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
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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
.
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 Spring '09
 ALI
 Math, Linear Algebra, Algebra, Multivariable Calculus, Vector Space, wedge product

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