Would I have to read a standard textbook in addition — i.e. Stewart, etc. — or would Kline's Calculus: an Intuitive and Physical Approach be sufficient? My interest is in applications: dynamical systems theory and physics in general. The book states that is is not rigorous, except for the last chapter.
Is Morris Kline's 'Calculus: An Intuitive and Physical Approach' a Good Book to Learn Calculus From?
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2 Answers
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Kline's book is not rigorous in the sense that it avoids the formal definitions and proofs that were developed so that calculus could be put on a "firm footing," at least until the last chapter. However, based on my experience, most calculus courses that use "standard" textbooks also don't emphasize these formalisms/don't do a good job of teaching them. As Tegh said, you would only be expected to know the "rigorous stuff" after learning real analysis, and many scientists never take a real analysis course.
See also this Reddit comment about the book.
Kline's book covers single-variable calculus pretty thoroughly. It does cover a decent chunk of multivariable calculus, but you would probably still want to supplement this with another text. For example, Kline doesn't cover vector calculus, which is important in many science fields.
After Kline's book, if you want to jump straight into dynamical systems, Steven Strogatz's Nonlinear Dynamics and Chaos is aimed at students that have completed a first course in calculus. Video lectures are available.
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2$\begingroup$ I found that Reddit comment interesting for how informed the writer appeared to be. And it reminded me of a couple of things @user2676187 might find useful -- my comments about "Calculus in Context" and this list of honors level calculus texts (also, the former lists several possibly useful "non-rigorous" calculus books and the latter lists several possibly useful "highly rigorous" calculus books). $\endgroup$ Commented Apr 17, 2023 at 17:15
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To rigorously learn calculus you would have to study real analysis. To study real analysis you would have to study some a little bit Discrete Math. (I recommend Suzzane E. P)
However for your case that is not necessary. I would recommend you study logic, set theory and proofs from discrete maths so you have fundamental knowledge of mathematics.
What I would suggest is study Stewart Calculus. With Stewart Calculus you have fully available solution online on Quizlet.
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3$\begingroup$ Another equally good option is Advanced Engineering Math by Dennis G Zill -- Zill's book is something one might study AFTER Kline's calculus or AFTER Stewart's calculus (or AFTER any of the hundreds of standard single/multi-variable calculus texts published since around 1960), not something one might study INSTEAD of Kline's calculus or INSTEAD Stewart's calculus. Also, in choosing a calculus book the OP will want to make sure it contains a lot of applications (center of mass, moments of inertia, etc.) and not worry excessively about how complete the text is (continued) $\endgroup$ Commented Apr 16, 2023 at 7:42
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2$\begingroup$ with respect to Spivak-calculus pure/rigor stuff. Supplement the multi-variable stuff with Shey's Div, Grad, Curl, and All That, then turn to a book like Zill's (e.g. Mathematical Methods in the Physical Sciences by Boas, among many other possibilities). It would also be helpful to have available one or more books devoted to vector analysis for supplementary reading/reference, such as: when at the (continued) $\endgroup$ Commented Apr 16, 2023 at 7:42
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2$\begingroup$ elementary multi-variable calculus level something like Schuster's Elementary Vector Geometry (beginning elementary muti-variable calculus level) and Davis's Introduction to Vector Analysis (middle to end of beginning multi-variable calculus level); and when at the Zill/Boas level, something like Spiegel's Vector Analysis and an Introduction to Tensor Analysis and Wrede's Introduction to Vector and Tensor Analysis. $\endgroup$ Commented Apr 16, 2023 at 7:43
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1$\begingroup$ @user2676187: I forgot you mentioned dynamical systems theory. After elementary calculus, for this you'll need less focus on traditional vector analysis and need more focus on ODE's and real analysis. But even here most any standard elementary calculus text (Stewart, Thomas/Finney, etc.) is fine -- even Kline would be fine, since afterwards (if going strongly into dynamical systems stuff) you'd likely take up advanced calculus or real analysis. $\endgroup$ Commented Apr 16, 2023 at 7:58
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$\begingroup$ @Dave you are correct about Zill textbook I made mistake. Will remove it. Thank you $\endgroup$– TeghCommented Apr 16, 2023 at 12:02