Calculus

Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. A study of limits and continuity in multiple dimensions yields many counter-intuitive and pathological results not demonstrated by single-variable functions.

The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. The multiple integral expands the concept of the integral to functions of many variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral.

It includes Limits and continuity, Partial differentiation and Multiple integration. In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the famous integral theorems of vector calculus:   Gradient theorem, Stoke’s theorem, Divergence theorem, and Green's theorem.

In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the Generalized Stokes' theorem, which applies to the integration of differential forms over manifolds. It's hard to find satisfactory books for an honors multivariate calculus course. In particular, the problem sets may not cover just what you want.

The problems may only ask for proof of more theorems, without first exercising the students in understanding the concepts and definitions. An honors course could have challenging computational problems as well as theory problems. Or there might not be enough theory problems. For whatever reasons, you may feel, as I did, that you need to provide extra problems. If any of the problem sets listed below fill you needs, you are welcome to use them.

Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.

Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. Non-deterministic or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus.