--- title: "VectorPlot" language: "en" type: "Symbol" summary: "VectorPlot[{vx, vy}, {x, xmin, xmax}, {y, ymin, ymax}] generates a vector plot of the vector field {vx, vy} as a function of x and y. VectorPlot[{{vx, vy}, {wx, wy}, ...}, {x, xmin, xmax}, {y, ymin, ymax}] plots several vector fields. VectorPlot[..., {x, y} \\[Element] reg] takes the variables {x, y} to be in the geometric region reg." keywords: - quiver - quiver plot - hedgehog field - vector field plot - slope plot - phase portrait canonical_url: "https://reference.wolfram.com/language/ref/VectorPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Vector Visualization" link: "https://reference.wolfram.com/language/guide/VectorVisualization.en.md" - title: "Function Visualization" link: "https://reference.wolfram.com/language/guide/FunctionVisualization.en.md" - title: "Vector Analysis" link: "https://reference.wolfram.com/language/guide/VectorAnalysis.en.md" - title: "Solvers over Regions" link: "https://reference.wolfram.com/language/guide/GeometricSolvers.en.md" - title: "Differential Equations" link: "https://reference.wolfram.com/language/guide/DifferentialEquations.en.md" related_functions: - title: "VectorDensityPlot" link: "https://reference.wolfram.com/language/ref/VectorDensityPlot.en.md" - title: "StreamPlot" link: "https://reference.wolfram.com/language/ref/StreamPlot.en.md" - title: "ListVectorPlot" link: "https://reference.wolfram.com/language/ref/ListVectorPlot.en.md" - title: "LineIntegralConvolutionPlot" link: "https://reference.wolfram.com/language/ref/LineIntegralConvolutionPlot.en.md" - title: "ParametricPlot" link: "https://reference.wolfram.com/language/ref/ParametricPlot.en.md" - title: "ContourPlot" link: "https://reference.wolfram.com/language/ref/ContourPlot.en.md" - title: "DensityPlot" link: "https://reference.wolfram.com/language/ref/DensityPlot.en.md" - title: "Plot3D" link: "https://reference.wolfram.com/language/ref/Plot3D.en.md" - title: "VectorPlot3D" link: "https://reference.wolfram.com/language/ref/VectorPlot3D.en.md" - title: "SliceVectorPlot3D" link: "https://reference.wolfram.com/language/ref/SliceVectorPlot3D.en.md" - title: "StreamPlot3D" link: "https://reference.wolfram.com/language/ref/StreamPlot3D.en.md" - title: "GeoVectorPlot" link: "https://reference.wolfram.com/language/ref/GeoVectorPlot.en.md" - title: "ComplexVectorPlot" link: "https://reference.wolfram.com/language/ref/ComplexVectorPlot.en.md" --- # VectorPlot VectorPlot[{vx, vy}, {x, xmin, xmax}, {y, ymin, ymax}] generates a vector plot of the vector field {vx, vy} as a function of x and y. VectorPlot[{{vx, vy}, {wx, wy}, …}, {x, xmin, xmax}, {y, ymin, ymax}] plots several vector fields. VectorPlot[…, {x, y}∈reg] takes the variables {x, y} to be in the geometric region reg. ## Details and Options * ``VectorPlot`` is also known as field plot, quiver plot and direction plot. * ``VectorPlot`` displays a vector field by drawing arrows normalized to a fixed length. The arrows are colored by default according to the magnitude $s=\left\| \left\{v_x,v_y\right\}\right\|$ of the vector field. * The plot visualizes the set $\left\{\left.\left\{v_x,v_y\right\}\right|\{x,y\}\in \text{reg}\right\}$. [image] * ``VectorPlot`` omits any arrows for which the ``vi`` etc. do not evaluate to real numbers. * The region ``reg`` can be any ``RegionQ`` object in 2D. * ``VectorPlot`` treats the variables ``x`` and ``y`` as local, effectively using ``Block``. * ``VectorPlot`` has attribute ``HoldAll``, and evaluates the ``vi`` 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 ``vi`` etc. symbolically first. * ``VectorPlot`` has the same options as ``Graphics``, with the following additions and changes: [] | | | | | --------------------------- | ----------------- | ---------------------------------------------------- | | AspectRatio | 1 | ratio of height to width | | ClippingStyle | Automatic | how to display arrows outside the vector range | | 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 | | PlotLayout | Automatic | how to position fields | | PlotLegends | None | legends to include | | PlotRange | {Full, Full} | range of x, y 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 | | RegionFillingStyle | Automatic | how to style plot region interiors | | RegionFunction | (True&) | determine what region to include | | ScalingFunctions | None | how to scale individual coordinates | | VectorAspectRatio | Automatic | width to length ratio for arrows | | VectorColorFunction | Automatic | how to color arrows | | VectorColorFunctionScaling | True | whether to scale the argument to VectorColorFunction | | VectorMarkers | Automatic | shape to use for arrows | | VectorPoints | Automatic | the number or placement of arrows | | VectorRange | Automatic | range of vector lengths to show | | VectorScaling | None | how to scale the sizes of arrows | | VectorSizes | Automatic | sizes of displayed arrows | | VectorStyle | Automatic | how to style arrows | | WorkingPrecision | MachinePrecision | precision to use in internal computations | * The individual arrows are scaled to fit inside bounding circles around each point. [image] * ``VectorScaling`` scales the magnitudes of the vectors into the range of arrow sizes ``smin`` to ``smax`` given by ``VectorSizes``. * ``VectorScaling -> Automatic`` will scale the arrow lengths depending on the vector magnitudes: [image] * Possible settings for ``PlotLayout`` that show single groups of vectors in multiple plot panels include: | | | | ------------------------------------- | ---------------------------------------------------- | | "Column" | use separate groups of vectors in a column of panels | | "Row" | use separate groups of vectors in a row of panels | | {"Column", k}, {"Row", k} | use k columns or rows | | {"Column", UpTo[k]}, {"Row", UpTo[k]} | use at most k columns or rows | [image] * The arrow markers given by ``VectorMarkers`` are drawn inside a box whose width and length are in the proportion $w/l$ specified by ``VectorAspectRatio``. * Common markers include: | | | | | ------- | --------- | ------------------------------------------- | | [image] | "Segment" | line segment aligned in the field direction | | [image] | "PinDart" | pin dart aligned along the field | | [image] | "Dart" | dart-shaped marker | | [image] | "Drop" | drop-shaped marker | * ``VectorColorFunction -> None`` will draw the arrows with the style specified by ``VectorStyle``. * The arguments supplied to functions in ``RegionFunction`` and ``VectorColorFunction`` are ``x``, ``y``, ``vx``, ``vy``, ``Norm[{vx, vy}]``. * Possible settings for ``ScalingFunctions`` are: {Subscript[``s``, ``x``], Subscript[``s``, ``y``]} scale ``x`` and ``y`` axes * Common built-in scaling functions ``s`` include: | | | | | ----------- | ------- | --------------------------------------------------- | | "Log" | [image] | log scale with automatic tick labeling | | "Log10" | [image] | base-10 log scale with powers of 10 for ticks | | "SignedLog" | [image] | log-like scale that includes 0 and negative numbers | | "Reverse" | [image] | reverse the coordinate direction | | "Infinite" | [image] | infinite scale | ### List of all options | | | | | -------------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | 1 | ratio of height to width | | Axes | False | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | ClippingStyle | Automatic | how to display arrows outside the vector range | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | Epilog | {} | primitives rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | FormatType | TraditionalForm | the default format type for text | | Frame | True | whether to draw a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame tick marks | | FrameTicksStyle | {} | style specifications for frame ticks | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | Method | Automatic | methods to use for the plot | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotLayout | Automatic | how to position fields | | PlotLegends | None | legends to include | | PlotRange | {Full, Full} | range of x, y values to include | | PlotRangeClipping | False | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotTheme | \$PlotTheme | overall theme for the plot | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RegionBoundaryStyle | Automatic | how to style plot region boundaries | | RegionFillingStyle | Automatic | how to style plot region interiors | | RegionFunction | (True&) | determine what region to include | | RotateLabel | True | whether to rotate y labels on the frame | | ScalingFunctions | None | how to scale individual coordinates | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | | VectorAspectRatio | Automatic | width to length ratio for arrows | | VectorColorFunction | Automatic | how to color arrows | | VectorColorFunctionScaling | True | whether to scale the argument to VectorColorFunction | | VectorMarkers | Automatic | shape to use for arrows | | VectorPoints | Automatic | the number or placement of arrows | | VectorRange | Automatic | range of vector lengths to show | | VectorScaling | None | how to scale the sizes of arrows | | VectorSizes | Automatic | sizes of displayed arrows | | VectorStyle | Automatic | how to style arrows | | WorkingPrecision | MachinePrecision | precision to use in internal computations | ## Examples (147) ### Basic Examples (5) Plot the vector field $\{y,-x\}$ with color indicating the vector magnitude: ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` --- Include a legend for the vector magnitudes: ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Use a drop-shaped marker to represent the vectors: ```wl In[1]:= VectorPlot[{x + y, y - x}, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> "Drop"] Out[1]= [image] ``` --- Use a multi-panel layout to show multiple vector fields at the same time: ```wl In[1]:= VectorPlot[{{x + y, y - x}, {y, x + y}}, {x, -3, 3}, {y, -3, 3}, PlotLayout -> "Row"] Out[1]= [image] ``` --- Use a log scale for the ``x`` axis: ```wl In[1]:= VectorPlot[{x, y}, {x, 1, 100}, {y, 1, 100}, ScalingFunctions -> {"Log", None}] Out[1]= [image] ``` Reverse the ``y`` scale so it increases toward the bottom: ```wl In[2]:= VectorPlot[{x, y}, {x, 1, 100}, {y, 1, 100}, ScalingFunctions -> {None, "Reverse"}] Out[2]= [image] ``` ### Scope (24) #### Sampling (9) Plot a vector field with vectors placed with specified densities: ```wl In[1]:= {VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> Coarse], VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> Fine]} Out[1]= {[image], [image]} ``` --- Sample the vector field on a regular grid of points: ```wl In[1]:= VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> "Regular"] Out[1]= [image] ``` Sample the vector field on an irregular mesh: ```wl In[2]:= VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> "Mesh"] Out[2]= [image] ``` --- Specify how many vector points to use in each direction: ```wl In[1]:= VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> 10] Out[1]= [image] ``` --- Plot the vectors that go through a set of seed points: ```wl In[1]:= points = RandomReal[{-2, 2}, {30, 2}]; In[2]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> points, Epilog -> {Black, PointSize[Medium], Point[points]}] Out[2]= [image] ``` --- Plot vectors over a specified region: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, RegionFunction -> Function[{x, y}, 1 ≤ x^2 + y^2 ≤ 9]] Out[1]= [image] ``` --- The domain may be specified by a region: ```wl In[1]:= VectorPlot[{y, -x}, {x, y}∈Disk[]] Out[1]= [image] ``` --- The domain may be specified by a ``MeshRegion`` : ```wl In[1]:= \[ScriptCapitalD] = MeshRegion[{{0, 0}, {5, -2}, {3, 0}, {5, 2}}, Polygon[{1, 2, 3, 4}]]; In[2]:= VectorPlot[{Sin[x], Cos[y]}, {x, y}∈\[ScriptCapitalD]] Out[2]= [image] ``` --- Plot multiple vector fields: ```wl In[1]:= VectorPlot[{{x, y}, {y, -x}}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None] Out[1]= [image] ``` --- Use ``Evaluate`` to evaluate the vector field symbolically before numeric assignment: ```wl In[1]:= VectorPlot[Evaluate[D[Sin[x y], {{x, y}}]], {x, 0, 1}, {y, 0, 1}] Out[1]= [image] ``` #### Presentation (15) Plot a vector field with automatically scaled arrows: ```wl In[1]:= VectorPlot[{x, -y ^ 2 + x + 2}, {x, -3, 3}, {y, -3, 3}, VectorScaling -> Automatic] Out[1]= [image] ``` --- Use a single color for the arrows: ```wl In[1]:= VectorPlot[{x, -y ^ 2 + x + 2}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None] Out[1]= [image] ``` --- Plot a vector field with arrows of specified size: ```wl In[1]:= Table[VectorPlot[{x, -y ^ 2 + x + 2}, {x, -3, 3}, {y, -3, 3}, PlotLabel -> s, VectorSizes -> s], {s, {Tiny, Small, Medium, Large}}] Out[1]= {[image], [image], [image], [image]} ``` --- Draw the arrows starting from the sample points: ```wl In[1]:= pts = Tuples[Range[-3, 3], 2]; In[2]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> Placed["Arrow", "Start"], VectorPoints -> pts, Epilog -> {AbsolutePointSize[5], Black, Point[pts]}] Out[2]= [image] ``` --- Draw the arrows without the arrowheads: ```wl In[1]:= VectorPlot[{x, -y}, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> "Segment"] Out[1]= [image] ``` --- Change the overall shape of the markers: ```wl In[1]:= VectorPlot[{x, -y ^ 2 + x + 2}, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> "Drop", VectorAspectRatio -> 0.5] Out[1]= [image] ``` --- Change the default color function: ```wl In[1]:= VectorPlot[{x, -y ^ 2 + x + 2}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> "AvocadoColors"] Out[1]= [image] ``` --- Include a legend: ```wl In[1]:= VectorPlot[{x, -y ^ 2 + x + 2}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> "AvocadoColors", PlotLegends -> Automatic] Out[1]= [image] ``` --- Use no color: ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None] Out[1]= [image] ``` --- Show multiple functions as densities in separate panels: ```wl In[1]:= VectorPlot[{{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {-(1 + x - y ^ 2), -1 - x ^ 2 + y}}, {x, -3, 3}, {y, -3, 3}, PlotLayout -> "Row"] Out[1]= [image] ``` Use a column instead of a row: ```wl In[2]:= VectorPlot[{{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {-(1 + x - y ^ 2), -1 - x ^ 2 + y}}, {x, -3, 3}, {y, -3, 3}, PlotLayout -> "Column"] Out[2]= [image] ``` --- Set the style for multiple vector fields: ```wl In[1]:= VectorPlot[{{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {-(1 + x - y ^ 2), -1 - x ^ 2 + y}}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None, VectorStyle -> {{Orange, "Drop"}, {Green, "Dart"}}] Out[1]= [image] ``` --- ``VectorColorFunction`` takes precedence over colors in ``VectorStyle`` : ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, VectorStyle -> Orange] Out[1]= [image] ``` Set ``VectorColorFunction -> None`` to specify colors with ``VectorStyle`` : ```wl In[2]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None, VectorStyle -> Orange] Out[2]= [image] ``` --- Include a legend for multiple vector fields: ```wl In[1]:= VectorPlot[{{y, -x}, {x, y}}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> {"one", "two"}, VectorColorFunction -> None] Out[1]= [image] ``` --- Use a theme with simple ticks and grid lines: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotTheme -> "Business"] Out[1]= [image] ``` --- Use a log scale for the ``x`` axis: ```wl In[1]:= VectorPlot[{x, y}, {x, 1, 100}, {y, 1, 100}, ScalingFunctions -> {"Log", None}] Out[1]= [image] ``` Reverse the ``y`` scale so it increases toward the bottom: ```wl In[2]:= VectorPlot[{x, y}, {x, 1, 100}, {y, 1, 100}, ScalingFunctions -> {None, "Reverse"}] Out[2]= [image] ``` ### Options (97) #### AspectRatio (4) By default, ``AspectRatio`` uses the same width and height: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -2, 2}] Out[1]= [image] ``` --- Use a numerical value to specify the height to width ratio: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -2, 2}, AspectRatio -> 1 / 2] Out[1]= [image] ``` --- ``AspectRatio -> Automatic`` determines the ratio from the plot ranges: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -2, 2}, AspectRatio -> Automatic] Out[1]= [image] ``` --- ``AspectRatio -> Full`` adjusts the height and width to tightly fit inside the image size: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, AspectRatio -> Full, ImageSize -> {300, 150}] Out[1]= [image] ``` #### Axes (4) By default, ``Axes`` are not drawn: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False] Out[1]= [image] ``` --- Use ``Axes -> True`` to turn on axes: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Use ``AxesOrigin`` to specify where the axes intersect: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True, AxesOrigin -> {1, 1}] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> {True, False}], VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (3) No axes labels are drawn by default: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True, AxesLabel -> y] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True, AxesLabel -> {x, y}] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Frame -> False, Axes -> True, AxesOrigin -> {-1, -1}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, 0, 3}, {y, 0, 3}, Frame -> False, Axes -> True, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, 0, 3}, {y, 0, 3}, Frame -> False, Axes -> True, AxesStyle -> {Red, Blue}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, 0, 3}, {y, 0, 3}, Frame -> False, Axes -> True, AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, 0, 3}, {y, 0, 3}, Frame -> False, Axes -> True, AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### Background (1) Use colored backgrounds: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, Background -> LightGreen] Out[1]= [image] ``` #### ClippingStyle (4) Vectors are clipped via ``VectorRange``. The default clipping style is automatically chosen: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> {0.2, 0.8}] Out[1]= [image] ``` --- Do not show clipped vectors: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> {0.2, 0.8}, ClippingStyle -> None] Out[1]= [image] ``` --- Specify the style for clipped vectors: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> {0.2, 0.8}, ClippingStyle -> LightGray] Out[1]= [image] ``` --- Color upper and lower clipped vectors differently: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> {0.2, 0.8}, ClippingStyle -> {LightGray, Cyan}] Out[1]= [image] ``` #### EvaluationMonitor (2) Show where the vector field function is sampled: ```wl In[1]:= ListPlot[Reap[VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, EvaluationMonitor :> Sow[{x, y}]]][[-1, 1]]] Out[1]= [image] ``` --- Count the number of times the vector field function is evaluated: ```wl In[1]:= Block[{k = 0}, VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, EvaluationMonitor :> k++];k] Out[1]= 250 ``` #### ImageSize (5) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Tiny], VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> 150], VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> {Automatic, 150}], VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> UpTo[200]], VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> {300, 200}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the image size: ```wl In[2]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, AspectRatio -> Full, ImageSize -> {300, 200}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, ImageSize -> {UpTo[150], UpTo[100]}], VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= {[image], [image]} ``` #### PlotLayout (2) Place each group of vectors in a separate panel using shared axes: ```wl In[1]:= VectorPlot[{{y, -x}, {x, y}}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Medium, PlotLayout -> "Column"] Out[1]= [image] ``` Use a row instead of a column: ```wl In[2]:= VectorPlot[{{y, -x}, {x, y}}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Medium, PlotLayout -> "Row"] Out[2]= [image] ``` --- Use multiple columns or rows: ```wl In[1]:= VectorPlot[{{y, -x}, {x, y}, {y, x + y}, {x, x - y}}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Medium, PlotLayout -> {"Column", 2}] Out[1]= [image] ``` Prefer full columns or rows: ```wl In[2]:= VectorPlot[{{y, -x}, {x, y}, {y, x + y}, {x, x - y}, {x + y, x}, {y - x, x + y}}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Medium, PlotLayout -> {"Column", UpTo[4]}] Out[2]= [image] In[3]:= VectorPlot[{{y, -x}, {x, y}, {y, x + y}, {x, x - y}, {x + y, x}, {y - x, x + y}}, {x, -3, 3}, {y, -3, 3}, ImageSize -> Medium, PlotLayout -> {"Column", 4}] Out[3]= [image] ``` #### PlotLegends (4) No legends are included by default: ```wl In[1]:= VectorPlot[{y, Exp[-x]}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` --- Include a legend to show the color range of vector norms: ```wl In[1]:= VectorPlot[{y, Exp[-x]}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Include a legend for multiple fields: ```wl In[1]:= VectorPlot[{{x, y}, {y, -x}}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> {"one", "two"}, VectorColorFunction -> None] Out[1]= [image] ``` --- Control the placement of the legend: ```wl In[1]:= VectorPlot[{y, Exp[-x]}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> Placed[Automatic, Below]] Out[1]= [image] ``` #### PlotRange (5) The full plot range is used by default: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotRange -> Full] Out[1]= [image] ``` --- Specify an explicit limit for both $x$ and $y$ ranges: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotRange -> 2] Out[1]= [image] ``` --- Specify an explicit $x$ range: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotRange -> {{0, 2}, Full}] Out[1]= [image] ``` --- Specify an explicit $x$ maximum range: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotRange -> {{Full, 4}, Full}] Out[1]= [image] ``` --- Specify different $x$ and $y$ ranges: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotRange -> {{-2, 1.5}, {-1, 2}}] Out[1]= [image] ``` #### PlotTheme (1) Use a theme with detailed ticks and axes: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotTheme -> "Scientific"] Out[1]= [image] ``` Add automatic ``GridLines`` : ```wl In[2]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotTheme -> "Scientific", GridLines -> Automatic] Out[2]= [image] ``` #### RegionBoundaryStyle (6) Show the region being plotted: ```wl In[1]:= VectorPlot[{x, y}, {x, y}∈Disk[]] Out[1]= [image] ``` --- Show the region defined by a region function: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, RegionFunction -> Function[{x, y}, x + y > 0]] Out[1]= [image] ``` --- The boundaries of full rectangular regions are not shown: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}] Out[1]= [image] ``` --- Use ``None`` to not show the boundary: ```wl In[1]:= VectorPlot[{x, y}, {x, y}∈Disk[], RegionBoundaryStyle -> None] Out[1]= [image] ``` Omit the interior filling as well: ```wl In[2]:= VectorPlot[{x, y}, {x, y}∈Disk[], RegionBoundaryStyle -> None, RegionFillingStyle -> None] Out[2]= [image] ``` --- Specify a style for the boundary: ```wl In[1]:= VectorPlot[{x, y}, {x, y}∈Disk[], RegionBoundaryStyle -> Red] Out[1]= [image] ``` --- Specify a style for full rectangular regions: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, RegionBoundaryStyle -> Red] Out[1]= [image] ``` #### RegionFillingStyle (6) Show the region being plotted: ```wl In[1]:= VectorPlot[{x, y}, {x, y}∈Disk[]] Out[1]= [image] ``` --- Show the region defined by a region function: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, RegionFunction -> Function[{x, y}, x + y > 0]] Out[1]= [image] ``` --- The interior of full rectangular regions are not shown: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}] Out[1]= [image] ``` --- Use ``None`` to not show the interior filling: ```wl In[1]:= VectorPlot[{x, y}, {x, y}∈Disk[], RegionFillingStyle -> None] Out[1]= [image] ``` Omit the boundary curve as well: ```wl In[2]:= VectorPlot[{x, y}, {x, y}∈Disk[], RegionBoundaryStyle -> None, RegionFillingStyle -> None] Out[2]= [image] ``` --- Specify a style for the interior filling: ```wl In[1]:= VectorPlot[{x, y}, {x, y}∈Disk[], RegionFillingStyle -> Gray] Out[1]= [image] ``` --- Specify a style for full rectangular region: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, RegionFillingStyle -> Gray] Out[1]= [image] ``` #### RegionFunction (4) Plot vectors only over a certain region: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, RegionFunction -> Function[{x, y, vx, vy, n}, 1 ≤ x^2 + y^2 ≤ 9]] Out[1]= [image] ``` --- Modify the method by which field points are generated to better reflect the boundary of the region: ```wl In[1]:= Table[VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, RegionFunction -> Function[{x, y, vx, vy, n}, 1 ≤ x^2 + y^2 ≤ 9], PlotLabel -> vp, VectorPoints -> vp], {vp, {"Hexagonal", "Regular", "Mesh", 25}}] Out[1]= [image] ``` --- Plot vectors only over regions where the field magnitude is above a given threshold: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, RegionFunction -> Function[{x, y, vx, vy, n}, n > 6]] Out[1]= [image] ``` --- Use any logical combination of conditions: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, RegionFunction -> Function[{x, y, vx, vy, n}, 4 < n < 9 || 0 < n < 2]] Out[1]= [image] ``` #### ScalingFunctions (2) Use a log scale for the ``x`` axis: ```wl In[1]:= VectorPlot[{x, y}, {x, 1, 100}, {y, 1, 10}, ScalingFunctions -> {"Log", None}] Out[1]= [image] ``` --- Reverse the ``y`` scale so it increases toward the bottom: ```wl In[1]:= VectorPlot[{x, y}, {x, 1, 100}, {y, 1, 100}, ScalingFunctions -> {None, "Reverse"}] Out[1]= [image] ``` #### VectorAspectRatio (2) The default aspect ratio for a vector marker is 1/4: ```wl In[1]:= VectorPlot[{y - x(1 - x^2 - y^2)^2, -x - y(1 - x^2 - y^2)^2}, {x, -3, 3}, {y, -3, 3}, VectorAspectRatio -> 1 / 4] Out[1]= [image] ``` --- Increase the relative width of a vector marker: ```wl In[1]:= VectorPlot[{y - x(1 - x^2 - y^2)^2, -x - y(1 - x^2 - y^2)^2}, {x, -3, 3}, {y, -3, 3}, VectorAspectRatio -> 1 / 2] Out[1]= [image] ``` #### VectorColorFunction (4) Color the vectors according to their norms: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> Hue] Out[1]= [image] ``` --- Use any named color gradient from ``ColorData``: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> "Rainbow"] Out[1]= [image] ``` --- Color the vectors according to their $x$ values: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> Function[{x, y, vx, vy, n}, ColorData["ThermometerColors"][x]]] Out[1]= [image] ``` --- Use ``VectorColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> Function[{x, y, vx, vy, n}, ColorData[22][Round[n]]], VectorColorFunctionScaling -> False] Out[1]= [image] ``` #### VectorColorFunctionScaling (4) By default, scaled values are used: ```wl In[1]:= VectorPlot[{y, -x }, {x, -4, 4}, {y, -4, 4}, VectorColorFunction -> Hue] Out[1]= [image] ``` --- Use ``VectorColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= VectorPlot[{y, -x }, {x, -4, 4}, {y, -4, 4}, VectorColorFunction -> Hue, VectorColorFunctionScaling -> False] Out[1]= [image] ``` --- Use unscaled coordinates in the $x$ direction and scaled coordinates in the $y$ direction: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> Function[{x, y, vx, vy, n}, Hue[x, y, 1]], VectorColorFunctionScaling -> {False, True}] Out[1]= [image] ``` --- Explicitly specify the scaling for each color function argument: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> Function[{x, y, vx, vy, n}, Hue[n, y, 1]], VectorColorFunctionScaling -> {True, True, True, True, False}] Out[1]= [image] ``` #### VectorMarkers (4) Vectors are drawn as arrows by default: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` --- Use a named appearance to draw the vectors: ```wl In[1]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> "Drop"] Out[1]= [image] ``` --- Use different markers for different vector fields: ```wl In[1]:= VectorPlot[{{y, -x }, {x, y}}, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> {"Segment", "Dart"}] Out[1]= [image] ``` --- By default, markers are centered on vector points: ```wl In[1]:= pts = Tuples[Range[-3, 3], 2]; In[2]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorPoints -> pts, Epilog -> {AbsolutePointSize[5], Black, Point[pts]}] Out[2]= [image] ``` Start the vectors at the points: ```wl In[3]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> Placed["Arrow", "Start"], VectorPoints -> pts, Epilog -> {AbsolutePointSize[5], Black, Point[pts]}] Out[3]= [image] ``` End the vectors at the points: ```wl In[4]:= VectorPlot[{y, -x }, {x, -3, 3}, {y, -3, 3}, VectorMarkers -> Placed["Arrow", "End"], VectorPoints -> pts, Epilog -> {AbsolutePointSize[5], Black, Point[pts]}] Out[4]= [image] ``` #### VectorPoints (10) Use automatically determined vector points: ```wl In[1]:= VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> Automatic] Out[1]= [image] ``` --- Use symbolic names to specify the set of field vectors: ```wl In[1]:= Table[VectorPlot[{y, -x}, {x, -2, 2}, {y, -2, 2}, PlotLabel -> k, VectorPoints -> k], {k, {Fine, Coarse}}] Out[1]= {[image], [image]} ``` --- Create a hexagonal grid of field vectors with the same number of arrows for $x$ and $y$ : ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorPoints -> 5] Out[1]= [image] ``` --- Create a hexagonal grid of field vectors with a different number of arrows for $x$ and $y$ : ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorPoints -> {4, 7}] Out[1]= [image] ``` --- Specify a list of points for showing field vectors: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, VectorPoints -> RandomReal[{-3, 3}, {300, 2}]] Out[1]= [image] ``` --- Use a different number of field vectors on a hexagonal grid: ```wl In[1]:= Table[VectorPlot[{y ^ 2, x ^ 2 - y ^ 2}, {x, -3, 3}, {y, -3, 3}, PlotLabel -> n, VectorPoints -> n], {n, 5, 20, 5}] Out[1]= {[image], [image], [image], [image]} ``` --- The location for vectors is given in the middle of the drawn vector: ```wl In[1]:= points = {{-1, -1}, {-1, 1}, {0, 0}, {1, -1}, {1, 1}}; In[2]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> points, Epilog -> {Black, PointSize[Medium], Point[points]}] Out[2]= [image] ``` --- Use a rectangular mesh instead of a hexagonal mesh: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> "Regular"] Out[1]= [image] ``` --- Use a mesh generated from a triangularization of the region: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2}, {y, -2, 2}, VectorPoints -> "Mesh"] Out[1]= [image] ``` --- Use "Mesh" to better represent the boundary of a region: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2}, {y, -2, 2}, RegionFunction -> Function[{x, y}, x^2 + (y^2/4) ≥ 1], VectorPoints -> "Mesh"] Out[1]= [image] ``` #### VectorRange (4) The clipping of vectors with very small or very large magnitudes is done automatically: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> Automatic] Out[1]= [image] ``` --- Specify the range of vector norms: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> {0.1, 2}] Out[1]= [image] ``` --- Suppress the clipped vectors: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> {0.1, 2}, ClippingStyle -> None] Out[1]= [image] ``` --- Show all the vectors: ```wl In[1]:= VectorPlot[({x, y}/Norm[{x, y}]^3), {x, -3, 3}, {y, -3, 3}, VectorRange -> All] Out[1]= [image] ``` #### VectorScaling (2) Use automatically determined vector scales: ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorScaling -> Automatic] Out[1]= [image] ``` --- With the vector scaling function set to ``None``, all vectors have the same size: ```wl In[1]:= VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorScaling -> None] Out[1]= [image] ``` #### VectorSizes (2) The sizes of the displayed vectors are determined automatically: ```wl In[1]:= VectorPlot[{y - x(1 - x^2 - y^2)^2, -x - y(1 - x^2 - y^2)^2}, {x, -3, 3}, {y, -3, 3}, VectorScaling -> Automatic, VectorSizes -> Automatic] Out[1]= [image] ``` --- Specify the range of arrow lengths: ```wl In[1]:= VectorPlot[{y - x(1 - x^2 - y^2)^2, -x - y(1 - x^2 - y^2)^2}, {x, -3, 3}, {y, -3, 3}, VectorScaling -> Automatic, VectorSizes -> {0, 1}] Out[1]= [image] ``` #### VectorStyle (6) Set the style for the displayed vectors: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None, VectorStyle -> Red] Out[1]= [image] ``` --- Set the style for multiple vector fields: ```wl In[1]:= VectorPlot[{{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {-(1 + x - y ^ 2), -1 - x ^ 2 + y}}, {x, -3, 3}, {y, -3, 3}, VectorColorFunction -> None, VectorStyle -> {Red, Blue}] Out[1]= [image] ``` --- Use ``Arrowheads`` to specify an explicit style of the arrowheads: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, VectorPoints -> 8, PlotRange -> All, VectorStyle -> Arrowheads[{{0.03, Automatic, Graphics[Circle[]]}}]] Out[1]= [image] ``` --- Specify both arrow tail and head: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, VectorPoints -> 8, PlotRange -> All, VectorStyle -> Arrowheads[{{0.03, Automatic, Graphics[Rectangle[{-0.5, -0.5}, {0.5, 0.5}]]}, {0.03, Automatic, Graphics[Circle[]]}}]] Out[1]= [image] ``` --- Graphics primitives without ``Arrowheads`` are scaled based on the vector scale: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, VectorPoints -> 8, PlotRange -> All, VectorStyle -> Graphics[Circle[]]] Out[1]= [image] ``` --- Change the scaling using the ``VectorScaling`` option: ```wl In[1]:= VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1}, VectorPoints -> 8, PlotRange -> All, VectorStyle -> Graphics[Circle[]], VectorScaling -> Automatic] Out[1]= [image] ``` ### Applications (8) Hamiltonian vector field of $\sin (x y)$ : ```wl In[1]:= VectorPlot[Evaluate@{D[Sin[x y], {y}], -D[Sin[x y], {x}]}, {x, 0, 2Pi}, {y, 0, 2Pi}, VectorRange -> All] Out[1]= [image] ``` #### Direction Fields (2) Plot the direction field for the logistic differential equation $y'=y(1-y)$ : ```wl In[1]:= VectorPlot[{1, y(1 - y)}, {x, 0, 2}, {y, 0, 2}, VectorMarkers -> "Segment"] Out[1]= [image] ``` Give the segments a common color: ```wl In[2]:= VectorPlot[{1, y(1 - y)}, {x, 0, 2}, {y, 0, 2}, VectorMarkers -> "Segment", VectorColorFunction -> None, VectorStyle -> Blue] Out[2]= [image] ``` Add several solution curves with a uniform color: ```wl In[3]:= VectorPlot[{1, y(1 - y)}, {x, 0, 2}, {y, 0, 2}, VectorMarkers -> "Segment", VectorColorFunction -> None, VectorStyle -> Blue, StreamPoints -> Coarse, StreamColorFunction -> None, StreamStyle -> Red, StreamScale -> None] Out[3]= [image] ``` --- Plot the direction field for a differential equation that is only defined over a specific subset of the $x y$-plane, $y'=\sqrt{1-x^2-y^2}$ : ```wl In[1]:= VectorPlot[{1, Sqrt[1 - x^2 - y^2]}, {x, -1, 1}, {y, -1, 1}, RegionFunction -> Function[{x, y}, x^2 + y^2 ≤ 1], VectorMarkers -> "Segment"] Out[1]= [image] ``` #### Stability (2) Characterize linear planar systems interactively: ```wl In[1]:= Manipulate[VectorPlot[Evaluate[m.{x, y}], {x, -1, 1}, {y, -1, 1}, PlotLabel -> Text["m"] == MatrixForm[m]], {{m, (| | | | - | - | | 1 | 0 | | 0 | 2 |)}, {(| | | | - | - | | 1 | 0 | | 0 | 2 |) -> "Nodal source", (| | | | - | - | | 1 | 1 | | 0 | 1 |) -> "Degenerate source", (| | | | -- | - | | 0 | 1 | | -1 | 1 |) -> "Spiral source", (| | | | -- | -- | | -1 | 0 | | 0 | -2 |) -> "Nodal sink", (| | | | -- | -- | | -1 | 1 | | 0 | -1 |) -> "Degenerate sink", (| | | | -- | -- | | 0 | 1 | | -1 | -1 |) -> "Spiral sink", (| | | | -- | - | | 0 | 1 | | -1 | 0 |) -> "Center", (| | | | - | -- | | 1 | 0 | | 0 | -2 |) -> "Saddle"}}] Out[1]= DynamicModule[«8»] ``` --- Use vectors to indicate the stability of a limit cycle: ```wl In[1]:= field[{x_, y_}] = {α x - y - α x(x^2 + y^2), x + α y - α y(x^2 + y^2)}; In[2]:= Table[VectorPlot[Evaluate[field[{x, y}]], {x, -2, 2}, {y, -2, 2}, VectorPoints -> Flatten[Table[p = {Cos[2π t], Sin[2π t]};{(3 / 4) p, (5 / 4)p}, {t, Range[20] / 20.}], 1], StreamPoints -> {{{0.5, 0.5}}}, StreamStyle -> Black, StreamColorFunction -> None, StreamScale -> None], {α, -3, 3, 2}] Out[2]= {[image], [image], [image], [image]} ``` #### Gradients (1) Gradient field of $\sin (x y)$ over the unit square: ```wl In[1]:= VectorPlot[Evaluate[D[Sin[x y], {{x, y}}]], {x, 0, 1}, {y, 0, 1}] Out[1]= [image] ``` #### Physics (2) Show the direction field for a frictionless pendulum satisfying $\left( \begin{array}{c} \theta \\ v \\ \end{array} \right)'=\left( \begin{array}{c} v \\ -\sin (\theta ) \\ \end{array} \right)$ : ```wl In[1]:= VectorPlot[{v, -Sin[θ]}, {θ, -3π, 3π}, {v, -4, 4}, VectorPoints -> 24, StreamPoints -> Fine, StreamColorFunction -> None, StreamStyle -> {Black, "Segment"}, AspectRatio -> Automatic, ImageSize -> 400] Out[1]= [image] ``` --- Display the electric field for two point charges: In[1]:= Manipulate[VectorPlot[(q1{x-1,y})/Norm[{x-1,y}]^3+(q2{x+1,y})/Norm[{x+1,y}]^3,{x,-5,5},{y,-5,5}],{{q1,-1,"Subscript[q, 1]"},-3,3},{{q2,2.5,"Subscript[q, 2]"},-3,3}] Out[1]= Manipulate[VectorPlot[q1\*({x - 1, y}/Norm[{x - 1, y}]^3) + q2\*({x + 1, y}/Norm[{x + 1, y}]^3), {x, -5, 5}, {y, -5, 5}], {{q1, -2.7199999999999998, "\!\(\*SubscriptBox[\(q\), \(1\)]\)"}, -3, 3}, {{q2, 2.17, "\!\(\*SubscriptBox[\(q\), \(2\)]\)"}, -3, 3}] ### Properties & Relations (13) Use ``StreamPlot`` to plot with streamlines instead of vectors : ```wl In[1]:= StreamPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` --- Use ``ListVectorPlot`` or ``ListStreamPlot`` for plotting data: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= {ListVectorPlot[data], ListStreamPlot[data]} Out[2]= {[image], [image]} ``` --- Use ``VectorDensityPlot`` to add a density plot of the scalar field: ```wl In[1]:= VectorDensityPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` Use ``StreamDensityPlot`` to plot streamlines instead of vectors: ```wl In[2]:= StreamDensityPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[2]= [image] ``` Use ``ListVectorDensityPlot`` or ``ListStreamDensityPlot`` for plotting data: ```wl In[3]:= data = Table[{{x, y} = RandomReal[{-1.5, 1.5}, {2}], {{y, x - x ^ 3}, Abs[x + y]}}, {300}]; In[4]:= {ListVectorDensityPlot[data], ListStreamDensityPlot[data]} Out[4]= [image] ``` --- Use ``LineIntegralConvolutionPlot`` to plot the line integral convolution of a vector field: ```wl In[1]:= LineIntegralConvolutionPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` --- Use ``VectorDisplacementPlot`` to visualize the effect of a displacement vector field on a specified region: ```wl In[1]:= VectorDisplacementPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, y}∈Annulus[], VectorPoints -> Automatic] Out[1]= [image] ``` --- Use ``ListVectorDisplacementPlot`` to visualize the effect of displacement field data on a region: ```wl In[1]:= data = Table[{{x, y}, {x + y, 1 + x - y}}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDisplacementPlot[data, Polygon[{{3, 3}, {0, 1}, {-3, 3}, {-3, -3}, {0, -1}, {3, -3}}], VectorPoints -> Automatic] Out[2]= [image] ``` --- Use ``VectorPlot3D`` and ``StreamPlot3D`` to visualize 3D vector fields: ```wl In[1]:= {VectorPlot3D[{z, x, x + y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}], StreamPlot3D[{z, x, x + y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]} Out[1]= {[image], [image]} ``` Use ``ListVectorPlot3D`` or ``ListStreamPlot3D`` to plot with data: ```wl In[2]:= data = Table[{z, x, x + y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]; In[3]:= {ListVectorPlot3D[data], ListStreamPlot3D[data]} Out[3]= {[image], [image]} ``` --- Plot vectors on surfaces with ``SliceVectorPlot3D`` : ```wl In[1]:= SliceVectorPlot3D[{-y, x, z}, Sphere[], {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Plot data vectors on surfaces with ``ListSliceVectorPlot3D`` : ```wl In[1]:= data = Table[{z, x, x + y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]; In[2]:= ListSliceVectorPlot3D[data, "CenterPlanes"] Out[2]= [image] ``` --- Plot complex functions as a vector field with ``ComplexVectorPlot`` : ```wl In[1]:= ComplexVectorPlot[z + 1 / z, {z, 2}] Out[1]= [image] ``` Plot streams instead of vectors with ``ComplexStreamPlot`` : ```wl In[2]:= ComplexStreamPlot[z^2 + 1, {z, 2}] Out[2]= [image] ``` --- Use ``VectorDisplacementPlot3D`` to visualize the effect of a displacement vector field on a specified 3D region: ```wl In[1]:= VectorDisplacementPlot3D[{-y, x, x + z}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, VectorPoints -> "Boundary"] Out[1]= [image] ``` --- Use ``ListVectorDisplacementPlot3D`` to visualize the effect of 3D displacement vector field data on a specified region: ```wl In[1]:= data = Table[{{x, y, z}, {-y, x, x + z}}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]; In[2]:= ListVectorDisplacementPlot3D[data, VectorPoints -> "Boundary"] Out[2]= [image] ``` --- Use ``GeoVectorPlot`` to generate a vector plot over the Earth: ```wl In[1]:= coords = Flatten[Table[{lat, long}, {long, -180, 180, 30}, {lat, -75, 75, 15}], 1]; In[2]:= vects = Table[{RandomReal[], Quantity[RandomReal[{0, 360}], "AngularDegrees"]}, {Length[coords]}]; In[3]:= GeoVectorPlot[GeoVector[GeoPosition[coords] -> vects]] Out[3]= [image] ``` Use ``GeoStreamPlot`` to use streams instead of vectors: ```wl In[4]:= GeoStreamPlot[GeoVector[GeoPosition[coords] -> vects]] Out[4]= [image] ``` ## See Also * [`VectorDensityPlot`](https://reference.wolfram.com/language/ref/VectorDensityPlot.en.md) * [`StreamPlot`](https://reference.wolfram.com/language/ref/StreamPlot.en.md) * [`ListVectorPlot`](https://reference.wolfram.com/language/ref/ListVectorPlot.en.md) * [`LineIntegralConvolutionPlot`](https://reference.wolfram.com/language/ref/LineIntegralConvolutionPlot.en.md) * [`ParametricPlot`](https://reference.wolfram.com/language/ref/ParametricPlot.en.md) * [`ContourPlot`](https://reference.wolfram.com/language/ref/ContourPlot.en.md) * [`DensityPlot`](https://reference.wolfram.com/language/ref/DensityPlot.en.md) * [`Plot3D`](https://reference.wolfram.com/language/ref/Plot3D.en.md) * [`VectorPlot3D`](https://reference.wolfram.com/language/ref/VectorPlot3D.en.md) * [`SliceVectorPlot3D`](https://reference.wolfram.com/language/ref/SliceVectorPlot3D.en.md) * [`StreamPlot3D`](https://reference.wolfram.com/language/ref/StreamPlot3D.en.md) * [`GeoVectorPlot`](https://reference.wolfram.com/language/ref/GeoVectorPlot.en.md) * [`ComplexVectorPlot`](https://reference.wolfram.com/language/ref/ComplexVectorPlot.en.md) ## Related Guides * [Vector Visualization](https://reference.wolfram.com/language/guide/VectorVisualization.en.md) * [Function Visualization](https://reference.wolfram.com/language/guide/FunctionVisualization.en.md) * [Vector Analysis](https://reference.wolfram.com/language/guide/VectorAnalysis.en.md) * [Solvers over Regions](https://reference.wolfram.com/language/guide/GeometricSolvers.en.md) * [Differential Equations](https://reference.wolfram.com/language/guide/DifferentialEquations.en.md) ## History * [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md) \| [Updated in 2012 (9.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn90.en.md) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md) ▪ [2018 (11.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn113.en.md) ▪ [2020 (12.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn121.en.md) ▪ [2021 (13.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn130.en.md) ▪ [2022 (13.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn131.en.md)