Let $(M,g)$ be a smooth Riemannian manifold embedded in $\mathbb{R}^m$. I would like to understand the transformation formula which will allow me to pass from the integral $\int_M \dots dV_g(x)$ to $\int_M \dots dV_g(y)$ where $x \mapsto y:=\exp_x(\epsilon v)$ for $v \in \mathbb{S}^{m-1}$ and $\epsilon>0$, and $dV_g$ represents the integration w.r.t. the Riemann-Lebesgue volume measure.
This looks to me as we need to understand $\frac{d}{dx} \exp_x(\epsilon v)$. According to the comments on this question it should be expressed in terms of $J(1)$, where $J$ is a Jacobi field satisfying certain initial conditions. Since I am beginner in this, I partially understand the connection with Jacobi field, but not really what are these initial conditions and how should the change of variables formula look like. Let me point out that I am interested in this question for small $\epsilon$, so I would be happy to have something of the form $dV_g(x)=(1+O(\epsilon))dV_g(y)$.
Could someone direct me to the right answer?
