SUMMARY OF CURRENT MATH METHODS MODULES

1. Elementary notions of multivariable calculus

(a) Cartesian, cylindrical and spherical coordinate systems. Visualization, and mathematical description, of simple objects in these coordinates. Scaling factors, line elements, surface elements and volume elements.

(b) Various kinds of rates of change (partial derivatives, directional derivatives and gradients) of scalar functions of two and three variables. The interpretation of these rates of change in geometric contexts, and their significance in a variety of applications (heat and mass transport, solid and fluid mechanics, electromagnetics).

2. Constrained Optimization .

Optimization of objective functions of more than one variables under single or multiple constraints. Use of Lagrange multipliers. An interactive module is developed to illustrate these concepts.

3. Vector Fields and the Notions of Flux, Divergence and Curl.

A library of prototypical vector fields (constant, rotational, dilatational), and their visualization via direction fields and flow lines. The notions of divergence and curl; their computation and visualization by means of JAVA applets. Applications to electromagnetics.

4. Integrals and the Theorems of Green, Gauss and Stokes.

Computation of line, volume and surface integrals, and their relationship to physical quantities such as work, circulation, vorticity, dilatation, flux, potential, sources and sinks. Concepts are developed in a text-based medium with links to JAVA applets. The concepts are illustrated using electromagnetic fields.