cylindertop=5;cylinderbottom=0;
cylinderradius=3;
cylinder=ParametricPlot3D[{cylinderradius*Cos[theta],cylinderradius*Sin[theta],z},{theta,0,2*Pi},{z,cylinderbottom,cylindertop},Mesh->False];
vectorfield=VectorPlot3D[{2x,2y,2z},{x,-3.,3.},{y,-3.,3.},{z,cylinderbottom,cylindertop-0.8}, VectorPoints->Coarse,VectorStyle->"Arrow3D"];

plot=Show[{cylinder,vectorfield},Axes->True,AxesLabel->{"X","Y","Z"},PlotRange->{{-3.6,3.6},{-3.6,3.6},{cylinderbottom,cylindertop}}]

Export["cylinder_flux_out.png", plot,ImageSize->300];Export["cylinder_flux_out_thumbnail.png", plot,ImageSize->200];