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I'm looking for a textbook that introduces Taylor's theorem for multivariable functions. Specifically, I'm looking for a textbook that explains it in a way that is similar to as is done here, where it explains it clearly using vectors and Hessians (see the second half of the page).

I've tried using keywords to search through Google Books, but to no success. Every textbook explanation of this concept that I have come across (and there don't seem to be many) seems to be lacking detail and clarity, and is therefore seemingly far inferior to the one given on the aforementioned website.

I've also noticed that it is uncommon for this to be covered in multivariable calculus and, more generally, real analysis textbooks. I suspect that I should instead be looking for vector analysis textbooks? Even so, I'm not concerned with the specific type of mathematics textbook, as long as the concepts are explained in a clear and detailed manner, in a way that is similar to what is done on the aforementioned website.

I would greatly appreciate it if people could please help me out with this.

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IMHO, a book called "Vector Calculus" by Marsden and Tromba is a good choice. It uses intuition, especially geometrical one, to explain and illustrate various notions in multivaraible calculus. It is not too rigorous as it aims for first year students (and IMHO mainly for Engineer majors), however it is absolutely not a hand-waving book that omits proofs. I think it has the right balance between rigor and geometrical intuition that coincides with the site that you have linked.

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  • $\begingroup$ This is what I was looking for. Thank you for the recommendation! $\endgroup$ – The Pointer Sep 29 '18 at 6:45

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