chakravala/cartan.jl

Maurer-Cartan-Lie frame connections ∇ Grassmann.jl TensorField derivations

Cartan.jl

Maurer-Cartan-Lie frame connections ∇ Grassmann.jl TensorField derivations

Provides TensorField{B,F,N} <: GlobalFiber{LocalTensor{B,F},N} implementation for both a local ProductSpace topology and the simplicial mesh topologies imported with Grassmann.jl. Many of these modular methods can work on input meshes or product topologies of any dimension, although there are some methods which are specialized. Building on this, Cartan provides an algebra for any GlobalSection and associated bundles on a manifold, such as general Connection and CovariantDerivative operators in terms of Grassmann elements. Utility package for differential geometry and tensor calculus intended for packages such as Adapode.jl.

The Cartan package is intended to standardize the composition of various methods and functors applied to specialized categories transformed with a unified representation over a product topology, especially having fibers of the Grassmann algebra. Initial topologies include ProductSpace types and implicitly simplicial manifolds.

RealRegion{V, T} where {V, T<:Real} (alias for ProductSpace{V, T, N, N, S} where {V, T<:Real, N, S<:AbstractArray{T, 1}})
Interval (alias for ProductSpace{V, T, 1, 1} where {V, T})
Rectangle (alias for ProductSpace{V, T, 2, 2} where {V, T})
Hyperrectangle (alias for ProductSpace{V, T, 3, 3} where {V, T})

Visualizing TensorField reperesentations can be standardized in combination with Makie.jl or UnicodePlots.jl.

Due to the versatility of the TensorField type instances, it's possible to disambiguate them into these type alias specifications with associated methods:

MeshFunction (alias for TensorField{B, F, 1, BA} where {B, F<:AbstractReal, BA<:ChainBundle})
ElementFunction (alias for TensorField{B, F, 1, PA} where {B, F<:AbstractReal, PA<:(AbstractVector)})
IntervalMap (alias for TensorField{B, F, 1, PA} where {B, F, PA<:(AbstractArray{<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}, 1})})
RectangleMap (alias for TensorField{B, F, 2, BA} where {B, F, BA<:(ProductSpace{V, T, 2, 2} where {V, T})})
HyperrectangleMap (alias for TensorField{B, F, 3, PA} where {B, F, PA<:(ProductSpace{V, T, 3, 3} where {V, T})})
ParametricMap (alias for TensorField{B, F, N, PA} where {B, F, N, PA<:(ProductSpace{V, T, N, N, S} where {V, T<:Real, N, S<:AbstractArray{T, 1}})})
RealFunction (alias for TensorField{B, F, 1, PA} where {B, F<:AbstractReal, PA<:(AbstractVector{<:AbstractReal})})
PlaneCurve (alias for TensorField{B, F, 1, PA} where {B, F<:(Chain{V, G, Q, 2} where {V, G, Q}), PA<:(AbstractVector{<:AbstractReal})})
SpaceCurve (alias for TensorField{B, F, 1, PA} where {B, F<:(Chain{V, G, Q, 3} where {V, G, Q}), PA<:(AbstractVector{<:AbstractReal})})
SurfaceGrid (alias for TensorField{B, F, 2, PA} where {B, F<:AbstractReal, PA<:(AbstractMatrix)})
VolumeGrid (alias for TensorField{B, F, 3, PA} where {B, F<:AbstractReal, PA<:(AbstractArray{P, 3} where P)})
ScalarGrid (alias for TensorField{B, F, N, PA} where {B, F<:AbstractReal, N, PA<:AbstractArray})
GlobalFrame{B, N, N} where {B<:(LocalFiber{P, <:TensorNested} where P), N, N} (alias for Cartan.GlobalSection{B, N, N1, BA, FA} where {B<:(LocalFiber{P, <:TensorNested} where P), N, N1, BA, FA<:AbstractArray{N, N1}})
DiagonalField (alias for TensorField{B, F} where {B, F<:DiagonalOperator})
EndomorphismField (alias for TensorField{B, F} where {B, F<:(TensorOperator{V, V, T} where {V, T<:(TensorAlgebra{V, <:TensorAlgebra{V}})})})
OutermorphismField (alias for TensorField{B, F} where {B, F<:Outermorphism})
CliffordField (alias for TensorField{B, F} where {B, F<:Multivector})
QuaternionField (alias for TensorField{B, F} where {B, F<:(Quaternion)})
ComplexMap (alias for TensorField{B, F} where {B, F<:(Union{Complex{T}, Single{V, G, B, Complex{T}} where {V, G, B}, Chain{V, G, Complex{T}, 1} where {V, G}, Couple{V, B, T} where {V, B}, Phasor{V, B, T} where {V, B}} where T<:Real)})PhasorField (alias for TensorField{B, T, F} where {B, T, F<:Phasor})
SpinorField (alias for TensorField{B, F} where {B, F<:AbstractSpinor})
GradedField{G} where G (alias for TensorField{B, F} where {G, B, F<:(Chain{V, G} where V)})
ScalarField (alias for TensorField{B, F} where {B, F<:Union{Real, Single{V, G, B, <:Real} where {V, G, B}, Chain{V, G, <:Real, 1} where {V, G}}})
VectorField (alias for TensorField{B, F} where {B, F<:(Chain{V, 1} where V)})
BivectorField (alias for TensorField{B, F} where {B, F<:(Chain{V, 2} where V)})
TrivectorField (alias for TensorField{B, F} where {B, F<:(Chain{V, 3} where V)})

In the Cartan package, a technique is employed where an identity TensorField is constructed from an interval or product manifold, to generate an algebra of sections which can be used to compose parametric maps on manifolds. Constructing a TensorField can be accomplished in various ways, there are explicit techniques to construct a TensorField as well as implicit methods. Additional packages such as Adapode build on the TensorField concept by generating them from differential equations. Many of these methods can automatically generalize to higher dimensional manifolds and are compatible with discrete differential geometry.

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developed by chakravala with Grassmann.jl