Having a list of solutions handy is helpful in rectifying such misconceptions and allows the student to solve other problems correctly, while the lack of such external assessment may solidify the mistake and harm the understanding of the material as the student repeats it unchecked Perhaps the student misinterpreted the conditions for the validity of a certain theorem or applied it in some wrong manner. Any student is fundamentally limited in their ability to review their own solutions.įirst, even if one believes that one solved a problem in the most rigorous way possible, mistakes and misconceptions are unavoidable. I disagree with the notion that rigor somehow negates the utility of a solution manual.
I may just read Spivak or go with Munkres Analysis. Not as rigorous as say Spivak, but I gained more from that book than Calculus on Manifolds. Moreover, it is much different than the standard Cal 3 course (usually aimed at engineers/physics and other science majors). The problem with rigorous multivariable calculus books is that they generally use the language of linear algebra and introduces concepts of topology (closed/open sets, boundary/interior etc). The OP requiring a solutions manual greatly supports this. Although one can definitely learn from it, I don't think many will be able to absorb any meaningful mathematics. Let alone the other proof based math classes. This was having things like Complex Analysis, Intro Analysis, and Topology under my belt. I myself recently took a class using that book, and found it difficult, although the teacher was no help and utterly uses. How would one want rigor, and yet also ask for solutions manual? It defeats the purpose.
Moreover, he asked for solution manual + rigorous multi-calculus book. There is a major leap going from something like Spivak Calculus to Spivak Calculus On Manifolds.
It is a very long book (perhaps 800 pages?) and I'm not sure that is what you are looking for. I don't know Spivak so cannot say if it is in the same vein, but the parts I have read were very well written and quite rigorous. It was written for the honors sophomore-level math at Cornell. If so, then one option worth a look is "vector calculus, linear algebra and differential forms" by Hubbard and Hubbard. That said, there are some amazing results that have been rigorously proven.īy "gone through" I'm assuming that means you worked through the proofs yourself and solved a reasonable fraction of the homework problems. And, to some extent, the proofs are extensions of the single-variable cases in any case - like the multi-variable Taylor series.Īs an aside, I watched a brilliant set of lectures on mathematical physics by Carl Bender (they are on YouTube) and he stressed how much of the work in his field has not been rigorously proven - and that looking for rigorous proofs is a major drawback and ultimately a limiting factor to what you can achieve. Multi-variable calculus is generally a tool for applied maths and physics, where there is less concern with rigorously proving everything. Rigorous single-variable calculus is essential for pure mathematics, of course, but after than you are more likely to move on to other pure mathematical disciplines, such as algebra, complex analysis, functional analysis and linear algebra etc. Just a thought, but there may be a limited market for fully rigorous multi-variable calculus.
So is there any book on the subject that will be in the same vein as Spivak (and preferably with ? I've also considered Courant & John Vol-II but having red the beginning of Vol-I, I dislike the approach the authors take, and the explanations feel somewhat more hand-wavy to me (at least compared to Spivak). At first I considered "Calculus on manifolds" but from what I've been told it's too dense and will be better appreciated as a second exposure. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution manual. I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus.