Interactive learning : mathematica enhanced vector calculus (ILMEV)

Abstract

This dissertation presents a computer learning assistant ILMEV (Interactive Learning -Mathematica Enhanced Vector calculus), that helps students understand the theory and applications of integration in vector calculus. Many students in engineering, the physical sciences and mathematics are required to learn vector calculus and find the subject challenging. Learning assistants in such advanced applied mathematics are rare and present many challenges. First, the presentation of this material in textbooks is not suitable for an automated assistant and thus, had to be substantially reorganized. Additionally, no computer algebra system has algorithms powerful enough to automatically solve all but the most elementary problems of this type that appear in textbooks. To overcome this, we implemented a geometric modeler and a simplified version of the cylindrical algebraic decomposition (CAD) algorithm so that ILMEV can now compute closed form solutions to many two dimensional textbook examples as well as examples far more complex. ILMEV succeeds because it contains algorithms for reducing the integrals appearing in vector calculus to sums of iterated integrals and all computer algebra systems have powerful algorithms for computing iterated integrals in closed form. Vector calculus is complicated, requiring the correct application of many formulas which, in turn, requires extensive calculations involving calculus and algebra. It is vital that students acquire good intuition about complex geometric objects. Consequently, the design of an assistant must be based on the best pedagogical principles (interactivity, visualization and experimentation) to assist in the teaching and learning. One important aspect of ILMEV is that it proceeds in simple logical steps and explains what it is doing. All of this would be for naught if ILMEV does not have an easy to use interface that enables student access to the power of a computer algebra system.