DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Contravariant Piola map

Definition

Let $\hat{K}, K \subset R^{d}$ be nondegenerate open simplices and let $F_{K} : \hat{K} \to K$ be the orientation-preserving affine bijection $F_{K} (\hat{x}) = A_{K} \hat{x} + b_{K}$ , where $J_{K} : = \det A_{K} > 0$ . For a measurable vector field $\hat{v}$ on $\hat{K}$ , define its contravariant Piola push-forward by

(P_{K} \hat{v}) (x) : = J_{K}^{- 1} A_{K} \hat{v} (F_{K}^{- 1} (x)), x \in K .

The inverse pull-back is $\hat{v} (\hat{x}) = J_{K} A_{K}^{- 1} v (F_{K} (\hat{x}))$ . Facet normals are oriented consistently with the chosen orientation of $F_{K}$ ; the associated oriented surface vector transforms by $n d s = J_{K} A_{K}^{- T} \hat{n} d \hat{s}$ .

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part III Chapter 14: H(div) finite elements (standard reference, not scraped)
Daniele Boffi, Franco Brezzi and Michel Fortin, Mixed Finite Element Methods and Applications (standard reference, not scraped)