I know that for a function $f$ on $\mathbb{R}^n$ and a diffeomorphism $\phi:\mathbb{R}^n\to\mathbb{R}^n$, the change of variables (or pushforward) $f_\phi$ of $f$ along $\phi$ is given by
\begin{equation*} f_\phi(x)=f(\phi^{-1}(x))\det{\frac{\partial \phi^{-1}(x)}{\partial x}}. \end{equation*}
Does anyone know how to define this for general manifolds? That is, when $f$ is a function $f:M\to\mathbb{R}$ and $\phi$ is a diffeomorphism from $M$ to $M$.