Overview of the OneForm Object
Construction of a OneForm object
List of OneForm object methods
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:
Construct a differential one-form from given components.
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.
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.
List of methods available for a OneForm object is available via the static exports of the object:
Construction by direct specification...
ω ≔ OneForm⁡x⁢y⁢dx−y2⁢dy
Construction with Differential...
df ≔ Differential⁡x2+y2
A OneForm is of type OneForm...
Extract data that make up a OneForm...
Overloaded Maple operators...
Scalar multiplication and 1-form addition:
A OneForm object acts as an operator (via contraction) on a VectorField object
X ≔ VectorField⁡−y⁢Dx+x⁢Dy
Overloaded Maple functions work as expected...
Operators for OneForm
Maple builtins for OneForm
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