  We address the problem of computing the thermodynamic properties of the
repulsive one-dimensional two-component Fermi gas with contact interaction,
also known as the Gaudin-Yang model. Using a specific lattice embedding and the
quantum transfer matrix we derive an exact system of only two nonlinear
integral equations for the thermodynamics of the homogeneous model which is
valid for all temperatures and values of the chemical potential, magnetic field
and coupling strength. This system allows for an easy and extremely accurate
calculation of thermodynamic properties circumventing the difficulties
associated with the truncation of the thermodynamic Bethe ansatz system of
equations. We present extensive results for the densities, polarization,
magnetic susceptibility, specific heat, interaction energy, Tan contact and
local correlation function of opposite spins. Our results show that at low and
intermediate temperatures the experimentally accessible contact is a
non-monotonic function of the coupling strength. As a function of the
temperature the contact presents a pronounced local minimum in the
Tonks-Girardeau regime which signals an abrupt change of the momentum
distribution in a small interval of temperature. The density profiles of the
system in the presence of a harmonic trapping potential are computed using the
exact solution of the homogeneous model coupled with the local density
approximation. We find that at finite temperature the density profile presents
a double shell structure (partially polarized center and fully polarized wings)
only when the polarization in the center of the trap is above a critical value
which is monotonically increasing with temperature.
