Suppose you have a compact Riemannian manifold $(M,g)$ with or without boundary, and $\Omega \subset M$ is an embedded submanifold of the same dimension as $M$, which is compact. Suppose $V$ is a vector field on $M$ that induces a flow $\theta^t:M \rightarrow M$. Let $\Omega_t = \theta^t(\Omega)$. I want to find the derivative
$$\frac{d}{dt}\bigg|_{t=0} \text{Vol}(\Omega_t) \ .$$
Intuition tells me that this should be equal to the integral of the integral of $V$ along the unit normal of $\partial \Omega$. How would I prove such a result?