VectorField & OneForm Object Operator Methods
overview of VectorField and OneForm object operators
The + and - operators provide addition and subtraction of two objects X and Y, and returns a new object Z. These objects are either all VectorField or all OneForm objects.
These objects can also be added using the existing commands add and sum.
The * operator provides scalar multiplication of an object X, and returns a new object Y. These two objects are either both VectorField or both OneForm objects.
The = operator tests equality of two VectorField or OneForm objects. The two objects are regarded as equal if they live on the same space and have the same components.
The ? operator provides a look-up of components of a VectorField or OneForm object. The object is indexed by its space coordinates, see examples below.
These operators are associated with the VectorField and OneForm objects. For more detail, see Overview of the VectorField object, Overview of the OneForm object.
The +, -, * Operators
We first construct some suitable VectorField objects: rotation, translations in (x,y) coordinates
R ≔ VectorField⁡x⁢Dy−y⁢Dx,space=x,y
Tx ≔ VectorField⁡Dx,space=x,y
Ty ≔ VectorField⁡Dy,space=x,y
One-form arithmetic works same as vector field arithmetic.
ωx ≔ OneForm⁡dx,space=x,y,z
ωy ≔ OneForm⁡dy,space=x,y,z
ωz ≔ OneForm⁡dz,space=x,y,z
Taking a linear combination of these one-forms using add (or sum) command.
The = Operator
X ≔ VectorField⁡x⁢x−1⁢Dx+y⁢Dy,space=x,y
Y ≔ VectorField⁡x2−x⁢Dx+y⁢Dy,space=x,y
θ1 ≔ OneForm⁡dx,space=x
θ2 ≔ OneForm⁡dx,space=x,y
Although θ1,θ2 look the same, they are not equal because they live on different spaces.
The ? Operator
X ≔ VectorField⁡x2⁢Dz+Dy,space=x,y,z
The VectorField & OneForm Object Operator Methods command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
VectorField (Object overview)
OneForm (Object overview)
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