In the paper I am reading, (http://arxiv.org/abs/1308.5376), they solve an integral and I am trying to replicate the results.
This question is a simplified version of the integral they calculate, I am just trying to get an idea of how to approach it. If we consider discrete time points and the vector field: $$ U_{\mu}(\pi) =\pi-\mu $$ where $\pi, \mu$ are vectors in $\mathbb R^n$. Then how does one interpret the integral: $$ \int_0^b U_{\mu(t+1)} (\pi(u) )~~du $$ where $b \in \mathbb R$.
What I am finding confusing is that when integrating over a vector field, this would be a line integral and we would be given some path/curve to integrate over, but this is not the case (as far as I can tell) here. I am interpreting it as:
$$ \int_0^b U_{\mu(t+1)} (\pi(u) )~~du = \sum_{u=0}^b \big[ \pi(u) - \mu(t+1) \big]\\ =\sum_{u=0}^b \big[ \pi(u) \big] - b~\mu(t+1) $$ is this the correct interpretation? As I cannot replicate their results. I posted another question that had the actual integral in their paper but didn't get much help, here is a link for interest: Integrating over a specific vector field
It is not a line integral. It is a single integral on individual component of the vector field. Using your notation:
$$\int_0^b U_{\mu(t+1)} (\pi(u) )\,du=\left(\int_0^b (\pi_1(u)-\mu_1(t+1)) \,du, \int_0^b (\pi_2(u)-\mu_2(t+1)) \,du, ..., \int_0^b (\pi_n(u)-\mu_n(t+1)) \,du\right)$$