I need some help for the following problem.

Let $M$ a riemannian manifold and $f$ a smooth differential function, then consider the following integral $$\int_M \Gamma(x,y)(f(y)-f(x))dV_y$$

where $dV_y$ is the measure on the manifold and $\Gamma$ is a positive function. Now, my question is how can do a taylor expansion of the integral, i.e., for example in the case of $M=\mathbb R$, we have with $y=x-\epsilon z$, $$\int_\mathbb{R}\Gamma(x,x-\epsilon z)(f(x-\epsilon z)-f(x))dy=\int_\mathbb{R}\Gamma(x,x-\epsilon z)(-\epsilon z f'(x)+(\epsilon z)^2f''(x)+...)(-\epsilon dz)$$

But now if I'm on a Riemannian manifold, how can I do something like that? I don't know any taylor series like that. Anyway, for the case of the manifold I tried to occupy the exponential map and the polar coordinates.


Thanks any idea will be appreciated! Thanks!