The VectorField object is designed and created to represent a vector field as a mathematical object. It can be queried for basic properties of a vector field, and can be used in computing vector field arithmetic, Lie derivatives and Lie brackets. A VectorField object can also act as an operator.

Some existing Maple builtins have been overloaded so that they work for a VectorField object.

All methods of the VectorField object become available only once a valid VectorField object is constructed successfully. See LieAlgebrasOfVectorFields[VectorField] command for more detail about constructing a VectorField object.

The VectorField object is one of the main Maple objects exported by the VectorField package. See Overview of the VectorField package for more detail.

For a space with coordinates $\left(x_1\,x_2\,\dots \,x_n\right)$ a vector field $X$ is an expression of the form $X\=\sum _{i\=0}^n{\mathrm{\xi }}^i\left(x_1\,x_2\,\dots \,x_n\right)\cdot \frac{a}{x^i}$. The ${\mathrm{\xi }}^i$ are referred to as components, and $x_1\,x_2\,\dots \,x_n$  are referred as space. Therefore, a VectorField object is mathematically represented by two data attributes: "components" and "space". The data attributes of a VectorField object can be accessed via the GetComponents and GetSpace methods.

After a VectorField object X is successfully constructed, each method in the VectorField object can be accessed by either the short form method(X, arguments) or the long form X:-method(X, arguments).

After a VectorField object is constructed, the following methods are available:

A VectorField object can also act as a derivation operator. See VectorField Object as Derivation Operator for more detail.

The following arithmetic operators (=, +, -, ?[]) are overloaded for use on VectorField object. See VectorField Object Operator Methods for more detail.

The following Maple builtins functions are extended so that they work for a VectorField object: type, expand, has, hastype, indets, map, normal, simplify, subs. See VectorField Object Overloaded Builtins for more detail.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{exports}\left(\mathrm{VectorField}\,'\mathrm{static}'\right)$

$\mathrm{GetComponents},\mathrm{GetSpace},\mathrm{AreSameSpace},\mathrm{LieDerivative},\mathrm{Commutator},\mathrm{LieBracket},\mathrm{dchange},\mathrm{DChange},\mathrm{`=`},\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{?[]},\mathrm{map},\mathrm{subs},\mathrm{normal},\mathrm{expand},\mathrm{simplify},\mathrm{indets},\mathrm{has},\mathrm{hastype},\mathrm{type},\mathrm{ModuleType},\mathrm{ModulePrint},\mathrm{ModuleCopy}$

$R≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$R≔-y\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{Tx}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$\mathrm{Tx}≔\frac{\ⅆ}{\ⅆx}$

$\mathrm{GetComponents}\left(R\right),\mathrm{GetSpace}\left(R\right)$

$\left[-y\,x\right],\left[x\,y\right]$

$R-2\mathrm{Tx}$

$\left(-2-y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)$

$R\left(x^2\right)$

$-2yx$

$\mathrm{LieBracket}\left(R\,\mathrm{Tx}\right)$

$-\frac{\ⅆ}{\ⅆy}$

$\mathrm{map}\left(z\mapsto f\left(x\right)\cdot z\,R\right)$

$-f\left(x\right)y\left(\frac{\ⅆ}{\ⅆx}\right)+f\left(x\right)x\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{type}\left(R\,'\mathrm{VectorField}'\right)$

$\mathrm{true}$

The Overview of the VectorField Object command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.