DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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.

Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence

Definition

An open set $U \subseteq R^{n}$ is piecewise- $C^{1}$ path-connected when it is nonempty and every two points of $U$ are joined in $U$ by a piecewise- $C^{1}$ path.

For a continuous vector field $F : U \to R^{n}$ , a $C^{1}$ function $ϕ : U \to R$ is a potential when $F = \nabla ϕ$ , with the gradient of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. The field is conservative when it has a potential. It is path-independent when any two piecewise- $C^{1}$ paths in $U$ with the same initial and terminal points have equal vector line integrals as defined in Scalar line integrals with respect to arc length and vector-field line integrals.

Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence

Definition

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