The classical version of Stokes' theorem revisited

Abstract

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 ℝ³ 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. However, the main point in the present article is first, that this latter fact usually does not get within reach for students in first year calculus courses and second, 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 this article uses only integration in 1, 2 and 3 variables together with a ‘fattening’ technique for surfaces and the inverse function theorem.

Keywords:

• Stokes' theorem
• Gauss' divergence theorem
• undergraduate mathematics
• curriculum

2000 AMS Subject Classifications:

• Primary 26

Acknowledgements

This work was carried out while the author was part time affiliated with the LearningLab DTU. The author would like to thank the staff and colleagues at the LearningLab DTU for their kind hospitality, rewarding and inspiring discussions on curriculum, teaching, and learning, and for technical and financial support.