--- title: "StreamPlot3D" language: "en" type: "Symbol" summary: "StreamPlot3D[{v x, v y, v z}, {x, xmin, xmax}, {y, ymin, ymax}, {z, zmin, zmax}] plots streamlines for the vector field {vx, vy, vz} as functions of x, y and z. StreamPlot3D[{vx, vy, vz}, {x, y, z} \\[Element] reg] takes the variables {x, y, z} to be in the geometric region reg." keywords: - stream plot 3D - stream line 3D - stream tube 3D - stream ribbons 3D - streak lines 3D - vector field plot 3D - vector field visualization canonical_url: "https://reference.wolfram.com/language/ref/StreamPlot3D.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" related_functions: - title: "VectorPlot3D" link: "https://reference.wolfram.com/language/ref/VectorPlot3D.en.md" - title: "ListStreamPlot3D" link: "https://reference.wolfram.com/language/ref/ListStreamPlot3D.en.md" - title: "StreamPlot" link: "https://reference.wolfram.com/language/ref/StreamPlot.en.md" - title: "SliceVectorPlot3D" link: "https://reference.wolfram.com/language/ref/SliceVectorPlot3D.en.md" - title: "VectorPlot" link: "https://reference.wolfram.com/language/ref/VectorPlot.en.md" - title: "NDSolve" link: "https://reference.wolfram.com/language/ref/NDSolve.en.md" - title: "NDSolveValue" link: "https://reference.wolfram.com/language/ref/NDSolveValue.en.md" --- # StreamPlot3D StreamPlot3D[{vx, vy, vz}, {x, xmin, xmax}, {y, ymin, ymax}, {z, zmin, zmax}] plots streamlines for the vector field {vx, vy, vz} as functions of x, y and z. StreamPlot3D[{vx, vy, vz}, {x, y, z}∈reg] takes the variables {x, y, z} to be in the geometric region reg. ## Details and Options * ``StreamPlot3D`` is known as a 3D stream plot or streamline plot. Besides lines, streamlines can also be displayed as tubes (stream tubes) and ribbons (stream ribbons). * ``StreamPlot3D`` plots streamlines $\sigma _i(t)$ defined by $\sigma _i'(t)=v\left(\sigma _i(t)\right)$ and $\sigma _i(0)=p_i$, where $v(\{x,y,z\})=\left\{v_x,v_y,v_z\right\}$ and $p_i$ is an initial stream point. The streamline $\sigma _i(t)$ is the curve passing through point $p_i$, and whose tangents correspond to the vector field $v$ at each point. * The streamlines are colored by default according to the magnitude $s$ of the vector field $\left\| \left\{v_x,v_y,v_z\right\}\right\|$. [image] * ``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 ``vi`` 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 ``vi`` 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 ``StreamColorFunction`` are ``x, y, z, vx, vy, vz, Norm[{vx, vy, vz}]``. * Possible settings for ``StreamMarkers`` include: | | | | | ----------- | ------------- | -------------------------------- | | [image] | "Arrow" | lines with 2D arrowheads | | [image] | "Arrow3D" | tubes with 3D arrowheads | | [image] | "Line" | lines | | **[image]** | "Tube" | tubes | | [image] | "Ribbon" | flat ribbons | | [image] | "ArrowRibbon" | ribbons with built-in arrowheads | * With ``StreamScale -> Automatic`` 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: {Subscript[``s``, ``x``], Subscript[``s``, ``y``], Subscript[``s``, ``z``]} scale ``x``, ``y`` and ``z`` 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 | Automatic | ratio of height to width | | Axes | False | whether to draw axes | | AxesEdge | Automatic | on which edges to put axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | graphics directives to specify the style for 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 | | Boxed | True | whether to draw the bounding box | | BoxRatios | {1, 1, 1} | ratio of height to width | | BoxStyle | {} | style specifications for the box | | ClipPlanes | None | clipping planes | | ClipPlanesStyle | Automatic | style specifications for clipping planes | | ContentSelectable | Automatic | whether to allow contents to be selected | | ControllerLinking | False | when to link to external rotation controllers | | ControllerPath | Automatic | what external controllers to try to use | | Epilog | {} | 2D graphics primitives to be rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | FaceGrids | None | grid lines to draw on the bounding box | | FaceGridsStyle | {} | style specifications for face grids | | FormatType | TraditionalForm | default format type for text | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels, etc. | | ImageSize | Automatic | absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | Lighting | Automatic | simulated light sources to use | | Method | Automatic | methods to use for the plot | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | a label for the plot | | 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 | | PlotRegion | Automatic | 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 | {} | 2D graphics primitives to be rendered before the main plot | | RegionBoundaryStyle | Automatic | how to style plot region boundaries | | RegionFunction | True& | determine what region to include | | RotationAction | "Fit" | how to render after interactive rotation | | ScalingFunctions | None | how to scale individual coordinates | | SphericalRegion | Automatic | whether to make the circumscribing sphere fit in the final display area | | 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 | | Ticks | Automatic | specification for ticks | | TicksStyle | {} | style specification for ticks | | TouchscreenAutoZoom | False | whether to zoom to fullscreen when activated on a touchscreen | | ViewAngle | Automatic | angle of the field of view | | ViewCenter | Automatic | point to display at the center | | ViewMatrix | Automatic | explicit transformation matrix | | ViewPoint | {1.3, -2.4, 2.} | viewing position | | ViewProjection | Automatic | projection method for rendering objects distant from the viewer | | ViewRange | All | range of viewing distances to include | | ViewVector | Automatic | position and direction of a simulated camera | | ViewVertical | {0, 0, 1} | direction to make vertical | | WorkingPrecision | MachinePrecision | precision to use in internal computations | ## Examples (108) ### Basic Examples (4) Plot streamlines through a vector field in 3D: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Use tubes to show the streamlines: ```wl In[1]:= StreamPlot3D[{y^2, 1, x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Tube"] Out[1]= [image] ``` --- Include a legend for the vector field magnitudes: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Plot streamlines over an arbitrary region: ```wl In[1]:= StreamPlot3D[{z, -x, y}, {x, y, z}∈Ball[]] Out[1]= [image] ``` ### Scope (12) #### Sampling (3) Specify the density of seed points for the streamlines: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamPoints -> Coarse] Out[1]= [image] ``` --- Specify specific seed points for the streamlines: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamPoints -> RandomReal[{-1, 1}, {30, 3}]] Out[1]= [image] ``` --- Plot streamlines over a specified region: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> Function[{x, y, z}, x^2 + y^2 ≤ 1]] Out[1]= [image] ``` #### Presentation (9) Streamlines are drawn as lines by default: ```wl In[1]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` Use 3D tubes for the streamlines: ```wl In[2]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Tube"] Out[2]= [image] ``` Use flat ribbons: ```wl In[3]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Ribbon"] Out[3]= [image] ``` --- Use "arrow" versions of the stream markers to indicate the direction of flow along the streamlines: ```wl In[1]:= StreamPlot3D[{z, -x, y + x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow"] Out[1]= [image] ``` Arrows on tubes: ```wl In[2]:= StreamPlot3D[{z, -x, y + x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow3D"] Out[2]= [image] ``` Ribbons are turned into arrows by tapering the heads and notching the tails of the streamlines: ```wl In[3]:= StreamPlot3D[{z, -x, y + x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "ArrowRibbon"] Out[3]= [image] ``` --- Use a single color for the streamlines: ```wl In[1]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> None] Out[1]= [image] ``` --- Use a named color gradient for the streamlines: ```wl In[1]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> "DarkRainbow"] Out[1]= [image] ``` --- Include a legend for the field magnitude: ```wl In[1]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Use ``StreamScale`` to split streamlines into multiple shorter line segments: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamScale -> 0.1] Out[1]= [image] In[2]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamScale -> 0.2] Out[2]= [image] ``` --- Increase the number of points in each segment and increase the marker aspect ratio: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> {Full, 20, 0.2}] Out[1]= [image] ``` --- Use a theme: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotTheme -> "Marketing"] Out[1]= [image] ``` --- Use a log scale for the ``x`` axis: ```wl In[1]:= StreamPlot3D[{x, y, z}, {x, 1, 100}, {y, 1, 100}, {z, 1, 100}, ScalingFunctions -> {"Log", None, None}] Out[1]= [image] ``` Reverse the ``y`` scale so it increases toward the bottom: ```wl In[2]:= StreamPlot3D[{x, y, z}, {x, 1, 100}, {y, 1, 100}, {z, 1, 100}, ScalingFunctions -> {None, None, "Reverse"}] Out[2]= [image] ``` ### Options (70) #### Axes (3) By default, ``Axes`` are drawn: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Use ``Axes -> False`` to turn off axes: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, Axes -> False] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, Axes -> {True, False, False}], StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, Axes -> {False, True, False}], StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, Axes -> {False, False, True}]} Out[1]= [image] ``` #### AxesLabel (3) No axes labels are drawn by default: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Place a label on the $z$ axis: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesLabel -> z] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesLabel -> {x, y, z}] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesOrigin -> {2, -1, 1}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesStyle -> {{Thick, Brown}, {Thick, Blue}, {Thick, Green}}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### BoxRatios (2) By default, ``BoxRatios`` is set to ``Automatic`` : ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Make the box twice as long in the ``x`` direction: ```wl In[1]:= StreamPlot3D[{y - z, z - x, x - y}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, BoxRatios -> {2, 1, 1}] Out[1]= [image] ``` #### ImageSize (7) Use named sizes, such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> Tiny], StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> Small]} Out[1]= [image] ``` --- Specify the width of the plot: ```wl In[1]:= {StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> 150], StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= [image] ``` Specify the height of the plot: ```wl In[2]:= {StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> {Automatic, 150}], StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= [image] ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> UpTo[200]], StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= [image] ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> {200, 300}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, AspectRatio -> Full, ImageSize -> {200, 300}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> {UpTo[150], UpTo[100]}], StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= [image] ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, ImageSize -> Full, Background -> LightBlue], {200, 200}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[StreamPlot3D[{y, -x, y - x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 200}]] Out[1]= [image] ``` #### PlotLegends (3) No legends are included by default: ```wl In[1]:= StreamPlot3D[{y z, x z, x y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Include a legend that indicates the vector field norm: ```wl In[1]:= StreamPlot3D[{y z, x z, x y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Specify the location of the legend: ```wl In[1]:= StreamPlot3D[{y z, x z, x y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotLegends -> Placed[Automatic, Left]] Out[1]= [image] ``` #### PlotTheme (1) Specify a theme: ```wl In[1]:= StreamPlot3D[{y z, 2x z, 3x y}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, PlotTheme -> "Scientific"] Out[1]= [image] ``` #### RegionBoundaryStyle (4) Show the region defined by a ``RegionFunction`` : ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> (#1^2 + #2^2 + #3^2 < 1&)] Out[1]= [image] ``` --- Use ``None`` to avoid showing the boundary: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> (#1^2 + #2^2 + #3^2 < 1&), RegionBoundaryStyle -> None] Out[1]= [image] ``` --- Specify the color of the region boundary: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> (#1^2 + #2^2 + #3^2 < 1&), RegionBoundaryStyle -> Orange] Out[1]= [image] ``` --- The boundaries of full rectangular regions are not shown: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` #### RegionFunction (4) Plot streamlines in a ball: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> (#1^2 + #2^2 + #3^2 < 1&)] Out[1]= [image] ``` --- Plot streams only where the field magnitude exceeds a given threshold: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> (#7 > 2&)] Out[1]= [image] ``` --- Region functions depend, in general, on seven arguments: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, RegionFunction -> Function[{x, y, z, vx, vy, vz, n}, vx vy > 0]] Out[1]= [image] ``` --- Use ``RegionBoundaryStyle -> None`` to avoid showing the boundary: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> (#1^2 + #2^2 + #3^2 < 1&), RegionBoundaryStyle -> None] Out[1]= [image] ``` #### ScalingFunctions (1) Use a log scale for the ``x`` axis: ```wl In[1]:= StreamPlot3D[{x, y, z}, {x, 1, 100}, {y, 1, 10}, {z, 1, 10}, ScalingFunctions -> {"Log", None, None}] Out[1]= [image] ``` Reverse the ``y`` scale so it increases toward the bottom: ```wl In[2]:= StreamPlot3D[{x, y, z}, {x, 1, 100}, {y, 1, 100}, {z, 1, 100}, ScalingFunctions -> {None, None, "Reverse"}] Out[2]= [image] ``` #### StreamColorFunction (4) Color the streams by their norm: ```wl In[1]:= StreamPlot3D[{x, 2y, 3z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> Hue] Out[1]= [image] ``` --- Use any named color gradient from ``ColorData`` : ```wl In[1]:= StreamPlot3D[{x, 2y, 3z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> "Rainbow"] Out[1]= [image] ``` --- Color the streamlines according to their ``x`` value: ```wl In[1]:= StreamPlot3D[{x, 2y, 3z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> Function[{x, y, z, vx, vy, vz, n}, Hue[x]]] Out[1]= [image] ``` --- Use ``StreamColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= StreamPlot3D[{x, 2y, 3z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> "Rainbow", StreamColorFunctionScaling -> False] Out[1]= [image] ``` #### StreamColorFunctionScaling (2) By default, scaled values are used: ```wl In[1]:= StreamPlot3D[{y, -x, 10z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> "Pastel", PlotLegends -> Automatic] Out[1]= [image] ``` --- Use ``StreamColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= StreamPlot3D[{y, -x, 10z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamColorFunction -> "BlueGreenYellow", PlotLegends -> Automatic, StreamColorFunctionScaling -> False] Out[1]= [image] ``` #### StreamMarkers (5) By default, lines are used: ```wl In[1]:= StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Draw the streamlines as tubes: ```wl In[1]:= StreamPlot3D[IconizedObject[«field»], IconizedObject[«vars»], StreamMarkers -> "Tube"] Out[1]= [image] ``` Draw them as flat ribbons: ```wl In[2]:= StreamPlot3D[IconizedObject[«field»], IconizedObject[«vars»], StreamMarkers -> "Ribbon"] Out[2]= [image] ``` --- ``"Arrow"`` stream markers automatically break the streamlines into shorter segments: ```wl In[1]:= StreamPlot3D[IconizedObject[«field»], IconizedObject[«vars»], StreamMarkers -> "Arrow"] Out[1]= [image] ``` Use 3D arrowheads on tubes: ```wl In[2]:= StreamPlot3D[IconizedObject[«field»], IconizedObject[«vars»], StreamMarkers -> "Arrow3D"] Out[2]= [image] ``` Use directional ribbons: ```wl In[3]:= StreamPlot3D[IconizedObject[«field»], IconizedObject[«vars»], StreamMarkers -> "ArrowRibbon"] Out[3]= [image] ``` --- Make segmented markers continuous: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow3D", StreamScale -> Full] Out[1]= [image] ``` --- Break continuous markers into segments: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Ribbon", StreamScale -> 0.1] Out[1]= [image] ``` #### StreamPoints (4) Use automatically determined stream points to seed the curves: ```wl In[1]:= StreamPlot3D[{y^2, 1, x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] ``` --- Specify a maximum number of streamlines: ```wl In[1]:= StreamPlot3D[{y^2, 1, x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamPoints -> 25] Out[1]= [image] ``` --- Give specific seed points for the streams: ```wl In[1]:= StreamPlot3D[{y^2, 1, x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamPoints -> RandomReal[{-1, 1}, {20, 3}]] Out[1]= [image] ``` --- Use coarsely spaced streamlines: ```wl In[1]:= StreamPlot3D[{y^2, 1, x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamPoints -> Coarse] Out[1]= [image] ``` Use more finely spaced streamlines: ```wl In[2]:= StreamPlot3D[{y^2, 1, x}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamPoints -> Fine] Out[2]= [image] ``` #### StreamScale (9) Segmented markers have default lengths, numbers of points and aspect ratios: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow"] Out[1]= [image] ``` --- Modify the lengths of the segments: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> 0.2] Out[1]= [image] ``` --- Specify the number of sample points in each segment: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> {0.2, 2}] Out[1]= [image] ``` --- Modify the aspect ratios for the stream markers: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> {0.2, 2, 0.2}] Out[1]= [image] ``` --- Make segmented markers continuous: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> Full] Out[1]= [image] ``` --- Break continuous markers into segments: ```wl In[1]:= StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Ribbon", StreamScale -> 0.2] Out[1]= [image] ``` --- The aspect ratio controls the thickness of ribbons and tubes: ```wl In[1]:= {StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Ribbon", StreamScale -> {Automatic, Automatic, 0.05}], StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Tube", StreamScale -> {Automatic, Automatic, 0.05}]} Out[1]= {[image], [image]} ``` Increase the width of the ribbons and tubes: ```wl In[2]:= {StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Ribbon", StreamScale -> {Automatic, Automatic, 0.1}], StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Tube", StreamScale -> {Automatic, Automatic, 0.1}]} Out[2]= {[image], [image]} ``` --- The aspect ratio controls the sizes of arrowheads: ```wl In[1]:= {StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> {Automatic, Automatic, 0.15}], StreamPlot3D[{1, y, ArcTan[x]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Arrow", StreamScale -> {Automatic, Automatic, 0.2}]} Out[1]= [image] ``` --- Control the number of points in each segment: ```wl In[1]:= {StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamScale -> {0.5, 5}, ...], StreamPlot3D[{-4 x^2 y - 4 y^3 - x z, 4 x^3 + 4 x y^2 - y z, x^2 + y^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamScale -> {0.5, 20}, ...]} Out[1]= {[image], [image]} ``` #### StreamStyle (3) Change the appearance of the streamlines: ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamStyle -> Thickness[0.02]] Out[1]= [image] ``` --- ``StreamColorFunction`` takes precedence over ``StreamStyle`` : ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamStyle -> Directive[Red, Dashing[{0.1, 0.01}]]] Out[1]= [image] ``` --- Use ``StreamColorFunction -> None`` to specify a streamline color with ``StreamStyle`` : ```wl In[1]:= StreamPlot3D[{x, y, 1}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamStyle -> Directive[Red, Dashing[{0.1, 0.01}]], StreamColorFunction -> None] Out[1]= [image] ``` #### Ticks (6) Ticks are placed automatically in each plot: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] Out[1]= [image] ``` --- Use ``Ticks -> None`` to not draw any tick marks: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Ticks -> None] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Ticks -> {{-1.5, 0, 1.5}, {-1.5, 0, 1.5}, {-1.5, 0, 1.5}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with the specified labels: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Ticks -> {{{-1.5, -a}, {0, 0}, {1.5, a}}, {{-1.5, -a}, {0, 0}, {1.5, a}}, {{-1.5, -a}, {0, 0}, {1.5, a}}}] Out[1]= [image] ``` --- Specify tick marks with scaled lengths: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Ticks -> {{{-1.5, -a, .1}, {0, 0, .1}, {1.5, a, .1}}, {{-1.5, -a, .05}, {0, 0, .05}, {1.5, a, .05}}, {{-1.5, -a, .15}, {0, 0, .15}, {1.5, a, .15}}}] Out[1]= [image] ``` --- Customize each tick with position, length, labeling and styling: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Ticks -> {{{-1.5, -a, .1, Directive[Red, Dashed, Thick]}, {0, 0, .1, Directive[Red, Dashed]}, {1.5, a, .1, Directive[Red]}}, {{-1.5, -a, .05, Directive[Black, Dashed, Thick]}, {0, 0, .05, Directive[Black, Dashed]}, {1.5, a, .05, Directive[Black]}}, {{-1.5, -a, .15, Directive[Darker@Green, Dashed, Thick]}, {0, 0, .15, Directive[Darker@Green, Dashed]}, {1.5, a, .15, Directive[Darker@Green]}}}] Out[1]= [image] ``` #### TicksStyle (3) By default, the ticks and tick labels use the same styles as the axis: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, AxesStyle -> Directive[Bold, Red]] Out[1]= [image] ``` --- Specify overall tick style, including the tick labels: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, TicksStyle -> Directive[Bold, Red]] Out[1]= [image] ``` --- Specify tick style for each of the axes: ```wl In[1]:= StreamPlot3D[{x, y, 1 / y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, TicksStyle -> {Directive[Black, Bold], Directive[Bold, Red], Directive[Bold, Blue]}] Out[1]= [image] ``` ### Applications (10) #### Basic Applications (1) Consider a vector differential equation $x'=f(x)$ where $f(x)=\text{If}[\| x\| \leq 1,x,\{1,0,0\}]$ is defined piecewise. Visualize solutions of $x'=f(x)$ using seed points for the streamlines that are inside the unit sphere: ```wl In[1]:= StreamPlot3D[If[x^2 + y^2 + z^2 < 1, {x, y, z}, {1, 0, 0}], {x, -1.2, 1.2}, {y, -1.2, 1.2}, {z, -1.2, 1.2}, StreamPoints -> 0.5SpherePoints[50]] Out[1]= [image] ``` #### Fluid Flow (3) Consider [Stokes flow](https://en.wikipedia.org/wiki/Stokes_flow) for a point force of the form $F \delta (x,y,z)$, where $F$ is a constant vector and $\delta (x,y,z)$ is a Dirac delta function. For example, a force pointing down: ```wl In[1]:= F = {0, 0, -1}; ``` Define the fluid velocity vector $u$, the pressure $p$ and the viscosity $\mu$ : ```wl In[2]:= vars = {x, y, z}; μ = 1; u = (1/8π μ)Table[Sum[( (vars[[i]]vars[[j]]/(vars.vars)^3 / 2) + (IdentityMatrix[3][[i, j]]/Sqrt[vars.vars]))F[[j]], {j, 1, 3}], {i, 1, 3}]; p = (1/4π)Sum[(vars[[j]]F[[j]]/(vars.vars)^3 / 2), {j, 1, 3}]; ``` Confirm that the equations for Stokes flow are satisfied so that $\nabla \cdot u=0$ and $\nabla p=\mu \nabla ^2u+F \delta (x,y,z)$ : ```wl In[3]:= {Div[u, vars] == 0, Grad[p, vars] == μ Laplacian[u, vars]}//Simplify Out[3]= {True, True} ``` Plot streamlines for the flow: ```wl In[4]:= StreamPlot3D[u, {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, StreamMarkers -> "Arrow3D", PlotLegends -> Automatic] Out[4]= [image] ``` --- Visualize Stokes flow around a unit sphere. Define the fluid velocity vector $u$, the pressure $p$, the viscosity $\mu$ and the far-field fluid speed $U$ : ```wl In[1]:= vars = {x, y, z}; μ = 1; U = 1; u = {U, 0, 0} - (3/4)(x (U vars/(vars.vars)^3 / 2) + ({U, 0, 0}/Sqrt[vars.vars])) - (1/4)(({U, 0, 0}/(vars.vars)^3 / 2) - (3U x vars/(vars.vars)^5 / 2)); p = -(3μ x/2(vars.vars)^3 / 2); ``` Confirm that the equations for Stokes flow are satisfied so that $\nabla \cdot u=0$ and $\nabla p=\mu \nabla ^2u$ : ```wl In[2]:= {Div[u, vars] == 0, Grad[p, vars] == μ Laplacian[u, vars]}//Simplify Out[2]= {True, True} ``` Plot streamlines for the flow: ```wl In[3]:= Show[StreamPlot3D[u, {x, -4, 4}, {y, -2, 2}, {z, -2, 2}, PlotLegends -> Automatic, StreamMarkers -> "Arrow", StreamPoints -> Flatten[Table[{-3, y, z}, {y, -1.75, 1.75, 0.5}, {z, -1.75, 1.75, 0.5}], 1]], Graphics3D[{Yellow, Sphere[]}], BoxRatios -> {2, 1, 1}] Out[3]= [image] ``` Plot the pressure for the flow: ```wl In[4]:= DensityPlot3D[p, {x, -4, 4}, {y, -2, 2}, {z, -2, 2}, PlotLegends -> Automatic, RegionFunction -> Function[{x, y, z}, x^2 + y^2 + z^2 > 1], BoxRatios -> {2, 1, 1}] Out[4]= [image] ``` --- Specify the Navier–Stokes equations for a fluid through a pipe with a bulge: ```wl In[1]:= velocities = {u[x, y, z], v[x, y, z], w[x, y, z]}; navierstokes = { Inactive[ Div][(-μ Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + ρ * velocities.Inactive[Grad][u[x, y, z], {x, y, z}] + p^(1, 0, 0)[x, y, z], Inactive[ Div][(-μ Inactive[Grad][v[x, y, z], {x, y, z}]), {x, y, z}] + ρ * velocities.Inactive[Grad][v[x, y, z], {x, y, z}] + p^(0, 1, 0)[x, y, z], Inactive[ Div][(-μ Inactive[Grad][w[x, y, z], {x, y, z}]), {x, y, z}] + ρ * velocities.Inactive[Grad][w[x, y, z], {x, y, z}] + p^(0, 0, 1)[x, y, z], u^(1, 0, 0)[x, y, z] + v^(0, 1, 0)[x, y, z] + w^(0, 0, 1)[x, y, z]} /. {μ -> 10 ^ -3, ρ -> 1}; ``` Specify the geometry for the flow: ```wl In[2]:= Ω = RegionUnion[Ball[], Cylinder[{{-2, 0, 0}, {2, 0, 0}}, 1 / 2]]; RegionPlot3D[Ω] Out[2]= [image] ``` Specify the boundary conditions for flow from left to right: ```wl In[3]:= inflowBC = DirichletCondition[{u[x, y, z] == 0.2, v[x, y, z] == 0, w[x, y, z] == 0}, x == -2]; outflowBC = DirichletCondition[p[x, y, z] == 0, x == 2]; wallBC = DirichletCondition[{u[x, y, z] == 0, v[x, y, z] == 0, w[x, y, z] == 0}, -2 < x < 2]; bcs = {inflowBC, outflowBC, wallBC}; ``` Solve for the flow velocities and pressure: ```wl In[4]:= {uVel, vVel, wVel, pressure} = NDSolveValue[{navierstokes == {0, 0, 0, 0}, bcs}, {u, v, w, p}, {x, y, z}∈Ω, Method -> {"FiniteElement", "InterpolationOrder" -> {u -> 2, v -> 2, w -> 2, p -> 1}, "MeshOptions" -> {"MaxCellMeasure" -> 0.01}}]; ``` Specify seed points for the streamlines: ```wl In[5]:= sp = Prepend[#, 0]& /@ Flatten[Table[CirclePoints[r, 10], {r, {0.3, 0.5, 0.7, 0.9}}], 1]; ``` Plot the streamlines for the flow: ```wl In[6]:= StreamPlot3D[{uVel[x, y, z], vVel[x, y, z], wVel[x, y, z]}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, RegionFunction -> Function[{x, y, z}, {x, y, z}∈Ω], StreamPoints -> sp, BoxRatios -> {2, 1, 1}, Method -> {"IntegrationLimits" -> 100}, StreamMarkers -> "Arrow", StreamScale -> {0.05, Automatic, 0.07}, ImageSize -> Medium] Out[6]= [image] ``` #### Electrical Systems (1) Visualize electric field lines for a dipole: ```wl In[1]:= StreamPlot3D[({x, y, z - 1}/Norm[{x, y, z - 1}]^3) - ({x, y, z + 1}/Norm[{x, y, z + 1}]^3), {x, -3, 3}, {y, -3, 3}, {z, -3, 3}] Out[1]= [image] ``` The streamlines appear to have uniform color because of the extremely rapid change in the vector field norm near the point charges at $z=\pm 1$. Bounding the magnitude of the vector field norm with a region function makes the colors visible: ```wl In[2]:= StreamPlot3D[({x, y, z - 1}/Norm[{x, y, z - 1}]^3) - ({x, y, z + 1}/Norm[{x, y, z + 1}]^3), {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, RegionFunction -> (#7 < 5&), RegionBoundaryStyle -> None, PlotLegends -> Automatic] Out[2]= [image] ``` Add spheres to indicate the positive (red) and negative (black) point charges: ```wl In[3]:= Show[StreamPlot3D[({x, y, z - 1}/Norm[{x, y, z - 1}]^3) - ({x, y, z + 1}/Norm[{x, y, z + 1}]^3), {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, RegionFunction -> (#7 < 5&), RegionBoundaryStyle -> None, PlotLegends -> Automatic], Graphics3D[{Red, Sphere[{0, 0, 1}, 0.2], Black, Sphere[{0, 0, -1}, 0.2]}]] Out[3]= [image] ``` 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: ```wl In[4]:= Show[StreamPlot3D[({x, y, z - 1}/Norm[{x, y, z - 1}]^3) - ({x, y, z + 1}/Norm[{x, y, z + 1}]^3), {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, RegionFunction -> (#7 < 5&), RegionBoundaryStyle -> None, PlotLegends -> Automatic, StreamMarkers -> "Arrow3D"], Graphics3D[{Red, Sphere[{0, 0, 1}, 0.2], Black, Sphere[{0, 0, -1}, 0.2]}]] Out[4]= [image] ``` Use a custom ``StreamColorFunction`` to exert more control over the colors: ```wl In[5]:= Show[StreamPlot3D[({x, y, z - 1}/Norm[{x, y, z - 1}]^3) - ({x, y, z + 1}/Norm[{x, y, z + 1}]^3), {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, RegionFunction -> (0.01 < #7 < 5&), RegionBoundaryStyle -> None, StreamColorFunction -> (Blend[{GrayLevel[0], GrayLevel[1], RGBColor[1, 0, 0]}, #3]&), StreamMarkers -> "Arrow"], Graphics3D[{Red, Sphere[{0, 0, 1}, 0.2], Black, Sphere[{0, 0, -1}, 0.2]}]] Out[5]= [image] ``` #### Miscellaneous (5) Lorenz attractor: ```wl In[1]:= StreamPlot3D[{10(y - x), x(28 - z) - y, x y - (8/3) z}, {x, -30, 30}, {y, -30, 30}, {z, 0, 50}, StreamPoints -> {{1, 1, 1}}] Out[1]= [image] ``` --- Use ribbons or arrow ribbons to visualize the torsion of a twisted cubic: ```wl In[1]:= StreamPlot3D[{1, 2x, 3x^2}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "ArrowRibbon", StreamPoints -> {{0, 0, 0}}, ViewPoint -> {-1.3, 2.4, 2}] Out[1]= [image] ``` --- Visualize solutions of differential equations on manifolds: ```wl In[1]:= StreamPlot3D[{-y, x, x^2 - y^2}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, StreamPoints -> Table[{r, 0, 0}, {r, 0, 2, 0.2}]] Out[1]= [image] ``` --- Visualize streamlines for Poiseuille flow. The fluid speed is fastest along the central axis: ```wl In[1]:= StreamPlot3D[{1 - y^2 - z^2, 0, 0}, {x, -2, 2}, {y, -1, 1}, {z, -1, 1}, BoxRatios -> {2, 1, 1}, PlotLegends -> Automatic, RegionFunction -> Function[{x, y, z}, y^2 + z^2 ≤ 1], StreamMarkers -> "Arrow", StreamPoints -> Flatten[Table[Prepend[#, 0]& /@ CirclePoints[r, 10], {r, 0.1, 1, 0.1}], 1]] Out[1]= [image] ``` --- Visualize solutions of Euler's equations for a rotating rigid body: ```wl In[1]:= moments = {1, 2, 3}; StreamPlot3D[Evaluate[(⁠| | | | | --- | --- | --- | | 0 | -x3 | x2 | | x3 | 0 | -x1 | | -x2 | x1 | 0 |⁠).({x1, x2, x3}/moments)], {x1, -1, 1}, {x2, -1, 1}, {x3, -1, 1}, StreamPoints -> SpherePoints[40]] Out[1]= [image] ``` ### Properties & Relations (9) Use ``VectorPlot3D`` to visualize a field with discrete arrows: ```wl In[1]:= VectorPlot3D[{z, -x, y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] Out[1]= [image] ``` Use ``ListStreamPlot3D`` or ``ListVectorPlot3D`` to generate plots based on data: ```wl In[2]:= data = Table[{z, -x, y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]; In[3]:= {ListStreamPlot3D[data], ListVectorPlot3D[data]} Out[3]= [image] ``` --- Use ``StreamPlot`` to plot streamlines of 2D vector fields: ```wl In[1]:= StreamPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` Use ``VectorPlot`` to plot with vectors instead of streamlines: ```wl In[2]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[2]= [image] ``` Generate plots based on data: ```wl In[3]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.1}, {y, -3, 3, 0.1}]; In[4]:= {ListStreamPlot[data], ListVectorPlot[data]} Out[4]= {[image], [image]} ``` --- Use ``StreamDensityPlot`` to add a density plot of the scalar field: ```wl In[1]:= StreamDensityPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` Use ``VectorDensityPlot`` to plot with arrows instead of streamlines: ```wl In[2]:= VectorDensityPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[2]= [image] ``` Generate plots based on data: ```wl In[3]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.1}, {y, -3, 3, 0.1}]; In[4]:= {ListStreamDensityPlot[data], ListVectorDensityPlot[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 deformation of a region associated with a displacement vector field: ```wl In[1]:= VectorDisplacementPlot[{y, x - x^3}, {x, y}∈Disk[], VectorPoints -> Automatic] Out[1]= [image] ``` Use ``ListVectorDisplacementPlot`` to visualize the same deformation based on data: ```wl In[2]:= data = Table[{{x, y} = RandomReal[{-1, 1}, {2}], {y, x - x ^ 3}}, {300}]; In[3]:= ListVectorDisplacementPlot[data, Disk[], VectorPoints -> Automatic] Out[3]= [image] ``` --- Plot vectors along surfaces with ``SliceVectorPlot3D`` : ```wl In[1]:= SliceVectorPlot3D[{x, y, z}, "CenterPlanes", {x, -1, 1}, {y, -1, 1}, {z, -1, 1}] Out[1]= [image] In[2]:= data = Table[{x, y, z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]; In[3]:= ListSliceVectorPlot3D[data, "CenterPlanes"] Out[3]= [image] ``` --- Use ``VectorDisplacementPlot3D`` to visualize the deformation of a 3D region associated with a displacement vector field: ```wl In[1]:= VectorDisplacementPlot3D[{y, x, -z}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, VectorPoints -> "Boundary"] Out[1]= [image] ``` Use ``ListVectorDisplacementPlot3D`` to visualize the same deformation based on data: ```wl In[2]:= data = Table[{{x, y, z} = RandomReal[{0, 1}, {3}], {y, x, -z}}, {300}]; In[3]:= ListVectorDisplacementPlot3D[data, Cuboid[], VectorPoints -> "Boundary"] Out[3]= [image] ``` --- Use ``ComplexVectorPlot`` or ``ComplexStreamPlot`` to visualize a complex function of a complex variable as a vector field or with streamlines: ```wl In[1]:= {ComplexVectorPlot[Sin[3z], {z, 2}], ComplexStreamPlot[Sin[3z], {z, 2}]} Out[1]= [image] ``` --- Use ``GeoVectorPlot`` to plot vectors on a map: ```wl In[1]:= GeoVectorPlot[GeoVector[GeoPosition[CompressedData["«6488»"]] -> {{0.5706379232485146, Quantity[354.01686143622794, "AngularDegrees"]}, {0.48751317221734114, Quantity[327.18610724051325, "AngularDegrees"]}, {0.42407845478418604, Quantity[199.859344783 ... 904, "AngularDegrees"]}, {0.9366099079317551, Quantity[357.70482317743733, "AngularDegrees"]}, {0.8256817194791912, Quantity[242.07227508889662, "AngularDegrees"]}, {0.2661401145642981, Quantity[152.90003252279325, "AngularDegrees"]}}]] Out[1]= [image] ``` Use ``GeoStreamPlot`` to plot streamlines instead of vectors: ```wl In[2]:= GeoStreamPlot[GeoVector[GeoPosition[CompressedData["«6488»"]] -> {{0.5706379232485146, Quantity[354.01686143622794, "AngularDegrees"]}, {0.48751317221734114, Quantity[327.18610724051325, "AngularDegrees"]}, {0.42407845478418604, Quantity[199.859344783 ... 904, "AngularDegrees"]}, {0.9366099079317551, Quantity[357.70482317743733, "AngularDegrees"]}, {0.8256817194791912, Quantity[242.07227508889662, "AngularDegrees"]}, {0.2661401145642981, Quantity[152.90003252279325, "AngularDegrees"]}}]] Out[2]= [image] ``` ### Possible Issues (3) Tube ``StreamMarkers`` can be distorted by the ``BoxRatios`` : ```wl In[1]:= StreamPlot3D[{2, -10 z - x y, 10 y - x z}, {x, -2π, 2π}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Tube"] Out[1]= [image] ``` --- Carefully adjusting the ``BoxRatios`` can eliminate the tube distortion: ```wl In[1]:= StreamPlot3D[{2, -10 z - x y, 10 y - x z}, {x, -2π, 2π}, {y, -1, 1}, {z, -1, 1}, StreamMarkers -> "Tube", BoxRatios -> {2π, 1, 1}] Out[1]= [image] ``` --- 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: ```wl In[1]:= StreamPlot3D[{ArcTan[z], y, -x}, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}] Out[1]= [image] In[2]:= StreamPlot3D[{ArcTan[z], y, -x}, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, StreamMarkers -> "Arrow3D", StreamScale -> Full] Out[2]= [image] ``` ## See Also * [`VectorPlot3D`](https://reference.wolfram.com/language/ref/VectorPlot3D.en.md) * [`ListStreamPlot3D`](https://reference.wolfram.com/language/ref/ListStreamPlot3D.en.md) * [`StreamPlot`](https://reference.wolfram.com/language/ref/StreamPlot.en.md) * [`SliceVectorPlot3D`](https://reference.wolfram.com/language/ref/SliceVectorPlot3D.en.md) * [`VectorPlot`](https://reference.wolfram.com/language/ref/VectorPlot.en.md) * [`NDSolve`](https://reference.wolfram.com/language/ref/NDSolve.en.md) * [`NDSolveValue`](https://reference.wolfram.com/language/ref/NDSolveValue.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) ## History * [Introduced in 2021 (12.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn123.en.md) \| [Updated in 2022 (13.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn131.en.md)