An Undergraduate Mathematics Curriculum

The indispensable lower division core consists of the following three courses.

• Calculus by Howard Anton
• Linear Algebra and Its Applications by David Lay
• Elementary Differential Equations by Derrick & Grossman

A standard problem in all mathematics curricula is how to transition from lower-division problem-based courses such as the three above to the upper-division proof-based courses. The following trio of books accomplishes this quite well.

• How to Prove It: A Structured Approach by Daniel Velleman
• Introduction to Analysis by Edward Gaughan
• Set Theory and Logic by Robert Stoll

The following constitute the standard trio of upper-division courses that all mathematics majors should cover to be considered mathematically mature (as well as mathematically literate).

• A First Course in Abstract Algebra by John Fraleigh
• Principles of Mathematical Analysis by Walter Rudin
• Topology by James Munkres

Some standard elective courses in the area of applied mathematics are the following.

• Freund's Mathematical Statistics by Miller & Miller
• Numerical Analysis by Burden & Faires
• Partial Differential Equations : An Introduction by Walter Strauss

Some standard elective courses in the area of pure mathematics are the following.

• Introduction to Modern Set Theory by Judith Roitman
• Introduction to Mathematical Logic by Mendelson & Mendelson
• Introductory complex analysis and applications by William Derrick

One topic that is seldom covered in the undergraduate curriculum is geometry. This course is required for most mathematics education majors, but not mathematics majors. This is a shame since this course, even though it doesn't lead into any of the major fields of mathematics research, provides a most insightful foray into how an axiomatic mathematical system should function. I found it utterly fascinating and am very glad I took the opportunity to include it in my coursework.

• Euclidean and non-Euclidean geometries: Development and history by Marvin Greenberg