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251 vs. 251H vs. 291.
Math 251 continues the sequence begun with Math 151-152, usually with the same textbook and at the same level of rigor. The honors sections labeled 251H of Math 251 are (in general) intended for honors students in disciplines other than mathematics and are more demanding versions of the same course. By contrast, Math 291 is deliberately intended as a course in honors mathematics for students whose primary interest in the course is the mathematics it contains. The textbook may not be that used in other calculus courses, and the choice of course material is at the instructor's discretion to a greater extent than in other lower-division courses. Theorems may be proved in class and required on examinations, and many variables may mean n variables, not just 2 or 3.
I never understood really what was the purpose of teaching a series like 151 (Calc 1) and the likes. There is no real science there, at any level, just formulas and rules for solving problems nobody knows where they come from.
I think it is a waste of students' time. If I was a student here in this system, I would have absolutely hated it and would never go into math!