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A 2D multiwavelength study of the ionized gas and stellar population in the Giant HII Region NGC 588

(ABRIDGED) We present an analysis of NGC588 based on IFS data with PMAS, together with Spitzer images at 8 mi and 24 mi. The extinction distribution in the optical shows complex structure, with maxima correlating in position with those of the emission at 24 mi and 8 mi. The Ha luminosity absorbed by the dust within the GHIIR reproduces the structure observed in the 24 mi image, supporting the use of this band as a tracer of recent star formation. A velocity difference of ~50 km/s was measured between the areas of high and low surface brightness, which would be expected if NGC588 were an evolved GHIIR. Line ratios used in the BPT diagnostic diagrams show a larger range of variation in the low surface brightness areas. The ranges are ~0.5 to 1.2 dex for [NII]/Ha, 0.7 to 1.7 dex for [SII]/Ha, and 0.3 to 0.5 dex for [OIII]/Hb. Ratios corresponding to large ionization parameter (U) are found between the peak of the emission in Hb and the main ionizing source decreasing radially outwards within the region. Differences between the integrated and local values of the U tracers can be as high as ~0.8 dex. [OII]/Hb and [OIII]/[OII] yield similar local values for U and consistent with those expected from the integrated spectrum of an HII region ionized by a single star. The ratio [SII]/Ha departs significantly from the range predicted by this scenario, indicating the complex ionization structure in GHIIRs. There is a significant scatter in derivations of Z using strong line tracers as a function of position, caused by variations in the degree of ionization. The scatter is smaller for N2O3 which points to this tracer as a better Z tracer than N2. The comparison between integrated and local line ratio values indicates that measurements of the line ratios of GHIIR in galaxies at distances >~25 Mpc may be dominated by the ionization conditions in their low surface brightness areas.