Hodge Laplacian in Spherical Coordinates¶

In $(r,\theta,\phi)$ coordinates, the Euclidean metric is

$$ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2.$$

We build $d$, $\star$, and the codifferential from this metric. With the codifferential convention used below, the scalar result is the negative of the usual positive-coordinate Laplace operator.

Spherical metric¶

The determinant is $r^4\sin^2\theta$, while the inverse metric contains the angular scale factors $r^{-2}$ and $(r^2\sin^2\theta)^{-1}$.

declare symbol r, θ, φ : MathValue

def N : Integer := 3
def x : Vector MathValue := [| r, θ, φ |]

def g_i_j : Matrix MathValue :=
  [| [| 1, 0, 0 |]
   , [| 0, r ^ 2, 0 |]
   , [| 0, 0, r ^ 2 * (sin θ) ^ 2 |] |]_i_j
def g~i~j : Matrix MathValue :=
  [| [| 1, 0, 0 |]
   , [| 0, 1 / r ^ 2, 0 |]
   , [| 0, 0, 1 / (r ^ 2 * (sin θ) ^ 2) |] |]~i~j

g_#_#

$\begin{pmatrix} 1 & 0 & 0 \\ 0 & r^{2} & 0 \\ 0 & 0 & \sin(θ)^{2} r^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$

Differential-form operators¶

The Hodge star raises the form indices with $g^{ij}$ and contracts them against the three-dimensional Levi-Civita tensor.

def d (A : Tensor MathValue) : Tensor MathValue :=
  !(flip ∂/∂) x A

def hodge (A : DiffForm MathValue) : DiffForm MathValue :=
  let k := dfOrder A
   in withSymbols [i, j]
        sqrt (abs (M.det g_#_#)) *
        foldl
          (.)
          ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k))
          (map (\n -> g~(i_n)~(j_n)) [1..k])

def δ (A : DiffForm MathValue) : DiffForm MathValue :=
  let k := dfOrder A
   in ((-1) ^ (N * (k + 1) + 1)) * hodge (d (hodge A))

The Hodge Laplacian is $d\delta+\delta d$, with shorter endpoint formulas for scalars and volume forms.

def Δ (A : DiffForm MathValue) : DiffForm MathValue :=
  match dfOrder A as integer with
  | #0 -> δ (d A)
  | #N -> d (δ A)
  | _  -> d (δ A) + δ (d A)

def f : MathValue := function (r, θ, φ)

Δ f

$-\frac{\partial^2 f}{\partial 1^2} - 2 \frac{\partial f}{\partial 1} r^{-1} - \frac{\partial^2 f}{\partial 2^2} r^{-2} - \cos(θ) \sin(θ)^{-1} \frac{\partial f}{\partial 2} r^{-2} - \sin(θ)^{-2} \frac{\partial^2 f}{\partial 3^2} r^{-2}$

After simplification, the output is

$$ \Delta f=-\left( f_{rr}+\frac{2}{r}f_r +\frac1{r^2}f_{\theta\theta} +\frac{\cos\theta}{r^2\sin\theta}f_\theta +\frac1{r^2\sin^2\theta}f_{\phi\phi} \right). $$

Each coordinate-dependent coefficient follows from the metric; the calculation contains no spherical-coordinate Laplacian formula as an input.

Hodge Laplacian in Spherical Coordinates¶

Spherical metric¶

Differential-form operators¶

Links