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.
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