This is the first of two courses covering advanced calculus, and provides a rigorous treatment of single-variable calculus, including limits, continuity, derivatives, integrals, and uniform convergence of sequences and series of functions. After taking Mathematics 4603, students often take Mathematics 4604 (Advanced Calculus II), which deals rigorously with topics and issues related to multivariable functions, including the the implicit function theorem as well as the integral theorems of Green and Stokes. Mathematics 4603 assumes a good knowledge of lower-division calculus as presented in courses such as Mathematics 2243 and 2263. Both Mathematics 4603 and 4604 can be counted toward a math major's analysis requirement as described here .
Students taking Mathematics 4603-4604 will learn to appreciate and understand the basic objects of calculus (functions, derivatives, integrals, series of functions) in a way that is not expected of students in our standard freshman-sophomore calculus sequences. In order to do this, they will need to come to grips with very precise definitions for these objects, and they will need to use those definitions to reason carefully and logically about important properties and relationships that are fundamental in the way that calculus is used. The concepts of calculus have been developed and clarified through of a long historical process of discovery. In freshman-sophomore calculus, students learned how to use many calculus procedures to solve problems involving such things as rates, velocities, areas, volumes, and approximations. In Math 4603-04, they learn why those procedures work and how to explain why they work.
Students who plan to do graduate study in mathematics often take the honors real analysis sequence Mathematics 5615-5616 at some point before graduation. Students who intend graduate study in math-related fields such as Economics and Statistics often take this sequence as well. Since Mathematics 5615-5616 is rigorous and deals with analysis from a rather general standpoint, students should usually have taken upper-division mathematics courses in the analysis category before doing Math 5615-16. The advanced calculus sequence Math 4603-4604 is one of many ways to prepare for Mathematics 5615-5616.
Mathematics 4603-4604 is recommended, though not required, for students in the Educational specialization who expect to teach calculus later.