xMin = -1;  (* Change the screen limits here *)
xMax = 1;
yMin = -1;
yMax = 1;
Print["For each of these linear vector fields, the divergence of F and the curl of F are constant functions.  The curl of F is a constant times k-hat, so curl_z F is that constant."]
Print["The flux of F out through the circle is the divergence of F times the area of the disk, and the circulation of F around the circle is curl_z F times the area of the disk."]
F[x_, y_ ] = {-y, x};  
Print["F(x, y) = ", F[x,y]]
Print["div F = 0, curl_z F = 2"]
VectorPlot[F[x,y], {x, xMin, xMax}, {y, yMin, yMax}, FrameLabel ->{x,y},VectorScale -> Automatic, Prolog -> Circle[]]
F[x_, y_ ] = {-x, y};  
Print["F(x, y) = ", F[x,y]]
Print["div F = 0, curl_z F = 0"]
VectorPlot[F[x,y], {x, xMin, xMax}, {y, yMin, yMax}, FrameLabel ->{x,y},VectorScale -> Automatic, Prolog -> Circle[]]
F[x_, y_ ] = {x, y};  
Print["F(x, y) = ", F[x,y]]
Print["div F = 2, curl_z F = 0"]
VectorPlot[F[x,y], {x, xMin, xMax}, {y, yMin, yMax}, FrameLabel ->{x,y},VectorScale -> Automatic, Prolog -> Circle[]]
F[x_, y_ ] = {y, 0};  
Print["F(x, y) = ", F[x,y]]
Print["div F = 0, curl_z F = -1"]
VectorPlot[F[x,y], {x, xMin, xMax}, {y, yMin, yMax}, FrameLabel ->{x,y},VectorScale -> Automatic, Prolog -> Circle[]]
F[x_, y_ ] = {-x -y, x-y};  
Print["F(x, y) = ", F[x,y]]
Print["div F = -2, curl_z F = 2"]
VectorPlot[F[x,y], {x, xMin, xMax}, {y, yMin, yMax}, FrameLabel ->{x,y},VectorScale -> Automatic, Prolog -> Circle[]]
F[x_, y_ ] = {2, 1};  
Print["F(x, y) = ", F[x,y]]
Print["div F = 0, curl_z F = 0"]
VectorPlot[F[x,y], {x, xMin, xMax}, {y, yMin, yMax}, FrameLabel ->{x,y},VectorScale -> Automatic, Prolog -> Circle[]]