integrals of exact forms

Let

\phi \in \Omega^k (U)

be an exact form and let

Unparseable latex formula:

\sigma \in C_k_+_1(U)

be a chain.
Calculate

\int _{\delta\sigma} \phi

.
I'm almost certain the answer's 0 but I'm not sure why, could someone please explain it.

(If it helps this possibly has something to do with stokes theorem and cohomology.)

(edited 11 years ago)

Reply 1

Mark85

Just use the definition of exact i.e.

\phi = \mathrm{d}\phi'

for some

\phi' \in \Omega^{k-1}(U)

and then Stokes' theorem which in this context says

\int_{\delta\sigma}\phi = \int_\sigma \mathrm{d}\phi

and then the basic property of the boundary

\mathrm{d}

that allows one to define cohomology.

Reply 2

thebadgeroverlord

Original post by abbii

Let

\phi \in \Omega^k (U)

be an exact form and let

Unparseable latex formula:

\sigma \in C_k_+_1(U)

be a chain.
Calculate

\int _{\delta\sigma} \phi

.
I'm almost certain the answer's 0 but I'm not sure why, could someone please explain it.

(If it helps this possibly has something to do with stokes theorem and cohomology.)