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Calculus: An Intuitive and Physical Approach
Application-oriented introduction relates the subject as closely as possible to science with explorations of the derivative; differentiation and integration of the powers of "x"; theorems on differentiation, antidifferentiation; the chain rule; trigonometric functions; more. Examples. 1967 edition.
962 pages, Kindle Edition
First published January 1, 1967
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About the author
Morris Kline
78 books103 followersMorris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.
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Displaying 1 - 12 of 12 reviews
October 21, 2019
Morris Kline’s book, Mathematics for the Nonmathematician, is my favorite book on the discipline. Kline showed an amazing ability to explain mathematical concepts intuitively, and to situate them within a sensible human context. In his hands, math was not simply a series of equations or deductive proofs, but an integral aspect of our civilization: a crucial tool in our species’ attempt to understand and manipulate the world. The book changed my view of the subject.
So when I found that Kline had written a book on the calculus, I knew that I had to read it. Calculus represents the furthest I have ever gone with mathematics in my formal schooling. By the time that I graduated high school, I was a problem-solving machine—with so many rules of algebra, trigonometry, derivation, and integration memorized that I could breeze through simple exercises. Yet this was a merely mechanical understanding. I was like a well-trained dog, obeying orders without comprehension; and this was apparent whenever I had to do any problems that required deeper thinking.
In time I lost even this, leaving me feeling like any ordinary mathematical ignoramus. My remedial education has been slow and painful. This was my primary object in reading this book: to revive whatever atrophied mathematical skills lay dormant, and to at least recover the level of ability I had in high school. Kline’s text was perfect for this purpose. His educational philosophy suits me. Rather than explain the calculus using formal proofs, he first tries to shape the student’s intuition. He does this through a variety of examples, informal arguments, and graphic representation, allowing the learner to get a “feel” for the math before attempting a rigorous definition.
He justifies his procedure in the introduction:
This rings true to my experience. In my first semester of university, when I thought that I was going to study chemistry, I took an introductory calculus course. It was divided into lectures with the professor and smaller “recitation” classes with a graduate student. In the lectures, the professor would inevitably take the class through long proofs, while the grad student would show us how to solve the problems in the recitatives. I inevitably found the professor’s proofs to be pointless, and soon decided to avoid them altogether, since they confused me rather than aided me. I got an A-minus in the class.
Though Kline forgoes the rigor one would expect in formal mathematics, this book is no breezy read. It is a proper textbook, designed to be used in a two-semester introductory course, complete with hundreds of exercises. And as fitting for such a purpose, this book is dry. Gone are the fascinating historical tidbits and gentle presentation of Kline’s book on popular mathematics. This book is meant for students of engineering and the sciences—students who need to know how to solve problems correctly, or planes will crash and buildings will collapse. But Kline is an excellent teacher in this context, too, and explains each concept clearly and concisely. It was often surprisingly easy to follow along.
The exercises are excellent as well, designed to progress in difficulty, and more importantly to encourage independent thinking. Rather than simply solving problems by rote, Kline encourages the student to apply the concepts creatively and in new contexts. Now, I admit that the sheer amount of exercises taxed my patience and interest. I wanted a refreshment, and Kline gave me a four-course meal. Still, I made sure to do at least a couple problems per section, to check whether I was actually understanding the basic idea. It helped immensely to have the solutions manual, which you can download from Dover’s website.
In the end, I am very glad to have read this book. Admittedly this tome did dominate my summer—as I plowed through its chapters for hours each day, trying to finish the book before the start of the next school year—and I undoubtedly tried to read it far too quickly. Yet even though I spent a huge portion of my time with this book scratching my head, getting questions wrong, it did help to restore a sense of intellectual confidence. Now I know for sure that I am still at least as smart as I was at age 18.
And the subject, if often tedious, is fascinating. Learning any branch of mathematics can be intensely satisfying. Each area interlocks with and builds upon the other, forming a marvelous theoretical edifice. And in the case of the calculus, this abstract structure contains the tools needed to analyze the concrete world—and that is the beauty of math.
So when I found that Kline had written a book on the calculus, I knew that I had to read it. Calculus represents the furthest I have ever gone with mathematics in my formal schooling. By the time that I graduated high school, I was a problem-solving machine—with so many rules of algebra, trigonometry, derivation, and integration memorized that I could breeze through simple exercises. Yet this was a merely mechanical understanding. I was like a well-trained dog, obeying orders without comprehension; and this was apparent whenever I had to do any problems that required deeper thinking.
In time I lost even this, leaving me feeling like any ordinary mathematical ignoramus. My remedial education has been slow and painful. This was my primary object in reading this book: to revive whatever atrophied mathematical skills lay dormant, and to at least recover the level of ability I had in high school. Kline’s text was perfect for this purpose. His educational philosophy suits me. Rather than explain the calculus using formal proofs, he first tries to shape the student’s intuition. He does this through a variety of examples, informal arguments, and graphic representation, allowing the learner to get a “feel” for the math before attempting a rigorous definition.
He justifies his procedure in the introduction:
Rigor undoubtedly refines the intuition but does not supplant it. . . . Before one can appreciate a precise formulation of a concept or theorem, he must know what idea is being formulated and what exceptions or pitfalls the wording is trying to avoid. Hence he must be able to call upon a wealth of experience acquired before tackling the rigorous formulation.
This rings true to my experience. In my first semester of university, when I thought that I was going to study chemistry, I took an introductory calculus course. It was divided into lectures with the professor and smaller “recitation” classes with a graduate student. In the lectures, the professor would inevitably take the class through long proofs, while the grad student would show us how to solve the problems in the recitatives. I inevitably found the professor’s proofs to be pointless, and soon decided to avoid them altogether, since they confused me rather than aided me. I got an A-minus in the class.
Though Kline forgoes the rigor one would expect in formal mathematics, this book is no breezy read. It is a proper textbook, designed to be used in a two-semester introductory course, complete with hundreds of exercises. And as fitting for such a purpose, this book is dry. Gone are the fascinating historical tidbits and gentle presentation of Kline’s book on popular mathematics. This book is meant for students of engineering and the sciences—students who need to know how to solve problems correctly, or planes will crash and buildings will collapse. But Kline is an excellent teacher in this context, too, and explains each concept clearly and concisely. It was often surprisingly easy to follow along.
The exercises are excellent as well, designed to progress in difficulty, and more importantly to encourage independent thinking. Rather than simply solving problems by rote, Kline encourages the student to apply the concepts creatively and in new contexts. Now, I admit that the sheer amount of exercises taxed my patience and interest. I wanted a refreshment, and Kline gave me a four-course meal. Still, I made sure to do at least a couple problems per section, to check whether I was actually understanding the basic idea. It helped immensely to have the solutions manual, which you can download from Dover’s website.
In the end, I am very glad to have read this book. Admittedly this tome did dominate my summer—as I plowed through its chapters for hours each day, trying to finish the book before the start of the next school year—and I undoubtedly tried to read it far too quickly. Yet even though I spent a huge portion of my time with this book scratching my head, getting questions wrong, it did help to restore a sense of intellectual confidence. Now I know for sure that I am still at least as smart as I was at age 18.
And the subject, if often tedious, is fascinating. Learning any branch of mathematics can be intensely satisfying. Each area interlocks with and builds upon the other, forming a marvelous theoretical edifice. And in the case of the calculus, this abstract structure contains the tools needed to analyze the concrete world—and that is the beauty of math.
November 8, 2013
This book is great for students who want to really grasp the main ideas of calculus, it contains a great deal of technical vocabulary and concepts that gives the reader tremendous insight into problems that you may not be able to get in a classroom environment, but in order to fully understand advanced topics in this book, I would recommend learning how to do solve differential and integral problems ahead of time because this book will be telling you why you are doing something not just how to do it. If you need help in or just want to learn calculus then I strongly recommend this book for you because of its depth into all aspects of calculus, I would also suggest some of the other books Morris Cline as a reference for a wide variety of mathematical topics.
January 5, 2022
I must admit beforehand that this review may be biased as I consider Kline to be almost solely responsible for my career path and interest field in general. Kline's other book "Mathematics for the Nonmathematician" had an enormous influence on me and set me up on a lifelong journey to discover the nature of mathematics and the universe itself. A journey that I persistently follow to this day. Needless to say, I feel great respect for the author which might get in the way now as I try to objectively review his work. Nevertheless, I'll try to present my case as to why this book is the best calculus book out there.
The key aspect that differentiates this book from others is the approach Kline takes to teaching mathematics. Kline presupposes that mathematics is deeply rooted in the sense of intuition and it is from this side that any student should first approach this field. Rather than treating mathematics in an abstract form of symbol manipulation, Kline says that it is imperative to provide a meaning for those symbols first, to give a context for the equations to exist. Kline does all that by considering real-world phenomena: a falling object, a moving car, a swinging pendulum, etc. Now you might say that all calculus books have these kinds of examples and there's nothing special in that. But the difference here is that all those books consider physical reality afterwards, often leaving it as an exercise in theory application, while Kline places it front and centre. Looking from the historical perspective, this is exactly how mathematics and indeed calculus itself was developed - by trying to solve real-world problems. Rigour was added much later.
The benefits of this intuitive approach are twofold. First is the clarity and ease of understanding that the student gets by learning the subject with respect to something he can relate to. It's incredibly difficult (at least for me) to make sense of all these abstract quantities that mathematics is full of. But thankfully, it needn't be so as intuition when correctly developed can be just as good as a guide.
The second benefit of taking Kline's approach is that the book becomes actually readable. To understand this point, you just have to open any calculus textbook you'll find in any university recommended book list (Stewart's Calculus will do just fine) and attempt to read a page or two. If you're like me, the book will read like the driest technical manual devised for a single purpose only - to help students solve their calculus class problems and no more than that. We have the usual structure: a list of axioms, followed by a theorem, then its proof, then some abstract exercises and the pattern repeats. An intuitive understanding here is left for the student to develop as an additional exercise. You'll not find this in Kline's Calculus. He provides not only the reasoning behind the mathematical ideas but also their historical context and evolution which they underwent as they were being developed. For myself, being a history buff, it's immensely interesting.
Now, this book is not a conventional calculus textbook and not only because of all the reasons I've already listed. The range of topics covered in the book includes some topics of what these days is considered a part of algebra (plain geometry etc). But in my view, that's only an advantage as it's explained so masterfully well. Still, I'd say that the book is best suited for a person who's just starting to learn calculus and wants to do it the right way. Graduates might miss some more advanced topics like differential equations and Fourier series as they are not covered in the book, or are mentioned briefly.
In summary, I cannot praise Kline's Calculus highly enough. It is wholly unique in its approach and clarity. I cannot think of a better introduction to the ideas of calculus than what this book presents.
The key aspect that differentiates this book from others is the approach Kline takes to teaching mathematics. Kline presupposes that mathematics is deeply rooted in the sense of intuition and it is from this side that any student should first approach this field. Rather than treating mathematics in an abstract form of symbol manipulation, Kline says that it is imperative to provide a meaning for those symbols first, to give a context for the equations to exist. Kline does all that by considering real-world phenomena: a falling object, a moving car, a swinging pendulum, etc. Now you might say that all calculus books have these kinds of examples and there's nothing special in that. But the difference here is that all those books consider physical reality afterwards, often leaving it as an exercise in theory application, while Kline places it front and centre. Looking from the historical perspective, this is exactly how mathematics and indeed calculus itself was developed - by trying to solve real-world problems. Rigour was added much later.
The benefits of this intuitive approach are twofold. First is the clarity and ease of understanding that the student gets by learning the subject with respect to something he can relate to. It's incredibly difficult (at least for me) to make sense of all these abstract quantities that mathematics is full of. But thankfully, it needn't be so as intuition when correctly developed can be just as good as a guide.
The second benefit of taking Kline's approach is that the book becomes actually readable. To understand this point, you just have to open any calculus textbook you'll find in any university recommended book list (Stewart's Calculus will do just fine) and attempt to read a page or two. If you're like me, the book will read like the driest technical manual devised for a single purpose only - to help students solve their calculus class problems and no more than that. We have the usual structure: a list of axioms, followed by a theorem, then its proof, then some abstract exercises and the pattern repeats. An intuitive understanding here is left for the student to develop as an additional exercise. You'll not find this in Kline's Calculus. He provides not only the reasoning behind the mathematical ideas but also their historical context and evolution which they underwent as they were being developed. For myself, being a history buff, it's immensely interesting.
Now, this book is not a conventional calculus textbook and not only because of all the reasons I've already listed. The range of topics covered in the book includes some topics of what these days is considered a part of algebra (plain geometry etc). But in my view, that's only an advantage as it's explained so masterfully well. Still, I'd say that the book is best suited for a person who's just starting to learn calculus and wants to do it the right way. Graduates might miss some more advanced topics like differential equations and Fourier series as they are not covered in the book, or are mentioned briefly.
In summary, I cannot praise Kline's Calculus highly enough. It is wholly unique in its approach and clarity. I cannot think of a better introduction to the ideas of calculus than what this book presents.
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April 30, 2009It's interesting to contrast the approach of this book with that of Spivak's Calculus. Spivak takes a rigorous approach, claiming that this enhances intuition, rather than detracting from it. Kline says exactly the opposite: the rigor is a confusing distraction, and students must first learn concepts intuitively. Spivak says that the chapters in his book should be read in order, as the book is designed to show that Calculus is a coherent subject rather than a collection of techniques. Kline says you can read his chapters in whatever order you please (almost), because he thinks it's important for instructors to be able to teach just the techniques they think are appropriate.
Overall, I prefer Spivak's approach. But this book is good for practicing some elementary Calculus problems to prepare for the Math Subject GRE, which is why I bought it. And the price is right.
Oh, one thing: you can email Dover to get a PDF of the solutions manual, but it's filled with errors.
Overall, I prefer Spivak's approach. But this book is good for practicing some elementary Calculus problems to prepare for the Math Subject GRE, which is why I bought it. And the price is right.
Oh, one thing: you can email Dover to get a PDF of the solutions manual, but it's filled with errors.
March 27, 2025
This book was a fun exploration of calculus and it showed me more of the beauty of math.
May 10, 2017
Very good for learning, but I need more practice. Especially in Integration by Parts and Partial Differentiation.
I only got up to Calculus II in school, but I did enjoy this regardless.
Through a second reading, I can say that I really like the practical approach taken for this book. One chapter will introduce a concept and the next chapter will apply it to a physical situation. The text goes from the introduction to functions and limits to partial differentiation.
I only got up to Calculus II in school, but I did enjoy this regardless.
Through a second reading, I can say that I really like the practical approach taken for this book. One chapter will introduce a concept and the next chapter will apply it to a physical situation. The text goes from the introduction to functions and limits to partial differentiation.
February 22, 2013
Sometimes painfully repetitious, but it takes you to an intuitive understanding of calculus you may have forgotten over the years. My last calculus class was in 1965, hence the need to refresh.
Currently reading
April 22, 2011very readable, very accessible, an excellent refresher that deepens my understanding to boot.
January 19, 2019
a very good supplement for calculus course. It explains very well the logic of Calculus. Textbooks nowadays don't bother to do that anymore.
November 25, 2024
Excellent for a first exposition to calculus, but not for those who have never studied math before. I love the physical approach that doesn't sacrifice intuition. It teaches calculus the way it was developed, through physics! Great for a highschooler.
September 15, 2022
Outstanding textbook for anyone who wants to learn calculus. I've read through and studied most of the book for self-learning. I've found that I've understood the concepts and details of calculus much better compared to other common textbooks. Therefore, I recommend this book to any beginning (or more advanced) calculus student seeking a deep conceptual understanding of mathematics. I also appreciated Kline's physical approach in this book as well as including introductory chapters for advanced concepts (differential equations, and partial derivatives).
December 23, 2020
Just as the title implies, an intuitive approach to understanding Calculus. Covers the fundamentals of Single Variable Calculus (simple derivatives/differentiation and integrals) all the way to Differential Equations and Multivariable Calculus Polar/Spherical Coordinates.
Displaying 1 - 12 of 12 reviews