Plot domain in different regions!

Question

I would like to know how to plot in the same plot two king of equations. For example, I wanna plot the following field in the region that satisfies $x^{2}+y^{2} < 9$ the following equation to plot:

 StreamPlot[{x + y, -x + y}, {x, -8, 8}, {y, -8, 8}, 
 FrameLabel -> {"x(\[Lambda])", "y(\[Lambda])"}, 
 PlotTheme -> "Scientific"]

and in the region that satisfies $x^{2} +y^{2} \ge 9$ the following plot,

StreamPlot[{-x - y, x + y}, {x, -8, 8}, {y, -8, 8}, 
 FrameLabel -> {"x(\[Lambda])", "y(\[Lambda])"}, 
 PlotTheme -> "Scientific"]

How can I do that?

Thanks in Advance!

Answer 1

Show[
 StreamPlot[{x + y, -x + y}, {x, -8, 8}, {y, -8, 8},
  PlotTheme -> "Scientific",
  RegionFunction -> Function[{x, y}, x^2 + y^2 < 9]],
 StreamPlot[{-x - y, x + y}, {x, -8, 8}, {y, -8, 8},
  PlotTheme -> "Scientific",
  RegionFunction -> Function[{x, y}, x^2 + y^2 >= 9]],
 FrameLabel -> (Style[#, 14, Bold] & /@ {"x(λ)", 
     "y(λ)"})]