that it may confuse the test-taker, who inevitably takes his test unthinkingly. Kline's [the author] teaching philosophy is different. He sets out everything in detail and with clarity, so that the student may himself solve problems without resorting to formulas or hoping that test problems would not differ from the drills he has solved. Furthermore, the practice problems in this book are engaging, and Kline has take great pains to show how every mathematical technique can be used in real life. Some of his practice problems also asks that the student prove some thing or other (it is not all application), and this allows the principle to further sink in the brain. A solutions manual is not included in the book, but may be requested, for free, from Dover. The details of how to do this is described in one of the front pages where the copyright information is found. The solutions manual could also be found in Dover's website. It is in pdf form and is something like 250 pages. The solutions in the solution's manual do not merely list the correct answers. It also describes how to get to that answer. There are some typos, so there may be moments when no answer of yours would be correct now matter how and how many times you get at it. I do not believe the typos are anything major, since it can be obvious when something is a typo. Besides these atypical moments, the solutions manual could be regarded as having the infallibility of God, and with inducing the same soul-enriching effects. Some people may complain about the wordiness of this book, but the "wordiness," in my opinion, aids in the understanding: it is a blow-by-blow description of the proofs and model problems. It could be skipped; I mostly don't. I highly recommend this book. ***On Amazon: I have already bought a book from Amazon before, purportedly new, but when I received it, its spine was wrinkled, a fabrication error I suppose. It was also evident from a number of other minor damages that Amazon does not take good care of the books in its store. The book was far worse-looking than many of the used books I bought from outside sellers. Now, I bought another book from Amazon, this one, and it was no surprise that it was worn-looking and evidently manhandled by whoever was in charge of Amazon storage. I was willing to overlook that, what I can't abide by is the stains I found on pages 347-351, and goodness knows how many more stains and surprises lie in wait. Is Amazon trying to pass off used books as new? I am sick of how Amazon handles its products.

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.]

I felt obligated to write this review, because as a good but rusty math student, I would like to help others find the right book according their own needs. First, back in my day, I studied 4 semesters of Calculus in college and considered myself to be good at math at the time - but that was 30 years ago. After my high school daughter came to me asking for help with deriving functions, I was in search of a primer that would be more comprehensible to me than her school text book. I ordered this book and was sadly disappointed at first. It is long, with small print and almost philosophical about math. I found it to be no better than her class text book when it came to reminding me of the rules for deriving functions like the product, quotient and chain rule. I ended up ordering a book with the word "dummies" in the name though I won't say exactly which one and that suited my purpose immediately. In 30 minutes it all came back to me and I was able to sit and help her understand the basic concepts at a high school level and work through homework problems. Now back to this book - the next day I picked up this book again and began reading it without the pressure of needing a quick, to the point answer or lesson. I began to read it, not as a text book, but more like an academic novel about math and that is when things started to click. It gave me a new appreciation for why calculus is important, how the derivative is not just the slope of the tangent, but the direction of a line at any given moment and why that is important to motion, targeting moving objects (military applications) and light striking lenses (optics). Ah, that is very cool stuff to wrap ones head around after many years of work that required little more than a 5th grade education in my corporate IT job! Given my new appreciation of this book, I will be reading more as I hope to understand concepts and drivers behind more advanced math concepts - both as a reminder of what I forgot and for new enlightenment. I don't expect to use this book to advance my own problem solving capabilities - although it may help in that respect too - but rather to gain a more enlightened appreciation for Calculus. The book includes exercises and problems along the way as well as answers for about half of them - an excellent complement to help one confirm they're grasping the material. In summary, I would not recommend this book if you're new to Calculus or trying to get through a high school or college level course - but if you're a advanced math major or even an old guy who wants to get a better appreciation - almost an academic, theoretical or philosophical appreciation - for advanced mathematics then this is the book for you. Just don't use it as a primer to help your impatient high schoolers and who need quick answers and bulleted minimalism.

I first took calculus over 20 years ago and always did well in math, so I wanted a text that I could keep as a refresher (for studying more advanced statistics) and reference for things I may have forgotten. So my experience with the book may differ from someone who has never been exposed to calculus before. In my opinion, this book is an excellent introductory text for single variable calculus. While it is a bit dated (it's a reprint from 1977), given its low cost, compactness, and how comprehensive it is, I think its value is unbeatable (especially compared to current university calculus texts). While there *are* some drawbacks/negative aspects, I think they're pretty minor but they should be pointed out either way: 1) Typos -- there are a non-insignificant number of typos throughout. Some of these are trivial to spot, while a few were in the answers to select exercises. I spent time scratching my head trying to match the printed answer when it turns out there was a typo. This might be a bigger concern for absolute beginners. 2) Graphics -- the figures are clearly very dated and don't have captions. However, this isn't really a problem until the later chapters on functions of more than one variable, and even then I thought they were clear enough. 3) Content -- I found the book a bit verbose in some places, but this may be more appreciated by a beginner. Also, if you will never take introductory physics, you might struggle with or be frustrated by how many of the examples and exercises are specific to physics problems (even though this is, historically, how calculus came about). 4) References -- since the book is so long, it would have been nice if the author included page numbers when referring back to a particular chapter/section/equation. He sometimes does this, but it isn't a big issue. Overall, I don't think anyone who will need to use calculus would be disappointed in having this book around. There are chapters and sections marked as optional, a comprehensive list of integrals of different forms at the back, and the author explains concepts intuitively. For the most important theorems and ideas, I use post-it tabs for quick reference.

As a seventeen-year-old sitting in physics class, I remember staring at a board full of kinematic equations. The goal was to memorize the formulas and learn how to apply them. But when test day came around, I had forgotten all of them. Memorization has never worked well for me. What stuck with me wasn’t the formulas, but the moments when I actually understood why things worked. That’s why I really appreciated how this book handled similar topics. Unlike my high school physics class, Kline doesn't just present formulas—he builds them from first principles, showing you how they come to be. It’s not about memorizing a cookbook full of calculus recipes; it’s about really getting it. This book shines when it comes to differentiation and integration. Kline takes you on a journey through the ideas, not just the mechanics. He even gives you a taste of partial derivatives and vector calculus. You can think of it as covering traditional Calculus I and II, but with an intuitive, big-picture focus. If you’re like me—someone who learns by understanding rather than memorizing—this book might be exactly what you’re looking for. It skips some of the formalism, but in exchange, it gives you a lasting, intuitive grasp of calculus. Don’t expect Spivak level formalism though. It may cut corners to convey intuition.

The good thing about texts relating to mathematics is that it is hard for them to become out of date. This book was published in the 1960s, but I find it to be perfectly able at teaching the calculus. In one of my classes, I even saw another fellow with this very same book. A full solution manual may be easily obtained by emailing Dover Publications.

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.

I'm sad to say I didn't enjoy this book much. Before critiquing; I have plenty of experience with calculus before (~10yrs ago tho) when I went to college so if you're a person that wants to know about calculus for the first time through this book, my review won't help you. That aside; I'll mention the pros first: a) The content is _very_ easy to read. To give you an idea how easy it is; I was able to blaze through > 100 pages in a day and I solved multiple exercises throughout (again, remember, I wasn't a noob in calculus so I'm technically cheating). I'm impressed how easily the author can convey these ideas with simple examples that don't involve horrendous algebraic manipulations and overly complex physical examples. b) It also uses a bit of a historical approach that I usually find useful to build intuition, since generally that's how humanity makes mathematics progress. c) The book is self contained and very complete. You only need high-school math to read this and most of the missing things you may encounter are covered either on the spot, on an appendix or somewhere else inside the book. The bad parts: a) The examples and ideas on the book are very simple and easy so I feel you don't get challenged enough to think about what you're doing. Most of the examples and discussion are around polynomial functions, especially on the first chapters which overly simplify things IMHO. b) The notation is dated; there's a lot of use of deltas (to loosely represent infinitesimals which I find confusing and less rigorous) both in limits and definitions of derivatives and integrals. b) The book uses the fundamental theory of calculus VERY EARLY which I think is a mistake. The definition of integral in the book is the antiderivative of a function. I was completely thrown off since that's not the right way of introducing integration I think. Much later on you do see the fundamental theorem and also that the integral means a summation. c) All of the physics examples (classical mechanics) in the first few chapters use very simplified assumptions that are seldom representative of approaching a real classical mechanics problem. d) Huge lack of mathematical rigor. I might be biased since my first calculus course was based on Spivak's book which is completely on the other end but most of the concepts around continuity, proofs and other important properties of real numbers and limits were incredibly grossed over to a point I felt I was taking things based on dogma rather than fact. This was the most frustrating part for me. Conclusion: If you want to learn calculus just to apply it, get the typical popular textbooks (Stewart, Edwards, etc), the notation is much more modern and the progression is more in tune with the contemporary pedagogics of calculus. If you want to learn calculus like a mathematician, get the Spivak or the Courant, these are fantastic and have challenging problems and rigorous proofs of everything under the sun.