There are several mathematics courses that can be taken after Calculus III.
In the Spring you could take Calculus IV or Linear Algebra; in the Fall you could also take Introduction to Higher Mathematics.
Linear Algebra (V2010) develops ways to solve systems of many linear equations. These might seem rather irrelevant since most interesting equations are not linear. However, linear equations have the great virtue that they can be solved, at least in principle. Also their solutions have unexpectedly nice properties.
It is very important to understand them thoroughly because often what one does with intractible nonlinear equations is develop sophisticated ways to
approximate (or model) them by systems of linear equations.
Calculus IV does integration in many variables. Integration is a much richer subject when there are many variables since one can now integrate over curves or surfaces in space as well as over three dimensional regions. The relation between differentiation and integration also becomes more interesting, since there is some interplay between the dimension of the domain of integration and the kind of function one is dealing with. The mathematics developed in this course
is essential to understanding fluid flow and electricity, as well as multivariate statistics.
Of these two courses, Linear Algebra is probably the easier.It serves as an introduction to algebraic ways of thought, and so is a prerequisite for the Modern Algebra course as well as for courses that study more concrete applications of algebra such as Making/Breaking Codes.
It is also a prerequisite for V 2500 Analysis and Optimization and a co-requisite for V 3027 Ordinary Differential Equations.
Calculus IV develops techniques that are essential to all further applications of calculus.
It is a prerequisite for the Modern Algebra and Modern Analysis sequences, as well as for Complex Variables (V 3007).
Other intermediate level electives offered in Spring 2009
V 2500 Analysis and Optimization: mathematical methods for economics.
V 3027 Ordinary Differential Equations: a basic course in differential equations.
It is a prerequisite for V 3028: Partial differential equations (which studies the equations satisfied by waves and heat)
as well as a recommended preparation for V 3050 (Discrete time models in Finance.)
V 3020 Number Theory and Cryptography: this should a fun course; it has few formal prerequisites, but you should talk to the instructor to see if Linear Algebra or Introduction to Higher Mathematics might be advisable as preparation.