# Change of variables in path integrals

I need to evaluate a path integral which involves a set of fields $X=\left\{ \psi_i \right\}$:

$$I = \int \prod_i \mathcal{D} \psi_i e^{-S \left[ \left\{ \psi_i \right\} \right] }$$

In order to simplify the treatment I'd want to make a change of variables, and express the integrand as a function of a new set of fields, let's call it $Y=\left\{ \phi_i \right\}$. My problem is: how do I implement the change of variables? Can I extend the multi-dimensional case to the continuum and include the determinant of the Jacobian of the transformation in the integral, i.e.

$$J_{ij} = \frac{\delta X_i}{\delta Y_j}$$

Moreover I'd want to defined one or more of the $Y_i$ variables as containing some derivatives of the original $X_i$ ones. My intuition leads me to write down the Jacobian in momentum space, where each derivative is then replaced by a $- \mathrm{i} k_\mu$ then evaluate it, also in momentum space. I would also say that as long as the Jacobian does not depends on the fields, it can be taken outside of the integral sign, i.e. it is a multiplicative constant. Is my intuition right?

Any suggestion, or any reference to a textbook is welcomed. In particular I couldn't find in any textbook I own the cases in which one of the redefined fields depends upon the derivative of one the original fields.

I know that this a mathematical question more than a physics question; however I decided to post it here because (usually) physicist are more used to manipulating path integrals.