Streamlines of a Vector Field

The vectors represent the flow of the fluid at the tail of the vector and the length of the vector represents the speed of the flow and the direction of the vector represents the direction of the flow.
This should make you think tangent lines!

In fact, if every vector of a vector field F(x,y,z) is tangent to a curve passing through the point, then the curve is a streamline (flow line, line of force) of the vector field.


  F[x_,y_]:={x^2,2y}


  vf=PlotVectorField[F[x,y],{x,-2,2},{y,-2,2},VectorHeads->True];


  stream[t_,c_]:={t,c Exp[-2/t]}


  sp=ParametricPlot[{stream[t,.3],stream[-t,.3]},{t,.9,2},
  PlotStyle->{RGBColor[1,0,0]}];


  Show[sp,vf];

How do you find streamlines? Solve differential equations: find the curve given a point and the tangent.
If (x(t), y(t)) is a streamline, then (x'(t), y'(t)) is tangent to the curve, but so is F. This implies
(x'(t), y'(t))=k(f1,f2). If you are given a point, these initial value problems can be solved (maybe!)
In the example above,
x'(t)=kx^2
y'(t)=2ky
Combining yields
dy/dx=2y/x^2
and separation of variables yields
dy/2y=x^(-2)dx.
Now you should see where the streamline comes from.

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