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:
“The welfare, the happiness, the energy and spirit of the men and women who do the daily work ... is the underlying necessity of all prosperity.... There can be nothing wholesome unless their life is wholesome; there can be no contentment unless they are contented.”
—Woodrow Wilson (1856–1924)
“Thus it seemed that this one hillside illustrated the principle of all the operations of Nature. The Maker of this earth but patented a leaf. What Champollion will decipher this hieroglyphic for us, that we may turn over a new leaf at last?”
—Henry David Thoreau (1817–1862)