The OneForm object is designed and created to represent a differential 1-form as a mathematical object. It can be queried for basic properties of a 1-form, and can be used in computing 1-form arithmetic and Lie derivatives. A OneForm object can also act as an operator.

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

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

The OneForm object is one of the Maple objects exported by the LieAlgebrasOfVectorFields package.

For a space with coordinates $(x_1\,x_2\,\dots \,x_n)$ a 1-form $\mathrm{\omega }$ is an expression of the form $\mathrm{\omega }\=\sum _{i\=0}^n{\mathrm{\theta }}_i\left(x_1\,x_2\,\dots \,x_n\right)\cdot {\mathrm{dx}}_i$. The ${\mathrm{\theta }}_i$ are referred to as components, and $x_1\,x_2\,\dots \,x_n$  are referred as space. Therefore, a OneForm object is mathematically represented by two data attributes: "components" and "space". The data attributes of a OneForm object can be accessed via the GetComponents and GetSpace methods.

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

The DifferentialGeometry package provides a more thorough implementation of differential forms as geometric objects.

There are two commands that construct a OneForm object:

The following is a list of available methods for a OneForm object.

A OneForm object can also act as an operator on a VectorField object living on the same space. See OneForm Object as Operator for more detail.

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

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

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

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

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

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

$\mathrm{\omega }≔xy\mathrm{dx}-y^2\mathrm{dy}$

$\mathrm{df}≔\mathrm{Differential}\left(x^2+y^2\right)$

$\mathrm{df}≔2x\mathrm{dx}+2y\mathrm{dy}$

$\mathrm{type}\left(\mathrm{\omega }\,\mathrm{OneForm}\right)$

$\mathrm{true}$

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

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

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

$\left[xy\,-y^2\right]$

$\mathrm{\omega }\left[x\right]$

$xy$

$2\mathrm{\omega }+y\mathrm{df}$

$4xy\mathrm{dx}$

$X≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right]\right)$

$X≔-y\frac{\partial }{\partial x}+x\frac{\partial }{\partial y}$

$\mathrm{\omega }\left(X\right)$

$-2x{y}^2$

$\mathrm{has}\left(\mathrm{\omega }\,x\right)$

$\mathrm{true}$

$\mathrm{hastype}\left(\mathrm{\omega }\,\mathrm{trig}\right)$

$\mathrm{false}$