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Unformatted text preview: THE WEDGE PRODUCT FOR SOPHOMORE CALCULUS FRANK BIRTEL The exterior algebra offers a computational and conceptual tool which can be in- troduced in sophomore multivariable calculus with a minimum of formalism. The purpose of this note is to demonstrate how that can be done for the ordinary second year calculus student from the very beginning of his or her study. Although the formulations in this paper have been carried out in n-dimensional Euclidean space, which might strike some readers as notationally forbidding, each proposition can be stated and proved in three our four dimensions to avoid this notational generality, and except for notation, all statements and proofs will not differ from the n-dimensional version. When the exterior algebra is available from the outset in a sophomore calculus course, it can be used to discuss k-dimensional planes, simultaneous linear systems of equations, linear transformations and all aspects of multivariable integration, the gradient, divergence and curl culminating in a single Stokes’ theorem for differential forms which subsumes all of the separate Green, Gauss, divergent and classical Stokes results. Cartan originally introduced the exterior algebra in order to simplify calculations with integrals. Not only are calculations simplified, but also concepts become unified, geometric, and easy to remember. After fifty years it is surprising not to find these techniques incorporated into the standard calculus curriculum at an early stage. 1 2 FRANK BIRTEL In this paper we will presume the standard introduction to vectors and the inner (dot) product which appears in every multivariable calculus text. 1. Wedge Product Definition 1. Let v 1 , v 2 , . . . , v k be k vectors in R n . Define the wedge product v 1 ∧ v 2 ∧ ··· ∧ v k by stipulating that (i) v ∧ v = 0 for any vector v ∈ R n . (ii) v ∧ w = (- 1) w ∧ v for any vectors v and w in R n . (iii) With the exception of (i) and (ii) all algebraic rules which apply to “ordinary multiplication”, also apply to “ ∧ ”....
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