User talk:AlHutchins

re https://en.wikipedia.org/wiki/Exterior_derivative

The second definition given here, in terms of local coordinates, seems to be the same as the one given by Spivak:

CALCULUS ON MANIFOLDS (W. A. Benjamin 1965, page 91)

Spivak's next theorem, 4-10(2), then states

If \omega is a k-form and \eta is an l-form, then

d(\omega\wedge\eta) = d\omega \wedge \eta + (-1)^{k l} \omega \wedge d\eta.

This CONFLICTS with the first "axiomatic" definition on this web page. In it, using Spivak's notation, the sign of the second term is (-1)^k, not (-1)^{k l} .

I am not sure which of the first two definitions corresponds to the third "invariant formula" definition.

Possible sources of confusion:

- coordinates may be indexed from 1 to n, or from 0 to n-1

- in a monomial form "f dx_1 \wedge ... dx_n", the function f may appear at the end. Then its exterior derivative might be "dx_1 \wedge ... dx_n \wedge df".

- in the 3rd definition, in terms of invariant formula, the sign of all terms containing Lie bracket may be reversed.

Am I in a muddle, or is there real confusion here? AlHutchins (talk) 08:41, 14 May 2016 (UTC)