Calculus on Manifolds

Not the person you are responding, but I also used Spivak for single variable. For multivariable calculus I used Marsden and Tromba. The other Spivak is a great book, but maybe too much for a first approach to the subject.

Calculus on Manifolds by Michael Spivak. http://www.strangebeautiful.com/other-texts/spivak-calc-mani... (http://www.strangebeautiful.com/other-texts/spivak-calc-manifolds.pdf) For me, the eye opening thing was that any n-dimensional body is a transformation of the n-dimensional standard cube. This brings some mathematical consequences I don't understand, but what's important for me is that equations get much easier for the standard cube transformations because your corners are just ones and zeros. E. g. I wrote an app that "unbends" a book page from a photo so it looks flat. It can be done with a single but rather large transformation, or with a pair of transformations: to and from the standard cube, and each transformation is then much simpler than their composition. 15 years after I learned that trick and 10 years after I wrote an app, I wrote a book called Geometry for Programmers. If not for Spivak, I would have drowned in equations, never written the app and would have had nothing to write a book about.

Seconding all of Spivak's books. My favorite treatment of differential geometry.

If you want to get away from analysis, I've found that cryptography can be quite an engaging subject. If you have the right book, it can have the rigors of more mathematical subjects, while being accessible without extensive background and having visible real-world applications. I unfortunately don't have much experience with books in this area, but I do like https://files.boazbarak.org/crypto/lnotes_book.pdf (plus it's free ;) ). [EDIT: Previously I recommended Calculus on Manifolds here also, but on further reflection and reading some of the other responses I think I both misremembered the difficulty level of the book and overestimated what early-undergrad level means]

Calculus on Manifolds, by Michael Spivak

I'm pretty at Mathematics and doing that takes a lot of work. And the Mathematics books for noobs are no good. The better approach is to get a theoretical book, something like Spivak's Calculus or Linear Algebra by Friedberg, Insel and Spence. And then from there whenever you have difficulty with the material spread out laterally and you really start to gradually grow an understanding of mathematics. And then perhaps, one day you'll be up for Spivak's Calculus on Manifolds!