Overview of the OneForm Object
Description
Construction of a OneForm object
List of OneForm object methods
Examples
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 (x1,x2,…,xn) a 1-form ω is an expression of the form ω=∑i=0nθi⁡x1,x2,…,xn⋅dxi. The θi are referred to as components, and x1,x2,…,xn 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:
OneForm
Construct a differential one-form from given components.
Differential
Command for finding the differential df of function f, as a one-form.
The following is a list of available methods for a OneForm object.
AreSameSpace
DChange
GetComponents
GetSpace
LieDerivative
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.
with⁡LieAlgebrasOfVectorFields:
List of methods available for a OneForm object is available via the static exports of the object:
exports⁡OneForm,static
GetComponents,GetSpace,AreSameSpace,dchange,DChange,`=`,`+`,`-`,`*`,?[],map,subs,normal,expand,simplify,indets,has,hastype,type,ModuleType,ModulePrint,ModuleCopy
Construction by direct specification...
ω≔OneForm⁡x⁢y⁢dx−y2⁢dy
ω≔x⁢y⁢dx−y2⁢dy
Construction with Differential...
df≔Differential⁡x2+y2
df≔2⁢x⁢dx+2⁢y⁢dy
A OneForm is of type OneForm...
type⁡ω,OneForm
true
Extract data that make up a OneForm...
GetSpace⁡ω
x,y
GetComponents⁡ω
x⁢y,−y2
Overloaded Maple operators...
Component extraction...
ωx
x⁢y
Scalar multiplication and 1-form addition:
2⁢ω+y⁢df
4⁢x⁢y⁢dx
A OneForm object acts as an operator (via contraction) on a VectorField object
X≔VectorField⁡−y⁢Dx+x⁢Dy
X≔−y⁢ⅆⅆx+x⁢ⅆⅆy
ω⁡X
−2⁢x⁢y2
Overloaded Maple functions work as expected...
has⁡ω,x
hastype⁡ω,trig
false
See Also
VectorField (commands)
LieAlgebrasOfVectorFields[OneForm]
LieAlgebrasOfVectorFields[Differential]
Operators for OneForm
Maple builtins for OneForm
with
exports
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