Package allowing tensor calculations in arbitrary coordinate systems.
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This Julia package provides tools to perform tensor calculations of any order and any dimension in arbitrary coordinate systems (cartesian, polar, cylindrical, spherical, spheroidal or any user defined coordinate systems...). In particular differential operators are available: gradient, symmetrized gradient, divergence, Laplace, Hessian. The implementation of this library is much inspired by the Maple library Tens3d developed by Jean Garrigues.
This package and its manual are still under construction. The API may vary before official release.
The following example is provided to illustrate the purpose of the library
julia> using SymPy, TensND
julia> Spherical = coorsys_spherical() ; θ, ϕ, r = getcoords(Spherical) ; 𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ = unitvec(Spherical) ;
julia> @set_coorsys Spherical
julia> GRAD(𝐞ʳ) |> intrinsic
(1/r)𝐞ᶿ⊗𝐞ᶿ + (1/r)𝐞ᵠ⊗𝐞ᵠ
julia> DIV(𝐞ʳ ⊗ 𝐞ʳ) |> intrinsic
(2/r)𝐞ʳ
julia> LAPLACE(1/r) |> intrinsic
0
julia> f = SymFunction("f", real = true)
f
julia> DIV(f(r) * 𝐞ʳ ⊗ 𝐞ʳ) |> intrinsic
(Derivative(f(r), r) + 2*f(r)/r)𝐞ʳ
julia> LAPLACE(f(r)) |> intrinsic
d
2 2⋅──(f(r))
d dr
───(f(r)) + ──────────
2 r
dr
julia> for σⁱʲ ∈ ("σʳʳ", "σᶿᶿ", "σᵠᵠ") @eval $(Symbol(σⁱʲ)) = SymFunction($σⁱʲ, real = true)($r) end
julia> 𝛔 = σʳʳ * 𝐞ʳ ⊗ 𝐞ʳ + σᶿᶿ * 𝐞ᶿ ⊗ 𝐞ᶿ + σᵠᵠ * 𝐞ᵠ ⊗ 𝐞ᵠ ; intrinsic(𝛔)
(σᶿᶿ(r))𝐞ᶿ⊗𝐞ᶿ + (σᵠᵠ(r))𝐞ᵠ⊗𝐞ᵠ + (σʳʳ(r))𝐞ʳ⊗𝐞ʳ
julia> div𝛔 = simplify(DIV(𝛔)) ; intrinsic(div𝛔)
((-σᵠᵠ(r) + σᶿᶿ(r))/(r*tan(θ)))𝐞ᶿ + ((r*Derivative(σʳʳ(r), r) + 2*σʳʳ(r) - σᵠᵠ(r) - σᶿᶿ(r))/r)𝐞ʳThe package can be installed with the Julia package manager. From the Julia REPL, type ] to enter the Pkg REPL mode and run:
pkg> add TensNDOr, equivalently, via the Pkg API:
julia> import Pkg; Pkg.add("TensND")- DEV — development version of the documentation.
@misc{TensND.jl,
author = {Jean-François Barthélémy},
title = {TensND.jl},
url = {https://github.com/jfbarthelemy/TensND.jl},
version = {v0.1.3},
year = {2021},
month = {8}
}