How CAS and Visualization lead to a Complete Rethinking of an Introduction to Vector Calculus

This article demonstrates how newly available technology suggests to completely rethink how key concepts of (vector-) calculus are introduced. We consider this a particularly dramatic show-case, which also raises the question whether similarly radical rethinking applies to other areas throughout the mathematics curriculum. The new approach makes fairly advanced mathematical concepts accessible to a much broader population than the traditional treatment. Single variable calculus has seen some compelling changes in the last decade. We propose to proceed in complete analogy on all levels of calculus. The focus here is vector calculus. We demonstrate how to \see local linearity" (di erentiability) and local constancy (Riemann integrability). Highlights are: \How to see the divergence and curl by zooming", and a completely \intuitive" development of Stokes' theorem and all its cousins. The proposed approach is implemented in the software package MAPLE and has been classtested repeatedly.