- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
| Visualize[{expr1, expr2, ..., exprn}, {x1, a1, b1}, {x2, a2, b2}, ..., {xm, am, bm}] generates visual presentations of m-dimentional mapping |
| Visualize[{expr1, expr2, ..., exprn}] generates visual presentations of m-dimentional mapping, where m is number of indeterminates in {expr1, expr2,..., exprn} |
Visualize[ ]generates visual presentation of Mapping |
- Visualize function makes visual presentation of n-dimensional differential geometry objects. The following differential geometry objects can be visualized:
- n-dimentional coordinate systems
- curves in n-dimentional manifolds
- m-dimentional submanifolds in n-dimentional manifolds
A differential geometry object usually defined as a mapping. For instance an m-dimensional coordinate system can be defined as a mapping of one Rm domain into another Rm domain. Curves in an m-dimentional manifold can be presented locally as a set of parametric equations. An m-dimentional submanifold in an n-dimentional manifold can be defined locally by n parametric equations with m parameters.
- Generally speaking the function visualizes mappings between two manifolds or domains of manifolds.
- Let F be a mapping between manifolds or its domains such that: F: M
N. The mapping can be defined locally by the following equations:
where m=dim(M), n=dim(N); {x1,x2,..xm} are local coordinates on M and {y1,y2,..yn} are local coordinates on N. Local coordinates {x1,x2,..xm} is called domain of the mapping, and local coordinates {y1,y2,..yn} is called codomain of the mapping.
- One can visualize the mapping using the following:
Visualize[{y1, y2,..., yn}, {x1,a1,b1}, {x2,a2,b2},...,{xm,am,bm}]
- If {y1, y2,..., yn} depends on {x1, x2, ..., xm} only then the function can be used as follows:
Visualize[{y1, y2,..., yn}]. The parameters bounds will be selected automatically.
- Where one uses the following syntax Visualize[{y1, y2,..., yn}] the function detemins all indeterminates in the list of expressions {y1, y2, ..., yn} and treats them as the mapping variables. The variables bounds will be selected automatically.
- To visualize the domain of the mapping - Rm chart, we reduce the dimension by splitting Rm in 2D or/and 3D subspaces. In fact there are
subspaces of the Rm space, where k = {2, 3}. One has to choose a 2D or 3D planes in Rm. This planes are called sections. So one has:
- m(m-1)(m-2)/6 plots for 3-dimensional sections
- m(m-1)/2 plots for 2-dimensional sections
Sections are defined automaticly. It means that the visualize function splits Rm into all possible 2D or 3D subspaces which depend on another option VisualDimensions.
- To visualize the codomain of the mapping - Rn chart, we reduce the dimension by splitting Rn in 2D or/and 3D subspaces. In fact there are
subspaces of the Rn space, where k = {2, 3}. One has to choose a 2D or 3D planes in Rn. This planes are called projectors. So one has
- n(n-1)(n-2)/6 plots for 3-dimensional projectors
- n(n-1)/2 plots for 2-dimensional projectors
One can define projectors using visualize option VisualProjectors. By default VisualProjectors -> Automatic it means that the visualize function splits Rn into all possible 2D or 3D subspaces which depend on another option VisualDimensions.
- VisualDimensions is an option of the visualize function which defines the dementions of projectors and sections VisualDimensions -> {projectorsDimension, sectionsDimention}. By default VisualDimensions -> {3, 2}, it means that all projectors have dimensions 3 and all sections have dimensions 2. In that case one has 3 dimensional plots with 2 dimensional parameters. VisualDimensions can be: {i, j}, where i = {1, 2, 3} and j = {1, 2, 3}
- The Visualize function uses different Mathematica plot functions to visualize differential geometry objects. To specify the plot function explicitly one can use VisualPlot option.
| VisualDimensions | VisualPlot | Default | |
| {1,1} | Plot | Plot | |
| {1,2} | ContourPlot DensityPlot Plot3D | ContourPlot | |
| {1,3} | ContourPlot3D RegionPlot3D | ContourPlot3D | |
| {2,1} | ParametricPlot | ParametricPlot | |
| {2,2} | ParametricPlot StreamPlot StreamDensityPlot VectorPlot VectorDensityPlot | ParametricPlot | |
| {2,3} | NONE | NONE | |
| {3,1} | ParametricPlot3D | ParametricPlot3D | |
| {3,2} | ParametricPlot3D | ParametricPlot3D | |
| {3,3} | VectorPlot3D | VectorPlot3D |
- The following options can be given:
| VisualProjectors | Automatic | define projectors for a mapping codomain | |
| VisualDimensions | {3, 2} | how to split the codomain an domain of a mapping | |
| VisualManipulate | None | make Maniulate for specified parameters | |
| VisualPlot | Automatic | specify which Mathematica plot function to use | |
| VisualConstants | {} | specify constants in a mapping domain | |
| VisualShow | True | apply Show to the visualization results |
- Additionally for each Mathematica plot function (specified in VisualPlot option) corresponding options for the Mathematica plot function can be used. For instance for VisualPlot -> ParametricPlot one can use any option for ParametricPlot as an option for Visualize.
- The VisualConstants option allows one to define constants in a mapping domain.
For instants VisualConstants -> {a->1/2, b->1, z->2} sets corresponding parameters to the values from the rules.
- The VisualManipulate option allows one to make the Manipulate control.
For example VisualManipulate -> {{a, 1, 4}, {b, 0, 10}} will make Manipulate control with corresponding parameters. If no limits for variables are presented then the Visualize function will slect it automatically. One can define any parameters specification allowed in the Manipulate control. Any options from Manipulate can be used and understood by Visualize function.
- Visualize function has attribute Hold All.
EXAMPLES
Basic Examples 
Options 
