Built-in Mathematica Symbol

VectorPlot

VectorPlot[{v_x, v_y}, {x, x_min, x_max}, {y, y_min, y_max}]
generates a vector plot of the vector field {v_x, v_y} as a function of x and y.

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

MORE INFORMATION

VectorPlot by default shows vectors from the vector field at a regular grid of positions.

VectorPlot omits any vectors for which the v_i etc. do not evaluate to real numbers.

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

VectorPlot 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.

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

AspectRatio	1	ratio of height to width
EvaluationMonitor	None	expression to evaluate at every function evaluation
Frame	True	whether to draw a frame around the plot
FrameTicks	Automatic	frame tick marks
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
VectorColorFunction	None	how to color vectors
VectorColorFunctionScaling	True	whether to scale the argument to VectorColorFunction
VectorPoints	Automatic	the number or placement of vectors to plot
VectorScale	Automatic	the scale and size of vectors to plot
VectorStyle	Automatic	how to draw vectors
WorkingPrecision	MachinePrecision	precision to use in internal computations

The arguments supplied to functions in RegionFunction and VectorColorFunction are x, y, v_x, v_y, Norm[{v_x, v_y}].

EXAMPLES

CLOSE ALL

Basic Examples (2)

Plot the vector field

:

Plot multiple vector fields:

Plot the vector field

:

In[1]:=

Out[1]=

Plot multiple vector fields:

In[1]:=

Out[1]=

Scope (5)

Plot a vector field with constant size arrows:

Plot a vector field with arrows colored according to the magnitude of the field:

Plot a vector field with arrows of specified relative size scaled by field magnitude:

Plot a vector field with vectors at random positions:

Plot a vector field showing vectors of constant length at random positions:

Options (88)

By default, the aspect ratio is 1:

Set the aspect ratio:

Use colored backgrounds:

Show where the vector field function is sampled:

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

Label the axes:

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 vectors only over certain quadrants:

Plot vectors 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 x 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:

Scale the vector length by the

component of the vector field:

Set the style for the displayed 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:

Vector styles with dots:

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 (6)

Gradient field of

over the unit square:

Hamiltonian vector field of

:

Polya field plot of

[more info]:

Polya field plot of

:

Characterize linear planar systems interactively:

Use vectors to indicate the stability of a limit cycle: