TY - RPRT

T1 - The Classical Version of Stokes' Theorem Revisited

AU - Markvorsen, Steen

PY - 2005

Y1 - 2005

N2 - Using only fairly simple and elementary considerations - essentially
from first year undergraduate mathematics - we prove that the
classical Stokes' theorem for any given surface and vector field in
$\mathbb{R}^{3}$ follows from an application of Gauss' divergence
theorem to a suitable modification of the vector field in a tubular
shell around the given surface. The intuitive appeal of the
divergence theorem is thus applied to bootstrap a corresponding
intuition for Stokes' theorem. The two stated classical theorems are
(like the fundamental theorem of calculus) nothing but shadows of
the general version of Stokes' theorem for differential forms on
manifolds. The main points in the present paper, however, is firstly
that this latter fact usually does not get within reach for students
in first year calculus courses and secondly that calculus textbooks
in general only just hint at the correspondence alluded to above.
Our proof that Stokes' theorem follows from Gauss' divergence
theorem goes via a well known and often used exercise, which simply
relates the concepts of divergence and curl on the local
differential level. The rest of the paper uses only integration in
$1$, $2$, and $3$ variables together with a 'fattening' technique
for surfaces and the inverse function theorem.

AB - Using only fairly simple and elementary considerations - essentially
from first year undergraduate mathematics - we prove that the
classical Stokes' theorem for any given surface and vector field in
$\mathbb{R}^{3}$ follows from an application of Gauss' divergence
theorem to a suitable modification of the vector field in a tubular
shell around the given surface. The intuitive appeal of the
divergence theorem is thus applied to bootstrap a corresponding
intuition for Stokes' theorem. The two stated classical theorems are
(like the fundamental theorem of calculus) nothing but shadows of
the general version of Stokes' theorem for differential forms on
manifolds. The main points in the present paper, however, is firstly
that this latter fact usually does not get within reach for students
in first year calculus courses and secondly that calculus textbooks
in general only just hint at the correspondence alluded to above.
Our proof that Stokes' theorem follows from Gauss' divergence
theorem goes via a well known and often used exercise, which simply
relates the concepts of divergence and curl on the local
differential level. The rest of the paper uses only integration in
$1$, $2$, and $3$ variables together with a 'fattening' technique
for surfaces and the inverse function theorem.

KW - Stokes' theorem, Gauss' divergence theorem, level surfaces,

M3 - Report

T3 - Mat-Report

BT - The Classical Version of Stokes' Theorem Revisited

ER -