Bases

An arbitrary basis contains four matrices

• one in which columns correspond to the covariant vectors of new basis (𝐞ᵢ) with respect to the canonical one,
• one defining the contravariant (or dual) basis (𝐞ⁱ),
• one defining the metric tensor gᵢⱼ=𝐞ᵢ⋅𝐞ⱼ,
• one defining the inverse of the metric tensor gⁱʲ=𝐞ⁱ⋅𝐞ʲ.

and is built by one the following constructors

• Basis(eᵢ::AbstractMatrix{T},eⁱ::AbstractMatrix{T},gᵢⱼ::AbstractMatrix{T},gⁱʲ::AbstractMatrix{T}) where {T}
• Basis(ℬ::AbstractBasis{dim,T}, χᵢ::V) where {dim,T,V} where χᵢ is a list of scaling factors applied on the vectors of the basis ℬ
• Basis(v::AbstractMatrix{T}, var::Symbol)
• Basis(θ::T1, ϕ::T2, ψ::T3 = 0) where {T1,T2,T3}
• Basis(θ::T) where {T}
• Basis{dim,T}() where {dim,T}

Depending on the property of the basis (canonical, orthonormal, orthogonal...), the most relevant type (CanonicalBasis, RotatedBasis, OrthogonalBasis or Basis) is implicitly created by calling Basis.

julia> ℬ = Basis(Sym[1 0 0; 0 1 0; 0 1 1])
Basis{3, Sym}
→ basis: 3×3 Matrix{Sym}:
 1  0  0
 0  1  0
 0  1  1      
→ dual basis: 3×3 Matrix{Sym}:
 1  0   0
 0  1  -1
 0  0   1
→ covariant metric tensor: 3×3 Symmetric{Sym, Matrix{Sym}}:
 1  0  0
 0  2  1
 0  1  1
→ contravariant metric tensor: 3×3 Symmetric{Sym, Matrix{Sym}}:
 1   0   0
 0   1  -1
 0  -1   2

julia> ℬ₂ = Basis(symbols("θ, ϕ, ψ", real = true)...)
RotatedBasis{3, Sym}
→ basis: 3×3 Matrix{Sym}:
 -sin(ψ)⋅sin(ϕ) + cos(θ)⋅cos(ψ)⋅cos(ϕ)  -sin(ψ)⋅cos(θ)⋅cos(ϕ) - sin(ϕ)⋅cos(ψ)  sin(θ)⋅cos(ϕ)
  sin(ψ)⋅cos(ϕ) + sin(ϕ)⋅cos(θ)⋅cos(ψ)  -sin(ψ)⋅sin(ϕ)⋅cos(θ) + cos(ψ)⋅cos(ϕ)  sin(θ)⋅sin(ϕ)
                        -sin(θ)⋅cos(ψ)                          sin(θ)⋅sin(ψ)         cos(θ)
→ dual basis: 3×3 Matrix{Sym}:
 -sin(ψ)⋅sin(ϕ) + cos(θ)⋅cos(ψ)⋅cos(ϕ)  -sin(ψ)⋅cos(θ)⋅cos(ϕ) - sin(ϕ)⋅cos(ψ)  sin(θ)⋅cos(ϕ)
  sin(ψ)⋅cos(ϕ) + sin(ϕ)⋅cos(θ)⋅cos(ψ)  -sin(ψ)⋅sin(ϕ)⋅cos(θ) + cos(ψ)⋅cos(ϕ)  sin(θ)⋅sin(ϕ)
                        -sin(θ)⋅cos(ψ)                          sin(θ)⋅sin(ψ)         cos(θ)
→ covariant metric tensor: 3×3 TensND.Id2{3, Sym}:
 1  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1
→ contravariant metric tensor: 3×3 TensND.Id2{3, Sym}:
 1  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1

Predefined symbolic or numerical coordinates and basis vectors can be obtained from

• init_cartesian(dim::Integer)
• init_polar(coords = (symbols("r", positive = true), symbols("θ", real = true)); canonical = false)
• init_cylindrical(coords = (symbols("r", positive = true), symbols("θ", real = true), symbols("z", real = true)); canonical = false)
• init_spherical(coords = (symbols("θ", real = true), symbols("ϕ", real = true), symbols("r", positive = true)); canonical = false)
• init_rotated(coords = symbols("θ ϕ ψ", real = true); canonical = false)

The option canonical specifies whether the vector is expressed as a tensor with components in the canonical basis or directly in the rotated basis. The second option (ie canonical = false by default) is often preferable for further calculations in the rotated basis.

julia> (x, y, z), (𝐞₁, 𝐞₂, 𝐞₃), ℬ = init_cartesian() ;

julia> (r, θ), (𝐞ʳ, 𝐞ᶿ), ℬᵖ = init_polar() ;

julia> (r, θ, z), (𝐞ʳ, 𝐞ᶿ, 𝐞ᶻ), ℬᶜ = init_cylindrical() ;

julia> display(ℬᶜ)
RotatedBasis{3, Sym}
→ basis: 3×3 Matrix{Sym}:
 cos(θ)  -sin(θ)  0
 sin(θ)   cos(θ)  0
      0        0  1
→ dual basis: 3×3 Matrix{Sym}:
 cos(θ)  -sin(θ)  0
 sin(θ)   cos(θ)  0
      0        0  1
→ covariant metric tensor: 3×3 TensND.Id2{3, Sym}:
 1  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1
→ contravariant metric tensor: 3×3 TensND.Id2{3, Sym}:
 1  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1

julia> (θ, ϕ, r), (𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ), ℬˢ = init_spherical() ;

julia> components_canon(𝐞ʳ)
3-element Vector{Sym}:
 sin(θ)⋅cos(ϕ)
 sin(θ)⋅sin(ϕ)
        cos(θ)

julia> (θ, ϕ, ψ), (𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ), ℬʳ = init_rotated() ;

NOTE: it is worth noting the unusual order of coordinates and vectors of the spherical basis which have been chosen here so that θ = ϕ = 0 corresponds to the cartesian basis in the correct order.