Underlying Principle
To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an example for d = 2 dimensions. The essential idea can be understood by the diagram on the left, which shows that, in an oriented tiling of a manifold, the interior paths are traversed in opposite directions; their contributions to the path integral thus cancel each other pairwise. As a consequence, only the contribution from the boundary remains. It thus suffices to prove Stokes' theorem for sufficiently fine tilings (or, equivalently, simplices), which usually is not difficult.
Read more about this topic: Stokes' Theorem
Famous quotes containing the words underlying and/or principle:
“Every age develops its own peculiar forms of pathology, which express in exaggerated form its underlying character structure.”
—Christopher Lasch (b. 1932)
“No habit or quality is more easily acquired than hypocrisy, nor any thing sooner learned than to deny the sentiments of our hearts and the principle we act from: but the seeds of every passion are innate to us, and nobody comes into the world without them.”
—Bernard Mandeville (1670–1733)