Must know vector fields

Exploding vector field

VectorPlot[{x,y},
             {x, -6,  6}, {y, -6, 6}, 
VectorPoints->30,
	 Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

As you can see the default setting of this command does not give satisfactory results. Below is a modified version in which we specify scaling of vectors in a vector field.

VectorPlot[{x,y},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 35,
 VectorScale-> Medium, Axes->True, 
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{-x,-y},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 25,
 VectorScale-> {Small,Scaled[1],(#&)}, Axes->True, Frame->True,
		PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{(x/(Sqrt[(x^2)+(y^2)])),(y/(Sqrt[(x^2)+(y^2)]))},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 30,
 VectorScale-> Small, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

One component constant

VectorPlot[{x,0},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 20,
 VectorScale-> Small, Axes-> True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{0,x},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 20,
 VectorScale-> Small, Axes-> True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{0,y},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 20,
 VectorScale-> Small, Axes-> True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{y,0},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 20,
 VectorScale-> Small, Axes-> True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{x,1},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 20,
 VectorScale-> Medium, Axes-> True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{1,x},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 20,
 VectorScale-> Small, Axes-> True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

Rotational vector field

VectorPlot[{-y,x},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 30,
 VectorScale-> Medium, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{((-y)/(Sqrt[x^2+y^2])),(x/(Sqrt[x^2+y^2]))},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 30,
 VectorScale-> Small, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{y,-x},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 30,
 VectorScale-> Medium, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

Hyperbolic vector field

VectorPlot[{y,x},
             {x, -6,  6}, {y, -6, 6}, 
	VectorPoints -> 30,
 VectorScale-> Medium, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

More vector fields

VectorPlot[{Sin[y],Sin[x]},
             {x, -2π, 2 π}, {y, -2π, 2π}, 
	VectorPoints -> 30,
 VectorScale-> Small, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{2(y-1),1},
             {x, -5, 5}, {y, -5,5}, 
	VectorPoints -> 30,
 VectorScale->{Medium,Scaled[1],Automatic}, Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

VectorPlot[{Sin[y π],Sin[x π]},{x,-2,2},{y,-2,2},VectorPoints->40,VectorScale->Small,Axes->True,Frame->True,PlotRange->{{-2,2},{-2,2}}]

VectorPlot[{-y,x},{x,-6,6},{y,-6,6},VectorPoints->30,VectorScale->Medium,Axes->True,Frame->True,PlotRange->{{-5,5},{-5,5}},Epilog->{{Thickness[0.01`],Blue,Line[{{2,2},{-2,2}}],Arrowheads[{.075}],Arrow[{{2,2},{0,2}}]},{Thickness[0.01`],Green,Line[{{1,-1},{3,-3}}],Arrowheads[{.075}],Arrow[{{1,-1},{2.25,-2.25}}]}}]

VectorPlot[{(x^2)-(y^2),2 x y},
             {x, -6,  6}, {y, -6, 6}, 
VectorPoints->30, VectorScale->{Medium,Scaled[1.5],Automatic}, StreamPoints->30,
	 Axes->True, Frame->True,
	PlotRange->{{-5,5},{-5,5}}]

It turns out that the flow lines are given by the parametric equations (r is a real number)

fS[t_,r_]:={((-4 r^2 t)/(4 r^2  t^2+1)),((2 r)/(4 r^2  t^2+1))}

To verify the above claim calculate

FullSimplify[D[fS[t,r],t]-{(x^2)-(y^2),2 x y}/.{x-> fS[t,r][[1]],y-> fS[t,r][[2]]}]

As the vector field strongly suggests the stream lines are circles centered at the y-axis and passing through the origin. Below is a verification of this fact.

FullSimplify[(fS[t,r]-{0,r}).(fS[t,r]-{0,r})]