Fix a chain $c$, a homotopy $g^t,\ 0\leq t \leq \tau$ and take $Jc$ the track of $c$ under $g^t$. Arnold's mechanics text (pg 205) says it is easy to verify that $\partial (Jc)=g^\tau c-c-J\partial c.$
How can this formula be derived? I know $\partial Jc$ must be some formal linear combination of the chains $(g^\tau c, \ c,\ J\partial c)$ but what is the rationale for their signs?