  PbTiO$_3$ is a simple but very important ferroelectric oxide that has been
extensively studied and widely used in various technological applications.
However, most previous studies and applications were based on the bulk material
or the conventional [$001$]-orientated films. There are few studies on
PbTiO$_3$ films grown along other crystalline axes. In this study, a
first-principles calculation was performed to compute the polarization of
PbTiO$_3$ films strained by SrTiO$_3$ and LaAlO$_3$ substrates. Our results
show that the polarization of PbTiO$_3$ films strongly depends on the growth
orientation as well as the monoclinic angles. Further, it is suggested that the
ferroelectricity of PbTiO$_3$ mainly depends on the tetragonality of the
lattice, instead of the simple strain.
