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LieAlgebrasOfVectorFields

 OneForm
 constructing a OneForm object

 Calling Sequence OneForm( components = compList, space = varList) OneForm( dExpr, space = varList) OneForm( listOfPairs, space = varList ) OneForm( 0, space = varList)

Parameters

 compList - a list of scalar expressions [theta1, theta2,...,thetan] the components of the 1-form. varList - a list of names [x1, x2, ... ,xn], the coordinates of space DExpr - expression of the form theta1*d[x1] + theta2*d[x2] + ... + thetan*d[xn] listOfPairs - a list of ordered pairs [[theta1,x1], [theta2,x2], ..., [thetan,xn]] of component values and corresponding space coordinate

Description

 • The command OneForm(...) is a constructor method for creating a OneForm object. Once a valid OneForm object has been created, it has access to various methods which allow it to be manipulated and its contents queried. see Overview of OneForm object for more detail.
 • A 1-form $\mathrm{\omega }$ is an expression of the form $\mathrm{\omega }=\sum _{i=0}^{n}{\mathrm{\theta }}_{i}\cdot {\mathrm{dx}}_{i}$ living on a space with coordinates $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$. The ${\mathrm{\theta }}_{i}$ are referred to as components, and ${x}_{1},{x}_{2},\dots ,{x}_{n}$ are referred to as (coordinates of) space.
 • The OneForm command first validates the user input arguments and then constructs a 1-form object named OneForm. A valid OneForm object consists of two data attributes: components ${\mathrm{\theta }}_{1},{\mathrm{\theta }}_{2},\dots ,{\mathrm{\theta }}_{n}$ 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 1-form.  The space = varList argument is optional; if present, its specification of 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 1-form 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 LieAlgebrasOfVectorFields package.
 • This command can be used in the form OneForm(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-OneForm(...).

Examples

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

First calling sequence:

 > $\mathrm{ω}≔\mathrm{OneForm}\left(\mathrm{components}=\left[{x}^{2},xy\right],\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{\omega }}{≔}{{x}}^{{2}}{}{\mathrm{dx}}{+}{x}{}{y}{}{\mathrm{dy}}$ (1)

Second calling sequence:

 > $\mathrm{ω}≔\mathrm{OneForm}\left({x}^{2}{d}_{x}+xy{d}_{y}\right)$
 ${\mathrm{\omega }}{≔}{{x}}^{{2}}{}{\mathrm{dx}}{+}{x}{}{y}{}{\mathrm{dy}}$ (2)

Third calling sequence, 1-form specified by ordered pairs:

 > $\mathrm{ω}≔\mathrm{OneForm}\left(\left[\left[{x}^{2},x\right],\left[xy,y\right]\right]\right)$
 ${\mathrm{\omega }}{≔}{{x}}^{{2}}{}{\mathrm{dx}}{+}{x}{}{y}{}{\mathrm{dy}}$ (3)

Fourth calling sequence:

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

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

 > $\mathrm{φ}≔\mathrm{OneForm}\left({d}_{x},\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{\phi }}{≔}{\mathrm{dx}}$ (5)

Although the coordinates y,z are not visible in the printed form of this 1-form, they are present in the OneForm object:

 > $\mathrm{GetComponents}\left(\mathrm{φ}\right)$
 $\left[{1}{,}{0}{,}{0}\right]$ (6)
 > $\mathrm{GetSpace}\left(\mathrm{φ}\right)$
 $\left[{x}{,}{y}{,}{z}\right]$ (7)