Jacobian between $TM$ and $M\times M$

Let $ M$ be a closed riemannian manifold and $ \phi: \begin{array}{ccc} TM&\to &M\times M \ (x,v)&\mapsto & (\exp_x(v),\exp_x(-v)) \end{array}$

I need an asymptotic expression for the jacobian of $ \phi^{-1}$ as $ \|v\|\to 0$ , but I’m unable to compute it … I guess the order $ 0$ term is $ 2$ , but even this I’m unable to compute correctly… and I’m also interested in the quadratic term (the one of order $ \|v\|^2$ ), wish I guess depends on curvatures at $ x$ . I guess it’s someway linked to Jacobi field… If that simplify thing, the manifold $ M$ has dimension $ 3$ in my problem.

I’m interested as much in the result itsef as in the way to obtain (and understand) it.

Thanks a lot, have a good day