  We explore scaling relations between the physical properties of spatially
resolved regions within the galaxies that emerge in the Evolution and Assembly
of GaLaxies and their Environments (EAGLE) hydrodynamical, cosmological
simulations. Using 1 kpc-scale spaxels, we compute the relationships between
the star formation rate and stellar mass surface densities, i.e. the spatially
resolved star-forming main sequence (rSFMS), and between the gas metallicity
and the stellar mass surface density, i.e. the spatially resolved
mass-metallicity relation (rMZR). We compare to observed relations derived from
integral field unit surveys and imaging of galaxies. EAGLE reproduces the slope
of the local ($z\approx0.1$) rSFMS well, but with a $\approx-0.15$ dex offset,
close to that found for the galaxy-integrated relation. The shape of the rMZR
agrees reasonably well with observations, replicating the characteristic
turnover at high surface density, which we show is due to AGN feedback. The
residuals of the rSFMS and rMZR are negatively (positively) correlated at low
(high) surface density. The rSFMS becomes shallower as the simulation evolves
from $z=2$ to 0.1, a manifestation of inside-out galaxy formation. The shape of
the rMZR also exhibits dramatic evolution, from a convex profile at $z=2$ to
the observed concave profile at $z=0.1$, such that the gas in regions of high
stellar density is more enriched at higher redshift. The redshift independence
of the relationship between the galaxy-wide gas fraction and metallicity in
EAGLE galaxies is not preserved on 1 kpc scales, implying that chemical
evolution is non-local due to the transport of gas and metals within galaxies.
