  In Part I of this diptych, we outline the parallel mode of differential phase
contrast (TEM-DPC), which uses real-space distortion of Fresnel images arising
from electrostatic or magnetostatic fields to quantify the phase gradient of
samples with some degree of structural contrast. We present an analysis
methodology and the associated software tools for the TEM-DPC method and, using
them together with numerical simulations, compare the technique to the widely
used method of phase recovery based on the transport-of-intensity equation
(TIE), thereby highlighting the relative advantages and limitations of each.
The TEM-DPC technique is particularly suitable for \textit{in-situ} studies of
samples with significant structural contrast and, as such, complements the TIE
method since structural contrast usually hinders the latter, but is an
essential feature that enables the former. In Part II of this work, we apply
the theory and methodology presented to the analysis of experimental data to
gain insight into two-dimensional magnetic phase transitions.
