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docs: clarify curl(grad(f)) residual is discretization error #1523

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docs: clarify curl(grad(f)) residual is discretization error, not a bug
rajeeja Jun 11, 2026
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ruff format notebook
rajeeja Jun 12, 2026
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11 changes: 7 additions & 4 deletions docs/user-guide/vector_calculus.ipynb
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Expand Up @@ -321,7 +321,7 @@
"source": [
"### Example 1: Curl of Gradient Field\n",
"\n",
"A fundamental vector identity states that the curl of any gradient field should be zero (conservative field). Let's verify this:"
"In the continuous setting, curl(∇φ) = 0 exactly for any scalar field φ. On an unstructured mesh, however, gradient and curl are independent finite-volume stencils that do not form a discrete de Rham complex, so the identity holds only approximately. The residual is a **discretization error** — not a bug — and shrinks with grid refinement (O(Δx) for first-order stencils). For a Gaussian field on a coarse mesh the residual is typically O(1–10), consistent with the grid spacing.\n"
]
},
{
Expand Down Expand Up @@ -690,8 +690,8 @@
"\n",
"Let's verify some fundamental vector calculus identities using our computed fields:\n",
"\n",
"### Identity 1: Curl of Gradient is Zero\n",
"For any scalar field φ: ∇ × (∇φ) = 0"
"### Identity 1: Curl of Gradient (Discretization Residual)\n",
"In the continuous setting: ∇ × (∇φ) = 0. UXarray's finite-volume operators are not mimetic, so this holds only approximately. The residual below is discretization error, not a numerical bug.\n"
]
},
{
Expand All @@ -704,7 +704,10 @@
"# We already computed this above\n",
"max_curl_grad = np.abs(curl_of_gradient).max().values\n",
"print(f\"Maximum |curl(∇φ)|: {max_curl_grad:.2e}\")\n",
"print(f\"Identity verified: {max_curl_grad < 1e-10}\")"
"print(\n",
" \"Note: non-zero residual is expected discretization error from independent \"\n",
" \"finite-volume gradient and curl stencils (not a bug). Shrinks with grid refinement.\"\n",
")"
]
},
{
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