From Chapter 7 of Arnolds "Mathematical Methods in Classical Mechanics".
$Theorem:$ Given a manifold $M$, There is a unique (k+1)-form $\Omega$ on $TM_x$ which is the principal (k+1)-linear part at 0 of the integral over the boundary of a curvilinear parallelepiped, $F(v_1,...v_{k+1})$; i.e.
(1) $F(\epsilon v_1,.... \epsilon v_{k+1}) = \epsilon^{k+1} \Omega(v_1,.....,v_{k+1}) + o(\epsilon^{k+1})$ ($\epsilon \rightarrow 0$).
Where $F(v_1,...v_{k+1})=\int_{\delta \amalg} w^k$ where $\delta \amalg$ is the boundary of a k+1-parallelepiped
Can somebody help me understand this theorem?
What does it mean to be "the principal linear part"? And what is going on in (1)? What is the deal with Epsilons and what is $o(\epsilon^{k+1})$?