I'm reading this work.
Let $\Omega$ be a bounded (open) domain, and define $Q=(0,T)\times\Omega$. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. $\Omega(t)$ is divided by these two subsets.
Define $$Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$$ and $$S= \bigcup_{t \in (0,T)}S(t) \times \{t\}$$
Let $$\chi = \begin{cases}1 &\text{on $Q_1$}\\-1 &\text{on $Q_2$}\end{cases}.$$
(Page 4) Let $\varphi \in C_c^\infty(Q)$. We have by integration by parts in time: $$\int_Q \frac{\partial (\chi \varphi)}{\partial t} = \int_{Q_1} \frac{\partial (\chi \varphi)}{\partial t}+\int_{Q_2} \frac{\partial (\chi \varphi)}{\partial t} = -\int_{Q_1}\chi\varphi - \int_{Q_2}\chi\varphi + \int_S (1)\varphi \nu_t - \int_S (-1)\varphi \nu_t$$ where $\nu = (\nu_x, \nu_t)$ is the unit normal vector to $S$ pointing from $\Omega_1$ to $\Omega_2$.
Questions
Why is the last equality true in the calculation above? I don't follow what he does.
How do I visualise the vector $\nu = (\nu_x, \nu_t)$? What's the subscript supposed to tell me?
