Generate a plot over standard slice surfaces:

Standard axis-aligned stacked slice surfaces:

Standard boundary surfaces:

Plot over any surface region:

Plotting over a volume primitive is equivalent to plotting over RegionBoundary[reg]:

Plot over the surface :

Plot a vector field over multiple slice surfaces:

Specify the number of stacked planes:

Specify the cutting angle for a center-cut sphere slice:

Use VectorPoints to specify the number of arrows:

Use RegionFunction to expose obscured slices:

The domain may be specified by a region including Cone:

A formula region including ImplicitRegion:

A mesh-based region including BoundaryMeshRegion:

Use VectorScaling to show arrows scaled according to their magnitudes:

Use VectorSizes to prevent arrows from being too small:

Use VectorRange to control which vectors are plotted:

Use ClippingStyle to control the appearance of the clipped vectors:

Use PlotTheme to immediately get overall styling:

Use PlotLegends to get a color bar for the different values:

Control the display of axes with Axes:

Label axes using AxesLabel and the whole plot using PlotLabel:

Color the vectors by their magnitude with VectorColorFunction:

Use VectorMarkers to control the shape of the vectors:

Use VectorAspectRatio to modify the width-to-length ratio of the arrows:

Style the slice surface boundaries with BoundaryStyle:

Highlight a RegionFunction with RegionBoundaryStyle:

Style a RegionFunction with RegionBoundaryStyle:

TargetUnits specifies which units to use in the visualization:

Scale the axes in a plot:

Style the surface boundaries:

By default, the edges of the bounding box have the same length:

Use BoxRatios->Automatic to show the natural scale of the 3D coordinate values:

Use custom length ratios for each side of the bounding box:

By default, clipped vectors are given a constant color that is consistent with the minimum or maximum vector lengths given by VectorRange:

Suppress the clipped vectors:

Style the clipped vectors:

Style the short and long clipped vectors differently:

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

No legends are included by default:

Include a legend that indicates the vector norms by color:

With multiple fields and VectorColorFunction set to None, use a legend to identify each field:

Or use the fields in the legend:

Show All vectors by default:

Show a selected range:

Padding is computed automatically by default:

Specify no padding for all , , and ranges:

Specify an explicit padding for all , , and ranges:

Add 10% padding to all , , and ranges:

Specify different padding for , , and ranges:

Specify padding for the range:

Use different padding forms for each dimension:

Use a basic plot theme:

Override PlotTheme styles by explicitly setting options:

Compare different plot themes:

By default, a region function is not explicitly shown:

A similar effect can be created by combining VectorRange and ClippingStyle:

Show the boundary of a region defined by a region function:

Style the boundary of the region:

Plot vectors only over certain quadrants:

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

Use any logical combination of conditions:

By default, plots have linear scales in all directions:

Create a plot with a reversed axis:

Scaling functions are applied to slices that are defined in terms of the variables:

Slice surfaces that are defined relative to the bounding box are unaffected by scaling functions:

VectorAspectRatio specifies the width of the arrow over its length:

By default, vectors are colored according to their norm:

Change the color function:

Use any named color gradient from ColorData:

Color the vectors according to their value:

Use VectorColorFunctionScaling->False to get unscaled values:

By default, scaled values are used:

Use VectorColorFunctionScaling->False to get unscaled values:

Explicitly specify the scaling for each color function argument:

The default vector marker is "Arrow3D":

Use other named markers:

By default, arrows are centered at sampled points. Use Placed to start the arrow at the sampled point:

Use automatically determined vector points:

Use symbolic names to specify the set of field vectors:

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

Use a different number of vectors in each direction:

Specify the range of vector norms that are displayed with varying color:

Combine with ClippingStyle to remove the clipped vectors:

Or specify a different style for clipped vectors:

By default, arrows are displayed with a constant length:

Use Automatic to scale arrows proportionally to the corresponding vector norm:

Use VectorSizes to specify the range of relative lengths of the arrows:

Make the vectors half of the default size:

With VectorScaling, VectorSizes controls the range of the lengths of the arrows relative to the default size:

VectorColorFunction has precedence over VectorStyle:

Constant vector fields:

A circulating flow around the axis:

Divergent flow:

Convergent flow:

Illustrate the behavior of a linear system of differential equations in the case when is diagonal:

Solve the corresponding differential equation starting on the slice surfaces:

The differential equation solution follows the arrows in the plot all converging to the origin:

Automate the previous example and analyze all the possible sign combinations of the real eigenvalues for , where is a diagonal matrix with eigenvalues :

For , there is stability along the , , and directions:

For , there is stability along and , but instability along :

For , there is stability along and instability along and :

For , there is instability along , , and :

Define a matrix and compute its eigenvalues:

Since is symmetric, its eigenspaces , and are mutually orthogonal:

Generate the orthogonal complements of the eigenspaces:

Plot the vector field and observe the effects of the eigenvalues:

The plane orthogonal to contains and , so the origin is attractive in and repulsive in :

The plane orthogonal to contains and , so the field points directly toward :

The plane orthogonal to contains and , so the field points directly away from :

Define a matrix and compute its eigenvalues:

The eigenvalues and eigenvectors of indicate a sink at the origin for the vector field with spiral behavior around the axis:

Examine the vector field in planes orthogonal to :

Compute solutions of the linear system of differential equations for several initial conditions:

Add the solutions of to the vector field:

Define a new matrix and compute its eigenvalues and eigenvectors:

The eigenspace is a plane through the origin with normal , so solutions of are attracted to the origin while spiraling around :

Illustrate the behavior of a linear system of differential equations in the case when is block diagonal, with one real and two complex conjugate eigenvalues. The matrix has eigenvalues , , and :

Construct a matrix with eigenvalues , , and :

Find solutions to the corresponding differential equation:

Show vector field and solutions together:

Automate the previous example and analyze for different real and :

and :

and :

and :

and :

Solve an initial value problem with a solution that is contained in a cylinder:

Graph the surface, the vector field on the surface and the solution of the initial value problem:

Solve an initial value problem with a solution contained in a sphere:

Graph the surface, the vector field on the surface and the solution of the initial value problem:

Solve an initial value problem with a solution contained in a hyperbolic paraboloid:

Graph the surface, the vector field on the surface and the solution of the initial value problem:

Visualize Hill's spherical vortex, with vortex radius and velocity :

Compute vectors:

Visualize the vortex, with flow rotation highlighted in red:

Visualize the divergence-free field of a scalar function :

Visualize the vortices formed by these fields:

Visualize forces on surfaces.

A solid cylinder with a tensile load:

A solid cylinder with a compressive load:

A hollow cylinder with internal and external pressures:

An elastic bar in the shape of a circular cylinder with radius 1 has a net torque applied at both ends. The resulting displacement field is , where is the shear modulus, the nonzero stresses are and and the forces on the surfaces are the tractions:

Display the applied forces:

Display the displacement field that results from the applied forces:

The vector field from an electrostatic potential:

The resulting force vector field:

The force field on the center planes:

The force field on equipotential surfaces:

Visualize the outward pointing unit normal vectors to a surface:

Define vector field and compute the unit normals to the surface :

Visualize the vector field and the unit normals to :

The flux density of through is zero since is orthogonal to the normals of :

Define vector field and compute the unit normals to the surface :

Visualize the vector field and the unit normals to :

Compute the flux density of through :

The total flux is zero because the negative flux is canceled by the positive flux. Visualize this by coloring the surface by the flux density:

Visualize the "hairy ball theorem" (https://mathworld.wolfram.com/HairyBallTheorem.html) that, loosely speaking, says that you cannot comb the hair on a sphere without leaving a whorl:

Show a vector field in a tangent plane:

Choose 10 random points on the unit sphere:

Compute the angles ϕ and θ for the points at :

Plot geodesics from to the target points:

Show the sphere, geodesics, target points, and a vector field of tangents for the geodesics: