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(...).

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

$\mathrm{\omega }≔\mathrm{OneForm}\left(\mathrm{components}=\left[x^2\,xy\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$\mathrm{\omega }≔x^2\mathrm{dx}+xy\mathrm{dy}$

$\mathrm{\omega }≔\mathrm{OneForm}\left(x^{2}d\left[x\right]+xyd\left[y\right]\right)$

$\mathrm{\omega }≔x^2\mathrm{dx}+xy\mathrm{dy}$

$\mathrm{\omega }≔\mathrm{OneForm}\left(\left[\left[x^2\,x\right]\,\left[xy\,y\right]\right]\right)$

$\mathrm{\omega }≔x^2\mathrm{dx}+xy\mathrm{dy}$

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

$\mathrm{\zeta }≔0$

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

$\mathrm{\phi }≔\mathrm{dx}$

$\mathrm{GetComponents}\left(\mathrm{\phi }\right)$

$\left[1\,0\,0\right]$

$\mathrm{GetSpace}\left(\mathrm{\phi }\right)$

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