DirectSum.Basis{V} <: SubAlgebra{V} <: TensorAlgebra{V}

Grassmann basis container with cache of Submanifold elements and their Symbol names.

DirectSum.ExtendedBasis{V} <: SubAlgebra{V} <: TensorAlgebra{V}

Grassmann basis container without a dedicated Submanifold cache (only lazy caching).

Infinity{V} <: TensorGraded{V,0} <: TensorAlgebra{V}

Infinite quantity Infinity of the Grassmann algebra over V.

One{V} <: TensorGraded{V,0} <: TensorAlgebra{V}

Unit quantity One of the Grassmann algebra over V.

Single{V,G,B,T} <: TensorTerm{V,G,T} <: TensorGraded{V,G,T}

Single type with pseudoscalar V::Manifold, grade/rank G::Int, B::Submanifold{V,G}, field T::Type.

DirectSum.SparseBasis{V} <: SubAlgebra{V} <: TensorAlgebra{V}

Grassmann basis with sparse cache of Submanifold{G,V} elements and their Symbol names.

Submanifold{V,G,B} <: TensorGraded{V,G} <: Manifold{G}

Basis type with pseudoscalar V::Manifold, grade/rank G::Int, bits B::UInt64.

TensorBundle{n,ℙ,g,ν,μ} <: Manifold{n}

Let n be the rank of a Manifold{n}. The type TensorBundle{n,ℙ,g,ν,μ} uses byte-encoded data available at pre-compilation, where ℙ specifies the basis for up and down projection, g is a bilinear form that specifies the metric of the space, and μ is an integer specifying the order of the tangent bundle (i.e. multiplicity limit of Leibniz-Taylor monomials). Lastly, ν is the number of tangent variables.

Zero{V} <: TensorGraded{V,0} <: TensorAlgebra{V}

Null quantity Zero of the Grassmann algebra over V.

clifford(ω::TensorAlgebra)

Clifford conjugate of an element: clifford(ω) = involute(reverse(ω))

involute(ω::TensorAlgebra)

Involute of an element: ~ω = (-1)^grade(ω)*ω

reverse(ω::TensorAlgebra)

Reverse of an element: ~ω = (-1)^(grade(ω)(grade(ω)-1)/2)ω

~(ω::TensorAlgebra)

Reverse of an element: ~ω = (-1)^(grade(ω)(grade(ω)-1)/2)ω

imag(ω::TensorAlgebra)

The imag part (ω-(~ω))/2 is defined by abs2(imag(ω)) == -(imag(ω)^2).

real(ω::TensorAlgebra)

The real part (ω+(~ω))/2 is defined by abs2(real(ω)) == real(ω)^2.

alloc(V::Manifold,:V,"v","w","∂","ϵ")

Generates Basis declaration having Manifold specified by V. The first argument provides pseudoscalar specifications, the second argument is the variable name for the Manifold, and the third and fourth argument are variable prefixes of the Submanifold vector names (and covector basis names).

getbasis(V::Manifold,v)

Fetch a specific Submanifold{G,V} element from an optimal SubAlgebra{V} selection.

pseudoclifford(ω::TensorAlgebra)

Pseudo-clifford conjugate element: pseudoclifford(ω) = pseudoinvolute(pseudoreverse(ω))

pseudoinvolute(ω::TensorAlgebra)

Anti-involute of an element: ~ω = (-1)^pseudograde(ω)*ω

pseudoreverse(ω::TensorAlgebra)

Anti-reverse of an element: ~ω = (-1)^(pseudograde(ω)(pseudograde(ω)-1)/2)ω

@basis

Generates Submanifold elements having Manifold specified by V. As a result of this macro, all of the Submanifold{V,G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides pseudoscalar specifications, the second argument is the variable name for the Manifold, and the third and fourth argument are variable prefixes of the Submanifold vector names (and covector basis names). Default for @basis M is @basis M V v w ∂ ϵ.

@dualbasis

Generates Submanifold elements having Manifold specified by V'. As a result of this macro, all of the Submanifold{V',G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides pseudoscalar specifications, the second argument is the variable name for the dual Manifold, and the third and fourth argument are variable prefixes of the Submanifold covector names (and tensor field basis names). Default for @dualbasis M is @dualbasis M VV w ϵ.

@mixedbasis

Generates Submanifold elements having Manifold specified by V⊕V'. As a result of this macro, all of the Submanifold{V⊕V',G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides pseudoscalar specifications, the second argument is the variable name for the Manifold, and the third and fourth argument are variable prefixes of the Submanifold vector names (and covector basis names). Default for @mixedbasis M is @mixedbasis M V v w ∂ ϵ.