addition, subtraction, scalar multiplication, wedge product, tensor product
A &plus B - add two vectors, differential forms or tensors
A &minus B- subtract one vector, differential form or tensor from another
A &mult B - multiply a Maple expression by a vector, differential form or tensor
A &wedge B- form the wedge (or skew) product of a pair of differential forms or multi-vectors
A &tensor B- form the tensor product of a pair of tensors
A &algmult B - multiply two vectors in an algebra
Maple expressions, differential forms or tensors
In the DifferentialGeometry package the wedge product of 1-forms is defined in terms of the tensor product by α⁢∧β⁢=⁢α⁢⊗⁢β⁢−⁢β⁢⊗α.
When using these commands together within a single Maple expression, it is important to use parentheses to insure that the operations are executed in the correct order.
In an interactive Maple session, it is usually more convenient to use the commands evalDG and DGzip to perform these basic algebraic operations.
Here are the precise lists of admissible arguments for these commands.
A &plus B, A &minus B -- A and B: Maple expressions, vectors, differential forms of the same degree, differential biforms of the same bidegree, tensors with the same index type and density weights. A and B must be defined on the same frame.
A &mult B -- A: a Maple expression; B: a Maple expression, vector, differential form, differential biform, tensor. A and B must be defined on the same frame.
A &wedge B -- A and B: Maple expressions or differential forms, differential biforms. If A and B are forms, then the sum of their degrees cannot exceed the dimension of the frame on which they are defined. If A and B are bi-forms, then the sum of their horizontal degrees cannot exceed the dimension of the base manifold on which they are defined. A and B must be defined on the same frame.
A &tensor B -- A and B: Maple expressions, vectors, differential 1-forms, tensors. A and B must be defined on the same frame.
These commands are part of the DifferentialGeometry package, and so can be used in the forms given above only after executing the command with(DifferentialGeometry).
Use DGsetup to define a three-dimensional manifold M with coordinates [x, y, z].
The following coordinates have been protected:
The following vector fields have been defined and protected:
The following differential 1-forms have been defined and protected:
frame name: M
Create linear combinations of vector fields and differential 1-forms using &plus and &mult.
X1 ≔ D_x &plus D_z
X2 ≔ 3⁢z &mult D_x &plus −2⁢y &mult D_y
X3 ≔ X2 &minus 3⁢z &mult X1
α1 ≔ sin⁡z &mult dx &minus cos⁡y &mult dz
α2 ≔ cos⁡x &mult dy &plus cos⁡z &mult dz
Create differential 2-forms using &plus and &mult and &wedge.
α3 ≔ 2 &mult dx &wedge dy &plus 5 &mult dy &wedge dz
α4 ≔ α1 &wedge α2
α5 ≔ α1 &wedge α2 &minus α3
α6 ≔ α1 &wedge α3
Create various tensors using &plus, &mult and &tensor.
T1 ≔ X1 &tensor X1
T2 ≔ X1 &tensor α1
T3 ≔ 1 &tensor dx &wedge dy
T4 ≔ dx &tensor dx &tensor D_y &tensor D_z &tensor dz
T5 ≔ 1y2 &mult dx &t dx+dy &t dy
Create a multi-vector using &plus, &mult and &tensor.
V1 ≔ 2 &mult D_x &wedge D_y &plus 3 &mult D_y &wedge D_z
Use the command AlgebraLibraryData to retrieve the structure equations for the quaternions.
LA ≔ AlgebraLibraryData⁡Quaternions,Q
algebra name: Q
Calculate some simple sums and products of quaternions.
Q1 ≔ i &algmult j
Q2 ≔ e &plus i &plus j &plus k &algmult e &minus i &plus j &plus k
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