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Upper division undergraduate mathematics course structure

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This page gives information on the course structure for upper division undergraduate mathematics in universities. This is in contrast with lower division undergraduate mathematics.

The information on this page has been constructed based on an in-depth knowledge of the course structure of a few undergraduate programs and a broad survey of many others. The program surveyed have been based in the United States. The structure of undergraduate programs differs from country to country and even varies widely within country, so please do not treat this as a substitute for information about the course structure at a specific educational institution.

Basics

Goal of upper division mathematics courses

These courses are targeted at mathematics majors, though some of the courses may be taken by majors in other subjects. At top universities, these courses are designed to provide students with the basic background in knowledge and mathematical reasoning schools that would be necessary to prepare them for graduate school. It's worth keeping in mind that these programs do not serve the needs of other departments, in contrast with lower division undergraduate mathematics. Thus, the incentive structure behind the design of these courses differs considerably.

Analysis-first course structures

A number of universities begin their upper division undergraduate mathematics sequence with a year of undergraduate analysis. The goal of this year is three-fold:

  • It covers the key topics of basic real analysis.
  • It introduces a smattering of other topics seen throughout mathematics, such as metric spaces, topological notions, measure theory, set theory, and elementary linear algebra.
  • It includes general proof and reasoning skills.

Relation with multivariable calculus and linear algebra

In some universities, the analysis sequence is a superior parallel to the lower division courses in multivariable calculus and linear algebra. Students taking analysis in these universities do not need to take multivariable calculus or linear algebra, and the relevant topics from these subjects are included in the analysis course.

In other universities, students become eligible to take analysis only after completing multivariable calculus and linear algebra. The analysis courses in these universities are shorter since they rely on the multivariable calculus and linear algebra topics already covered.

Algebra as a second year of upper division study

Universities whose upper division undergraduate course structure begins with undergraduate analysis typically require abstract algebra for the second year of upper division study.