Alphabeta MathHow statement and proof provenance work
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Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence
Definition
An open set is piecewise- path-connected when it is nonempty and every two points of are joined in by a piecewise- path.
For a continuous vector field , a function is a potential when , 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- paths in 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.
Depends on
Used by
- A C¹ field with vanishing curl on a star-shaped open subset of ℝ³ is conservative Corollary
- Conservative fields are path-independent and have zero integral around every closed path Corollary
- On a star-shaped open domain, closed, exact, conservative, path-independent, and zero-loop are equivalent Corollary
- Two potentials of the same field differ by a constant on each piecewise-C1 path component Corollary
- A curl-free C¹ field on the complement of a line that is not conservative Counterexample
- A continuous path-independent field has a potential constructed by line integrals Theorem
- Path independence is equivalent to zero integral around every closed piecewise-C1 path Theorem
Dependency tree · two levels
7 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
- J. Lebl, Basic Analysis II, section 9.3 (standard reference, not scraped)
- J.-B. Campesato, Poincare Lemma, sections 1 and 2 (standard reference, not scraped)