Context
I am working through Polchinski's book on string theory (Volume 1 on the bosonic string). In his exercise question 3.5, one has to show how a gauge invariant action multiplied by a gauge invariant insertion, can be re-expressed as an integral of the original action, a gauge fixing action, and the Faddeev-Popov action over a functional. There is a solution manual (http://arXiv.org/abs/0812.4408v1), where this question is answered. However, in one step (equation 24 in the solutions maual), a relation between the determinant of a non-singular square matrix and an integral is used that I haven't seen before.
Question
Is the following true, and if so, why?
$$ \text{det}\left(\partial_\phi F^A\right)=\left(\int\!\left[\mathrm{d}\epsilon^B\right]\partial_\phi F^A\partial_{\epsilon^B}\phi\right)^{-1} $$
with the integrand being a non-singular square matrix and $\phi$ a field.