CoordinatesSystems

This package provides support to do vectorial computations. The first usage is the derivation of metric tensor and associated operators.

using CoordinatesSystems, Symbolics
𝐞̂ = UnitBasisVectors(CartesianCS)
@variables r Ψ θ
𝐑 = r * cos(Ψ) * sin(θ) * 𝐞̂.x + r * sin(Ψ) * sin(θ) * 𝐞̂.y + r * cos(θ) * 𝐞̂.z
𝐮 = BasisVectors{SphericalCS}(𝐑, [r, θ, Ψ])
g̅̅ = MetricTensor(𝐮)
r_ = 3.0
θ_ = π / 3
g = g̅̅(r_, θ_, 8.0)
g.r.r == 1.0 && g.θ.θ == r_^2 && g.Ψ.Ψ == r_^2 * sin(θ_)^2

using CoordinatesSystems, Symbolics
@variables r φ z
𝐞 = BasisVectors{CylindricalCS}(r * cos(φ), r * sin(φ),z, [r, φ, z])
𝐯 = PVector{CylindricalCS}(r^3, 0,0)
g̅̅ = MetricTensor(𝐞)
∇ = Divergence(g̅̅)
d = ∇ ⋅ 𝐯

r_ = abs(rand()) +0.0001
isapprox(d(r_, 2.0, 3.0),4 * r_^2, atol=1e-14)

The second usage is to perform vectorial calculations on mesh structures.

using CoordinatesSystems, Symbolics
nx = 100
ny = 90
arr_gen = ArrayGenerator(nx, ny)
a = 1.0
b = 2.0
𝐮 = PVector{CartesianCS}(arr_gen; fill=a)
𝐯 = PVector{CartesianCS}(arr_gen; fill=b)
M = 𝐮 ⋅ 𝐯
all(M .== 3 * a * b)

A = rand(nx, ny)
B = rand(nx, ny)
C = rand(nx, ny)
𝐮 = PVector{CartesianCS}(A, B, C)
all(norm(𝐮) .== sqrt.(A .^ 2 .+ B .^ 2 .+ C .^ 2))