  When path integrals are discussed in quantum field theory, it is almost
always assumed that the fields take values in a vector bundle. When the fields
are instead valued in a possibly-curved fiber bundle, the independence of the
formal path integral on the coordinates becomes much less obvious. In this
short note, aimed primarily at mathematicians, we first briefly recall the
notions of Lagrangian classical and quantum field theory and the standard
coordinate-full definition of the "formal" or "Feynman-diagrammatic" path
integral construction. We then outline a proof of the following claim: the
formal path integral does not depend on the choice of coordinates, but only on
a choice of fiberwise volume form. Our outline is an honest proof when the
formal path integral is defined without ultraviolet divergences.
