Built-in Mathematica Symbol

StreamDensityPlot

StreamDensityPlot[{{v_x, v_y}, s}, {x, x_min, x_max}, {y, y_min, y_max}]
generates a stream plot of the vector field {v_x, v_y} as a function of x and y, superimposed on a background density plot of the scalar field s.

StreamDensityPlot[{v_x, v_y}, {x, x_min, x_max}, {y, y_min, y_max}]
takes the scalar field to be the norm of the vector field.

StreamDensityPlot[{{v_x, v_y}, {w_x, w_y}, ..., s}, {x, x_min, x_max}, {y, y_min, y_max}]
generates plots of several vector fields.

MORE INFORMATION

StreamDensityPlot plots streamlines that show the local direction of the vector field at every point.

StreamDensityPlot by default shows enough streamlines to achieve a roughly uniform density throughout the plot.

StreamDensityPlot does not show streamlines at any positions for which the v_i etc. do not evaluate to real numbers.

StreamDensityPlot treats the variables x and y as local, effectively using Block.

StreamDensityPlot has attribute HoldAll, and evaluates the v_i, etc. only after assigning specific numerical values to x and y.

In some cases it may be more efficient to use Evaluate to evaluate the v_i, etc. symbolically before specific numerical values are assigned to x and y.

StreamDensityPlot has the same options as Graphics, with the following additions and changes:

AspectRatio	1	ratio of height to width
BoundaryStyle	None	how to draw RegionFunction boundaries
BoxRatios	Automatic	effective 3D box ratios for simulated lighting
ColorFunction	Automatic	how to color background densities
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every function evaluation
Frame	True	whether to draw a frame around the plot
FrameTicks	Automatic	frame tick marks
LightingAngle	None	effective angle for simulated lighting
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed for the scalar field
Mesh	None	how many mesh lines to draw in the background
MeshFunctions	{#5&}	how to determine the placement of mesh lines
MeshShading	None	how to shade regions between mesh lines
MeshStyle	Automatic	the style of mesh lines
Method	Automatic	methods to use for the plot
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotRange	{Full,Full}	range of x, y values to include
PlotRangePadding	Automatic	how much to pad the range of values
RegionFunction	True&	determine what region to include
StreamColorFunction	None	how to color streamlines
StreamColorFunctionScaling	True	whether to scale the argument to StreamColorFunction
StreamPoints	Automatic	determine number, placement and closeness of streamlines
StreamScale	Automatic	determine sizes and segmenting of individual streamlines
StreamStyle	Automatic	how to draw streamlines
WorkingPrecision	MachinePrecision	precision to use in internal computations

The arguments supplied to functions in MeshFunctions, RegionFunction, ColorFunction and StreamColorFunction are x, y, v_x, v_y, s.

The default setting MeshFunctions->{#5&} draws mesh lines for the scalar field s.

EXAMPLES

CLOSE ALL

Basic Examples (3)

Plot the streamlines for a vector field with background based on field magnitude:

Plot the streamlines for a vector field with background based on a logarithm of field magnitude:

Plot the streamlines for two fields with a background based on the first field's magnitude:

Plot the streamlines for a vector field with background based on field magnitude:

In[1]:=

Out[1]=

Plot the streamlines for a vector field with background based on a logarithm of field magnitude:

In[1]:=

Out[1]=

Plot the streamlines for two fields with a background based on the first field's magnitude:

In[1]:=

Out[1]=

Scope (5)

Plot a vector field with streamlines placed with specified densities:

Plot the streamlines that go through a set of seed points:

Use both automatic and explicit seeding with styles for explicitly seeded streamlines:

Plot the streamlines with lines for the field ${-x^2+y-1,x-y^2+1}$ with a background colored by the magnitude of the field:

Plot the streamlines with a background colored by a scaling function:

Plot the streamlines with arrows colored according to field magnitude:

Plot the streamlines with lines colored according to field magnitude:

Plot the streamlines with background colored according to field magnitude:

Options (114)

By default, the aspect ratio is 1:

Set the aspect ratio:

Use colored backgrounds:

By default, region boundaries have no style:

Apply a variety of styles to region boundaries:

Color the field magnitude using Hue:

Color using Hue based on

:

Use any named color gradient from ColorData:

Use ColorData for predefined color gradients:

Specify a color function that blends two colors by the

coordinate:

Use ColorFunctionScaling->False to get unscaled values:

By default, scaled values are used:

Use ColorFunctionScaling->False to get unscaled values:

Use unscaled coordinates in the

direction and scaled coordinates in the

directions:

Explicitly specify the scaling for each color function argument:

Show where the vector field function is sampled:

Count the number of times the vector field function is evaluated:

Label the axes:

Vary the lighting angle from 0 (right of the plot) to

/2 (top of the plot):

Fix the lighting angle at 0 (right of the plot), and vary the altitude from 0 to

/2:

Refine the plot where it changes quickly:

By default, no mesh lines are displayed:

Show the final sampling mesh:

Use a specific number of mesh lines:

Specify specific mesh lines:

Use different styles for different mesh lines:

By default, mesh lines correspond to the magnitude of the field:

Use the

value as the mesh function:

Use mesh lines corresponding to fixed distances from the origin:

Use None to remove regions:

Styles are used cyclically:

Use indexed colors from ColorData cyclically:

Apply a variety of styles to the mesh lines:

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

The full plot range is used by default:

Specify an explict limit for both

and

ranges:

Specify an explicit range that applies to both

and

ranges:

Specify an explicit

range:

Specify an explicit

minimum range:

Specify an explicit

range:

Specify an explicit

maximum range:

Specify different

and

ranges:

Plot streamlines only over certain quadrants:

Plot streamlines only over regions where the field magnitude is above a given threshold:

Use any logical combination of conditions:

Color streamlines according to the norm of the vector field:

Use any named color gradient from ColorData:

Use ColorData for predefined color gradients:

Specify a color function that blends two colors by the

coordinate:

Use StreamColorFunctionScaling->False to get unscaled values:

By default, scaled values are used:

Use StreamColorFunctionScaling->False to get unscaled values:

Use unscaled coordinates in the

direction and scaled coordinates in the

directions:

Explicitly specify the scaling for each color function argument:

Specify a specific maximum number of streamlines:

Use symbolic names to specify the number of streamlines:

Use both automatic and explicit seeding with styles for explicitly seeded streamlines:

Specify the minimum distance between streamlines:

Specify the minimum distance between streamlines at the start and end of a streamline:

Control the maximum length that each streamline can have:

Create full streamlines without segmentation:

Use curves for streamlines:

Use symbolic names to control the lengths of streamlines:

Specify segment lengths:

Specify an explicit dashing pattern for streamlines:

Specify number of points rendered on each streamline segment:

Specify absolute aspect ratios relative to the longest line segment:

Specify relative aspect ratios relative to each line segment:

Scale the length of the arrows by the

coordinate:

Apply a variety of styles to the streamlines:

Specify a custom arrow head:

Set the style for multiple vector fields:

Use named styles:

Named arrow styles:

Named dot styles:

Named pointer styles:

Named dart styles:

Color the vectors according to their norm:

Use any named color gradient from ColorData:

Color the vectors according to their

value:

Use VectorColorFunctionScaling->False to get unscaled values:

By default, scaled values are used:

Use VectorColorFunctionScaling->False to get unscaled values:

Use unscaled coordinates in the

direction and scaled coordinates in the

directions:

Explicitly specify the scaling for each color function argument:

Use automatically determined vector points:

Use symbolic names to specify the set of field vectors:

Create a regular grid of field vectors with the same number of arrows for

and

:

Create a regular grid of field vectors with a different number of arrows for

and

:

Specify a list of points for showing field vectors:

Use a different number of field vectors on a regular grid:

The location for vectors is given in the middle of the drawn vector:

Use automatically determined vector scale:

Specify the relative size of vectors as a fraction of the diagonal of the plot range:

Use symbolic names to control the size of vectors:

With the vector scaling function set to None, then all vectors have the same size:

Specify an absolute size for the arrowheads:

Specify arrowhead size relative to the length of the arrow:

Scale the vector length by the absolute value of the

coordinate:

By default, the vector length is scaled by the norm of the vector field:

Set the style for the vectors:

Set the style for multiple vector fields:

Plot the vector fields without arrowheads:

Plot the vector field with a named vector style:

Arrow vector styles:

Circular vector styles:

Dart vector styles:

Dot vector styles:

Pointer vector styles:

Use Arrowheads to specify an explicit style of the arrowheads:

Specify both arrow tail and head:

Graphics primitives without Arrowheads are scaled based on the vector scale:

Change the scaling using the VectorScale option:

Applications (1)

Show streamlines superimposed on the Hamiltonian for a 2D Hamiltonian system:

Check how closely NDSolve conserves the invariant:

Neat Examples (2)

Plot streamlines with the divergence as the background density:

Plot streamlines with the absolute value of the vorticity as the background density: