example 9.5   Mapping by w=z^2   p.217

z = x + I y
w = z^2

u = ComplexExpand[Re[w]]
v = ComplexExpand[Im[w]]

cn = Table[{u, v} /. x ->  c, {c, 0, 3, 1}]

zu1 = 
      Show[Table[
          ParametricPlot[Evaluate[cn[[n]]], {y, -5, 5}, 
            PlotRange ->  {{-20, 20}, {-20, 20}}, 
            AspectRatio ->  Automatic, 
            PlotStyle ->  Hue[0.9]], {n, 4}]]

kn = Table[{u, v} /. y ->  k, {k, 0, 3, 1}]

zu2 = 
      Show[Table[
          ParametricPlot[Evaluate[kn[[n]]], {x, -5, 5}, 
            PlotRange ->  {{-20, 20}, {-20, 20}}, 
            AspectRatio ->  Automatic, 
            PlotStyle ->  Hue[0.7]], {n, 4}]]

Show[zu1, zu2]