  A measurement is presented of differential cross sections for $t$-channel
single top quark and antiquark production in proton-proton collisions at a
centre-of-mass energy of 13 TeV by the CMS experiment at the LHC. From a data
set corresponding to an integrated luminosity of 35.9 fb$^{-1}$, events
containing one muon or electron and two or three jets are analysed. The cross
section is measured as a function of the top quark transverse momentum
($p_\mathrm{T}$), rapidity, and polarisation angle, the charged lepton
$p_\mathrm{T}$ and rapidity, and the $p_\mathrm{T}$ of the W boson from the top
quark decay. In addition, the charge ratio is measured differentially as a
function of the top quark, charged lepton, and W boson kinematic observables.
The results are found to be in agreement with standard model predictions using
various next-to-leading-order event generators and sets of parton distribution
functions. Additionally, the spin asymmetry, sensitive to the top quark
polarisation, is determined from the differential distribution of the
polarisation angle at parton level to be 0.440 $\pm$ 0.070, in agreement with
the standard model prediction.
