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LieAlgebrasOfVectorFields

 VectorField
 constructing a VectorField object

 Calling Sequence VectorField( components = compList, space = varList) VectorField( DExpr, space = varList) VectorField( listOfPairs, space = varList ) VectorField( 0, space = varList)

Parameters

 compList - a list of scalar expressions [xi1, xi2,...,xin] the components of the vector field. varList - a list of names [x1, x2, ... ,xn], the coordinates of space DExpr - expression of the form xi1*D[x1] + xi2*D[x2] + ... + xin*D[xn] listOfPairs - a list of ordered pairs [[xi1,x1], [xi2,x2], ..., [xin,xn]] of component values and corresponding space coordinate

Description

 • The command VectorField(...) is a constructor method for creating a VectorField object. Once a valid VectorField object is created, it has access to various methods which allow it to be manipulated and its contents queried. see Overview of VectorField object for more detail.
 • A vector field $X$ is an expression of the form $X=\sum _{i=0}^{n}{\mathrm{\xi }}^{i}\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$ living on a space with coordinates $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$. The ${\mathrm{\xi }}^{i}$ are referred to as components, and ${x}_{1},{x}_{2},\dots ,{x}_{n}$  are referred to as (coordinates of) space.   $\frac{\partial }{\partial {x}_{i}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{\xi }}^{i}\right)\left({x}_{1},{x}_{2},\mathrm{..},{x}_{n}\right)$
 • The VectorField command first validates the user input arguments and then constructs a VectorField object. A valid VectorField object consists of two data attributes: components ${\mathrm{\xi }}^{1},{\mathrm{\xi }}^{2},\dots ,{\mathrm{\xi }}^{n}$ and space variables ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
 • In the first calling sequence, both arguments components = compList, space=varList are required. These two lists must be of the same length.
 • The second calling sequence is a textual representation of the usual appearance of a vector field.  The space = varList argument is optional; if present, its specification of the space overrides the space [x1, x2,..., xn] implied by DExpr.
 • In the third calling sequence, the space =varList argument is optional; if present, its specification of the space overrides the space [x1, x2,..., xn] implied by listOfPairs.
 • The fourth calling sequence is a special constructor for the zero vector field on the specified space; the space = varList argument is required.
 • This command is part of the VectorField package. For more detail, see Overview of the VectorField package.
 • This command can be used in the form VectorField(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-VectorField(...).

Examples

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

First calling sequence:

 > $X≔\mathrm{VectorField}\left(\mathrm{components}=\left[{x}^{2},xy\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{{x}}^{{2}}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)

Second calling sequence:

 > $X≔\mathrm{VectorField}\left({x}^{2}\mathrm{D}\left[x\right]+xy\mathrm{D}\left[y\right]\right)$
 ${X}{≔}{{x}}^{{2}}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)

Third calling sequence, vector field specified by ordered pairs:

 > $X≔\mathrm{VectorField}\left(\left[\left[{x}^{2},x\right],\left[xy,y\right]\right]\right)$
 ${X}{≔}{{x}}^{{2}}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

Fourth calling sequence:

 > $Z≔\mathrm{VectorField}\left(0,\mathrm{space}=\left[x,y\right]\right)$
 ${Z}{≔}{0}$ (4)

The second calling sequence is especially useful as a sparse form entry, where only a few components are nonzero:

 > $\mathrm{Tx}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z,t\right]\right)$
 ${\mathrm{Tx}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (5)

Although the coordinates y,z,t are not visible in the printed form of this vector field, they are present in the VectorField object:

 > $\mathrm{GetComponents}\left(\mathrm{Tx}\right),\mathrm{GetSpace}\left(\mathrm{Tx}\right)$
 $\left[{1}{,}{0}{,}{0}{,}{0}\right]{,}\left[{x}{,}{y}{,}{z}{,}{t}\right]$ (6)