I would like to discuss this chapter with someone who has read the book.(adsbygoogle = window.adsbygoogle || []).push({});

From looking at other books, I realize that Spivak does things a little differently. He seems to be putting less structure on his chains (for instance, no mention of orientation, no 1-1 requirement and so on), and as a result, I find that things get a little weird.

For instance, the first thing I asked myself after reading the definition of the integral of a k-form over a k-chain is whether or not the result is independent of the chain. More precisely, if c and d are two k-chains with identical images, does

[tex]\int_c\omega=\int_d\omega[/tex]

as intuition demands??

I found a little guidance in answering this in the person of problem 4-25 (Independence of parametrization), but that's not entirely satisfying, because after all, coudn't it be that there is no 1-1 p such that c o p = d? If c and d are not injective for instance, the obvious p(t) := c^-1(d(t)) fails. And that det p'(x) >= 0 condition... what does it say about p? What characterize reparametrizations p with det p'(x) >= 0? (Does injectivity implies that the determinant does not chance sign? locally okay, but globally?!)

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integration on chains in Spivak's calculus on manifolds

Loading...

Similar Threads - Integration chains Spivak's | Date |
---|---|

A Exterior forms in wiki page | Dec 22, 2017 |

I Integration of a one form | Nov 5, 2017 |

A Integration along a loop in the base space of U(1) bundles | Oct 1, 2017 |

I Integration of a manifold | Sep 11, 2017 |

Boundary of a chain, Stokes' theorem. | Jan 17, 2015 |

**Physics Forums - The Fusion of Science and Community**