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I want to plot some flow lines for a certain vector field, but Mathematica doesn't seem to plot the entire flow line, only a small piece of it. How can I fix this?

VectorPlot[{Re[Sqrt[x + I y]], Im[Sqrt[x + I y]]}, 
{x, -1, 1}, 
{y, -1, 1}, 
VectorPoints -> Fine,  
StreamPoints -> Coarse, 
StreamStyle -> Red, 
StreamScale -> None]

enter image description here

Update:

I feel dumb for not realizing the discontinuity, but I'll mention that what I really want are flow lines of slope fields, not vector fields. I didn't see how to do this in Mathematica "directly," and hence used VectorPlot. How can I plot the lines of a slope field?

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It plots from -1 to 1 in each direction. What do you want it to do? –  bill s Mar 27 '13 at 14:39
1  
If you mean the discontinuity along the negative x-axis, it's because the vector field is discontinuous and changes direction. –  Michael E2 Mar 27 '13 at 15:13
    
Chris, please register an account now so you can edit and manage your posts. –  Szabolcs Mar 27 '13 at 15:29
    
Could you explain what a "slope field" is? Is it the same as a covector field (a one-form)? Or a section of the circle bundle associated with the tangent bundle? Or a section of that bundle with opposite directions identified at each point? –  whuber Mar 27 '13 at 18:54
1  
@whuber: The last option you mentioned, so there's no notion of forwards/backwards direction, just of tangency. –  user62206 Mar 28 '13 at 17:38

2 Answers 2

up vote 4 down vote accepted

You could specify seed-points for streamlines directly using StreamPoints: only those stremlines will be plotted which pass through these points. Here I set up two point sets: one at $y = 1$ and another at $y = -1$. These share identical $x$ coordinates so they meet at the discontinuity at $y = 0$. One can control the streamline lenghts by the third argument to StremPoints (second argument controls the minimal distance between streamlines).

points = Join[{#, -1.} & /@ Range[-1, 1, .2], {#, 1.} & /@ Range[-1, 1, .2]];
VectorPlot[{Re[Sqrt[x + I y]], Im[Sqrt[x + I y]]}, {x, -1, 1}, {y, -1,
   1}, VectorPoints -> Fine, StreamPoints -> {points, Automatic, 2}, 
 StreamStyle -> Red, StreamScale -> None]

Mathematica graphics

Here is a small Manipulate interface to get a grip on how the streampoints are positioned and how length and distance can be controlled. Here a circular layout is used for the streamline-points to be able to capture some missing lines (thanks whuber!).

Manipulate[
 points = Table[{Cos@t, Sin@t}*r, {t, 0., 2. \[Pi], d}];
 VectorPlot[{Re[Sqrt[x + I y]], Im[Sqrt[x + I y]]}, {x, -1, 
   1}, {y, -1, 1}, VectorPoints -> Fine, 
  StreamPoints -> {points, Automatic, l}, StreamStyle -> Red, 
  StreamScale -> None, 
  Epilog -> {Green, PointSize@Large, Point@points}],
 {{d, \[Pi]/20., "point spacing"}, 0, 1, \[Pi]/40, 
  Appearance -> "Labeled"},
 {{l, 2, "stream length"}, 0, 2, Appearance -> "Labeled"},
 {{r, 1, "radius"}, 0, 1, Appearance -> "Labeled"}
 ]

Mathematica graphics

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1  
+1 How would you cope with the streamlines for positive $x$--the ones that ought to converge at the origin but instead are terminated within a distance of about $0.1$ of it? –  whuber Mar 27 '13 at 18:49
    
Thanks! This is exactly what I needed. –  user62206 Mar 28 '13 at 18:58

For a slope field $dy/dx = f(y,x)$, I suppose you have to make do with something like a vector field of the form {1, f[x,y]} or Normalize@{1, f[x,y]}. We might as well normalize the vector, since magnitude doesn't matter in a slope field. Ordinarily, the value of the slope field at a point indicates the tangent line to an integral curve without regard to a direction of motion. I don't think Mathematica can represent that (or at least not conveniently). You have to use a vector.

In your case, it seems better to treat $x$ as a function of $y$ and for $dx/dy = g(x,y)$, use Normalize@{g[x,y], 1}, where g[x,y] is the quotient of the x and y components of your vector field:

VectorPlot[
 Evaluate@Normalize@{Re[Sqrt[x + I y]]/Im[Sqrt[x + I y]], 1},
 {x, -1, 1}, {y, -1, 1},
 VectorPoints -> Fine, StreamPoints -> Coarse, 
 StreamStyle -> Red, StreamScale -> None, 
 VectorStyle -> Arrowheads[0]]

Vector field plot

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