## The classical version of Stokes' Theorem revisited

Publication: Communication › Journal article – Annual report year: 2008

Using only fairly simple and elementary
considerations - essentially
from first year undergraduate mathematics
- we show how 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 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.

Original language | English |
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Journal | International Journal of Mathematical Education in Science and Technology |

Volume | 39 |

Issue number | 7 |

Pages (from-to) | 879-888 |

ISSN | 0020-739X |

DOIs | |

State | Published - 2008 |

Citations | Error in DOI please contact orbit@dtu.dk |
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### Keywords

- Gauss' divergence theorem, undergraduate mathematics, Stokes' theorem, curriculum

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