LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The inverse-projected characteristic graph satisfies the quasilinear PDE

Statement

The function $u = Z \circ Ψ^{- 1}$ constructed in Local quasilinear characteristic graph construction satisfies $a (x, u (x)) \cdot D u (x) = b (x, u (x))$ on its local domain.

Facts & Assumptions

Given: The local inverse-projected graph and its characteristic identities $X_{s} = a (X, Z)$ and $Z_{s} = b (X, Z)$ .

Proof

technique · direct

1.1

The identity $u (X (s, y)) = Z (s, y)$ differentiates in $s$ to $D u (X) \cdot X_{s} = Z_{s}$ .

givenalgebra

2.1

Substitute the two characteristic identities to get $D u (X) \cdot a (X, Z) = b (X, Z)$ .

step 1.1givenalgebra

3.1

Since $X = Ψ (s, y)$ ranges over the local domain and $Z = u (X)$ , this is $a (x, u (x)) \cdot D u (x) = b (x, u (x))$ for every $x$ there.

step 2.1given∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Part I: Explicit methods — Lecture notes for MA342H (standard reference, not scraped)