I'm a math major who is about to have his first brush with transition to upper math courses this coming fall. I took the calculus sequence at a local community college, and when I'm looking at some of the problems here or looking at syllabi of courses I will be taking someday, I get the impression that a lot of necessary things may have been left out of my courses. I've picked up Apostol to look over this summer; hopefully that will bridge several of the more obvious gaps, particularly those involving any rigour. Are there any essential things that you'd imagine may be skipped or glossed over in an essentially cookie cutter college calculus course? Some things that are on my checklist to learn over the summer are A review of series More vector analysis Gaussian integral Liebniz's integral rule Is that last one usually covered in a traditional sequence or saved until later? I see it in the syllabus of my destination university's real analysis course. I perpetually feel underprepared, even though I usually wind up doing very well; any additions to a list of necessities or even optional topics are greatly appreciated!