Coordinate systems and differential operators

Both symbolic (CoorSystemSym) and numerical (CoorSystemNum) coordinate systems are available. Symbolic systems support exact derivation; numerical systems evaluate differential operators pointwise via automatic differentiation — see the tutorial for examples.

julia> using TensND, SymPy
julia> Polar = coorsys_polar() ; r, θ = getcoords(Polar) ; 𝐞ʳ, 𝐞ᶿ = unitvec(Polar) ;
julia> @set_coorsys Polar
julia> LAPLACE(SymFunction("f", real = true)(r, θ))                              2
                             ∂
               ∂             ───(f(r, θ))
 2             ──(f(r, θ))     2
∂              ∂r            ∂θ
───(f(r, θ)) + ─────────── + ────────────
  2                 r              2
∂r                                r
julia> n = symbols("n", integer = true)n
julia> simplify(HESS(r^n))2×2 TensND.TensRotated{2, 2, Sym{PyCall.PyObject}, Tensors.SymmetricTensor{2, 2, Sym{PyCall.PyObject}, 3}}:
 n^2*r^n/r^2 - n*r^n/r^2          0
                       0  n*r^n/r^2

julia> using TensND, SymPy
julia> Spherical = coorsys_spherical() ; θ, ϕ, r = getcoords(Spherical) ; 𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ = unitvec(Spherical) ;
julia> @set_coorsys Spherical
julia> Christoffel(Spherical)3×3×3 Array{Sym{PyCall.PyObject}, 3}:
[:, :, 1] =
   0            0  1/r
   0  -sin(2*θ)/2    0
 1/r            0    0

[:, :, 2] =
        0  1/tan(θ)    0
 1/tan(θ)         0  1/r
        0       1/r    0

[:, :, 3] =
 -r            0  0
  0  -r*sin(θ)^2  0
  0            0  0
julia> ℬˢ = normalized_basis(Spherical)3×3 RotatedBasis{3, Sym{PyCall.PyObject}}:
 cos(θ)⋅cos(ϕ)  -sin(ϕ)  sin(θ)⋅cos(ϕ)
 sin(ϕ)⋅cos(θ)   cos(ϕ)  sin(θ)⋅sin(ϕ)
       -sin(θ)        0         cos(θ)
julia> σʳʳ = SymFunction("σʳʳ", real = true)(r) ;
julia> σᶿᶿ = SymFunction("σᶿᶿ", real = true)(r) ;
julia> σᵠᵠ = SymFunction("σᵠᵠ", real = true)(r) ;
julia> 𝛔 = σʳʳ * 𝐞ʳ ⊗ 𝐞ʳ + σᶿᶿ * 𝐞ᶿ ⊗ 𝐞ᶿ + σᵠᵠ * 𝐞ᵠ ⊗ 𝐞ᵠ3×3 TensND.TensRotated{2, 3, Sym{PyCall.PyObject}, Tensors.SymmetricTensor{2, 3, Sym{PyCall.PyObject}, 6}}:
 σᶿᶿ(r)       0       0
      0  σᵠᵠ(r)       0
      0       0  σʳʳ(r)
julia> div𝛔 = simplify(DIV(𝛔))3-element TensND.TensRotated{1, 3, Sym{PyCall.PyObject}, Tensors.Vec{3, Sym{PyCall.PyObject}}}:
 r*(-σᵠᵠ(r)*sin(2*θ)/(2*r^2*sin(θ)^2) + σᶿᶿ(r)/(r^2*tan(θ)))
                                                           0
    Derivative(σʳʳ(r), r) + 2*σʳʳ(r)/r - σᵠᵠ(r)/r - σᶿᶿ(r)/r