-- Spherical Laplacian in 3D using tensor notation

declare symbol r, θ, φ : MathExpr

def f := function (r, θ, φ)

def x := [| r, θ, φ |]

def X := [| r * sin θ * cos φ, r * sin θ * sin φ, r * cos θ |]

-- Local basis
def e_i_j : Matrix MathExpr := ∂/∂ X_j x~i

-- Metric tensor
def g_i_j := generateTensor (\[a, b] -> V.* e_a e_b) [3, 3]
def g~i~j := M.inverse g_#_#

assertEqual "Metric tensor g_#_#"
  g_#_#
  [| [| 1, 0, 0 |], [| 0, r^2, 0 |], [| 0, 0, r^2 * (sin θ)^2 |] |]_#_#

-- Christoffel symbols
def Γ_i_j_k := withSymbols [j, k, l]
  (1 / 2) * (∂/∂ g_j_l x~k + ∂/∂ g_j_k x~l - ∂/∂ g_k_l x~j)

def Γ~i_j_k := withSymbols [i, j, k, l]
  g~i~j . Γ_j_k_l

-- Laplacian via Christoffel symbols
def Laplacian := withSymbols [i, j, k]
  g~i~j . ∂/∂ (∂/∂ f x~j) x~i - g~i~j . Γ~k_i_j . ∂/∂ f x~k

assertEqual "Laplacian in spherical coordinates"
  Laplacian
  (∂/∂ (∂/∂ f r) r + 2 * ∂/∂ f r / r + ∂/∂ (∂/∂ f θ) θ / r^2 + cos θ * ∂/∂ f θ / (r^2 * sin θ) + ∂/∂ (∂/∂ f φ) φ / (r^2 * (sin θ)^2))