StreamPlot3D
StreamPlot3D[{v_{x},v_{y},v_{z}},{x,x_{min},x_{max}},{y,y_{min},y_{max}},{z,z_{min},z_{max}}]
plots streamlines for the vector field {v_{x},v_{y},v_{z}} as functions of x, y and z.
StreamPlot3D[{v_{x},v_{y},v_{z}},{x,y,z}∈reg]
takes the variables {x,y,z} to be in the geometric region reg.
Details and Options
 StreamPlot3D plots streamlines that show the local direction of the vector field at each point, effectively solving the system of differential equations , , and then plotting .
 By default, the direction of the vector field is indicated by the paths of the streamlines, and the magnitude is indicated by the color of the streamlines.
 StreamPlot3D by default shows enough streamlines to achieve a roughly uniform density throughout the plot and shows no background scalar field.
 StreamPlot3D treats the variables x, y and z as local, effectively using Block.
 StreamPlot3D has attribute HoldAll and evaluates the v_{i} etc. only after assigning specific numerical values to x, y and z. In some cases, it may be more efficient to use Evaluate to evaluate the v_{i} etc. symbolically first.
 StreamPlot3D has the same options as Graphics3D, with the following additions and changes:

BoxRatios {1,1,1} ratio of height to width EvaluationMonitor None expression to evaluate at every function evaluation Method Automatic methods to use for the plot PerformanceGoal $PerformanceGoal aspects of performance to try to optimize PlotLegends None legends to include PlotRange {Full,Full,Full} range of x, y, z values to include PlotRangePadding Automatic how much to pad the range of values PlotTheme $PlotTheme overall theme for the plot RegionBoundaryStyle Automatic how to style plot region boundaries RegionFunction True& determine what region to include ScalingFunctions None how to scale individual coordinates StreamColorFunction Automatic how to color streamlines StreamColorFunctionScaling True whether to scale the argument to StreamColorFunction StreamMarkers Automatic shape to use for streams StreamPoints Automatic the number or placement of streamlines StreamScale None how to scale the sizes of streamlines StreamStyle Automatic how to draw streamlines WorkingPrecision MachinePrecision precision to use in internal computations  The arguments supplied to functions in RegionFunction and ColorFunction are x,y,z,v_{x},v_{y},v_{z},Norm[{v_{x},v_{y},v_{z}}].
 Possible settings for StreamMarkers include:

"Arrow" lines with 2D arrowheads "Arrow3D" tubes with 3D arrowheads "Line" lines "Tube" tubes "Ribbon" flat ribbons "ArrowRibbon" ribbons with builtin arrowheads  With StreamScaleAutomatic and "arrow" stream markers, the streamlines are split into segments to make it easier to see the direction of the streamlines.
 Possible settings for StreamScale are:

Automatic automatically determine the streamline segments Full show the streamline as one piece Tiny,Small,Medium,Large named settings for how long the segments should be {len,npts,ratio} use explicit specification of streamline segmentation  The length len of streamline segments can be one of the following forms:

Automatic automatically determine the length None show the streamline as one piece Tiny,Small,Medium,Large use named segment lengths s use a length s that is a fraction of the graphic size  The number of points npts used to draw each segment can be Automatic or a specific number of points.
 The aspect ratio ratio specifies how wide the cross section of a streamline is relative to the streamline segment.
 Possible settings for ScalingFunctions include:

{s_{x},s_{y},s_{z}} scale x, y and z axes  Common builtin scaling functions s include:

"Log" log scale with automatic tick labeling "Log10" base10 log scale with powers of 10 for ticks "SignedLog" loglike scale that includes 0 and negative numbers "Reverse" reverse the coordinate direction "Infinite" infinite scale
Examples
open allclose allBasic Examples (4)
Scope (12)
Sampling (3)
Presentation (9)
Streamlines are drawn as lines by default:
Use 3D tubes for the streamlines:
Use "arrow" versions of the stream markers to indicate the direction of flow along the streamlines:
Ribbons are turned into arrows by tapering the heads and notching the tails of the streamlines:
Use a single color for the streamlines:
Use a named color gradient for the streamlines:
Include a legend for the field magnitude:
Use StreamScale to split streamlines into multiple shorter line segments:
Increase the number of points in each segment and increase the marker aspect ratio:
Options (42)
BoxRatios (2)
PlotLegends (3)
RegionBoundaryStyle (4)
Show the region defined by a RegionFunction:
Use None to avoid showing the boundary:
RegionFunction (4)
Plot streams only where the field magnitude exceeds a given threshold:
Region functions depend, in general, on seven arguments:
Use RegionBoundaryStyleNone to avoid showing the boundary:
ScalingFunctions (1)
StreamColorFunction (4)
Color the streams by their norm:
Use any named color gradient from ColorData:
Color the streamlines according to their x value:
Use StreamColorFunctionScalingFalse to get unscaled values:
StreamColorFunctionScaling (2)
StreamMarkers (5)
StreamPoints (4)
StreamScale (9)
Segmented markers have default lengths, numbers of points and aspect ratios:
Modify the lengths of the segments:
Specify the number of sample points in each segment:
Modify the aspect ratios for the stream markers:
Make segmented markers continuous:
Break continuous markers into segments:
The aspect ratio controls the thickness of ribbons and tubes:
Increase the width of the ribbons and tubes:
StreamStyle (3)
Change the appearance of the streamlines:
StreamColorFunction takes precedence over StreamStyle:
Use StreamColorFunctionNone to specify a streamline color with StreamStyle:
Applications (10)
Basic Applications (1)
Fluid Flow (3)
Consider Stokes flow for a point force of the form , where is a constant vector and is a Dirac delta function. For example, a force pointing down:
Define the fluid velocity vector , the pressure and the viscosity :
Confirm that the equations for Stokes flow are satisfied so that and :
Plot streamlines for the flow:
Visualize Stokes flow around a unit sphere. Define the fluid velocity vector , the pressure , the viscosity and the farfield fluid speed :
Confirm that the equations for Stokes flow are satisfied so that and :
Plot streamlines for the flow:
Plot the pressure for the flow:
Specify the Navier–Stokes equations for a fluid through a pipe with a bulge:
Specify the geometry for the flow:
Specify the boundary conditions for flow from left to right:
Solve for the flow velocities and pressure:
Electrical Systems (1)
Visualize electric field lines for a dipole:
The streamlines appear to have uniform color because of the extremely rapid change in the vector field norm near the point charges at . Bounding the magnitude of the vector field norm with a region function makes the colors visible:
Add spheres to indicate the positive (red) and negative (black) point charges:
Arrows can be used to provide more information, but the colors change because arrow markers are colored by the field magnitude at the tip of the arrow:
Use a custom StreamColorFunction to exert more control over the colors:
Miscellaneous (5)
Properties & Relations (9)
Use VectorPlot3D to visualize a field with discrete arrows:
Use ListStreamPlot3D or ListVectorPlot3D to generate plots based on data:
Use StreamPlot to plot streamlines of 2D vector fields:
Use VectorPlot to plot with vectors instead of streamlines:
Use StreamDensityPlot to add a density plot of the scalar field:
Use VectorDensityPlot to plot with arrows instead of streamlines:
Use LineIntegralConvolutionPlot to plot the line integral convolution of a vector field:
Use VectorDisplacementPlot to visualize the deformation of a region associated with a displacement vector field:
Use ListVectorDisplacementPlot to visualize the same deformation based on data:
Plot vectors along surfaces with SliceVectorPlot3D:
Use VectorDisplacementPlot3D to visualize the deformation of a 3D region associated with a displacement vector field:
Use ListVectorDisplacementPlot3D to visualize the same deformation based on data:
Use ComplexVectorPlot or ComplexStreamPlot to visualize a complex function of a complex variable as a vector field or with streamlines:
Use GeoVectorPlot to plot vectors on a map:
Use GeoStreamPlot to plot streamlines instead of vectors:
Possible Issues (3)
Tube StreamMarkers can be distorted by the BoxRatios:
Carefully adjusting the BoxRatios can eliminate the tube distortion:
The colors of "Arrow" and "Arrow3D" stream markers are determined at the tip of the arrow, which can result in inconsistent colors for long arrows:
Text
Wolfram Research (2021), StreamPlot3D, Wolfram Language function, https://reference.wolfram.com/language/ref/StreamPlot3D.html (updated 2022).
CMS
Wolfram Language. 2021. "StreamPlot3D." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/StreamPlot3D.html.
APA
Wolfram Language. (2021). StreamPlot3D. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StreamPlot3D.html