Spivak is not an easy book and is probably not a great way to learn calculus. It is, however, an excellent start to learning rigorous mathematical techniques and gives you great foothold for for later topics in math such as analysis. The calculus course I took used “Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach” by Hubbard and Hubbard—it is essentially a slightly more modern and easier to read version of Spivak. The people I took that course with all had superb starting points for their later math courses compared to others (warning: selection bias! those who took the course were more interested in math, etc.), but we often lamented the fact that we weren’t very good at actually doing much with multivariable calculus when it came to, say, physics courses. That’s more because the differential forms introduced in Spivak are not the standard way of writing calculus in other domains than because what Spivak is doing is inherently harder.
In short: Spivak is hard. Taking a long time on it is not surprising in the least. It’s probably not your most ‘efficient’ route to learning multivariable calculus, but it is a good way to learn to do proofs and think mathematically. Keep chugging away at it! Doing all the problems is probably a bit much and I, at least, would be more likely to not complete it if I were doing all the problems. Maybe you can find the list of used problems from a course and use that instead. Good luck!
(Edit: disclaimer, I’ve only used Spivak as a supplement to Hubbard & Hubbard)
You seem to be talking about Spivak’s “Calculus on Manifolds”, while the post is about “Calculus”. Munkres’s “Analysis on Manifolds” is a more didactically forgiving book than “Calculus on Manifolds” (and Hubbard’s book works as preparation for both).
That’s quite possible—I spent some time trying to figure out which was which but gave up and no longer have my copy. Which one is known as “baby Spivak”? (That’s the one I was referring to.)
Spivak is not an easy book and is probably not a great way to learn calculus. It is, however, an excellent start to learning rigorous mathematical techniques and gives you great foothold for for later topics in math such as analysis. The calculus course I took used “Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach” by Hubbard and Hubbard—it is essentially a slightly more modern and easier to read version of Spivak. The people I took that course with all had superb starting points for their later math courses compared to others (warning: selection bias! those who took the course were more interested in math, etc.), but we often lamented the fact that we weren’t very good at actually doing much with multivariable calculus when it came to, say, physics courses. That’s more because the differential forms introduced in Spivak are not the standard way of writing calculus in other domains than because what Spivak is doing is inherently harder.
In short: Spivak is hard. Taking a long time on it is not surprising in the least. It’s probably not your most ‘efficient’ route to learning multivariable calculus, but it is a good way to learn to do proofs and think mathematically. Keep chugging away at it! Doing all the problems is probably a bit much and I, at least, would be more likely to not complete it if I were doing all the problems. Maybe you can find the list of used problems from a course and use that instead. Good luck!
(Edit: disclaimer, I’ve only used Spivak as a supplement to Hubbard & Hubbard)
You seem to be talking about Spivak’s “Calculus on Manifolds”, while the post is about “Calculus”. Munkres’s “Analysis on Manifolds” is a more didactically forgiving book than “Calculus on Manifolds” (and Hubbard’s book works as preparation for both).
That’s quite possible—I spent some time trying to figure out which was which but gave up and no longer have my copy. Which one is known as “baby Spivak”? (That’s the one I was referring to.)