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I have the summer to prepare myself for a high school calculus-based physics course; however, I will have to study calculus by myself. I've completed a course in precalculus and I'm beginning to read "How to Prove It," by Velleman.

I am looking for a textbook that has mathematical rigor (i.e. not memorizing formulas), yet also provides adequate foundations for a physics course. From similar stack exchange questions, my top candidates are Spivak's Calculus and Apostol's Calculus (Vol. 1). Which one of these textbooks, if any, would be the best choice for my situation?

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    I love Stewart Calculus myself, and I also recommend you concurrently do some reading in physics specifically with calculus along the way. I've always enjoyed Halliday/Resnick/Krane Physics, myself. – 2017-02-09
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    *Thomas'* Calculus/Calculus & Analytical Geometry could be helpful in this regard. – 2017-02-09
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    If you've taken a precalculus course already you will be more than prepared for the level of calculus in a highschool advanced physics course. Even in AP courses the stress on actual calculus is very low. Although the want to learn more should always be encouraged and I too like Stewart's Early Transcendentals Calculus book, while Halliday/Walker/Resnick is a reasonable physics book. – 2017-02-09
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    Not necessarily "rigorous" mathematically but a free resource is through Ohio State's Mooculus course: https://mooculus.osu.edu/handouts If you login via google you can also access their exercises and practice problems. – 2017-02-09

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I studied Physics and would definetely recommend you Apostol's. It is a good book, rigorous enough and complete, while at the same time keeps a casual layout with a lot of explanative text, which makes it a good transition between High School and college.

As a personal reference I can tell you it was the recommended book for my first Calculus course on Physics, I bought it, and I still occasionally consult it as a TA to see how it explains some concepts.

Also keep in mind that you may want to buy both volumes, the second is not a more advanced book but simply covering other topics on the same basic level (sequences, integrals,etc.)

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    The second volume is more advanced material insofar as it covers multivariable calculus. Apostol is head and shoulder above a book like Stewart's, but Stewart's does have a larger selection of computational tedium. Maybe I misunderstand what you mean by "same basic level". Regardless volume 2 is not single variable calculus, generally, and goes through a bit of linear algebra for the first few chapters. – 2018-04-07
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If you like "How To Prove It," you could try:

Velleman, Calculus: A Rigorous First Course, Dover Publications, 2016.

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While both Spivak's and Apostol's books are rigorous in that they include complete proofs, Spivak's has a heavier emphasis on theoretical questions, and its exercises are much harder. Spivak's book also has a complete solution manual. Spivak's book can be considered one of the best introductions to rigorous mathematics.

But on balance, if your real interest is physics, my recommendation would be Apostol's book. Apostol also covers much more material after the basic single-variable stuff (at the end of Vol. 1 and throughout Vol. 2), and all of this is important for physics later on.

Facility with rigorous math is useful for higher-level physics, but only rarely for introductory physics classes, so if you find you want (or need) to move faster for some reason, it would be okay to use a calculus book that is conceptual and provides proofs where they're easy, but avoids theoretical questions concerning limits and the like. One good option for this would be Lang's A First Course in Calculus.