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Exact and closed C1 vector fields
Definition
Let be open and let be . Coordinates and partial derivatives are indexed from throughout, as in The standard list with and for is an ordered basis of ; hence , and is the zero space with basis and dimension and The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. It is exact when for some scalar function , using the gradient of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. It is closed when
where the partial derivatives are those of Directional derivatives and partial derivatives of a map . The requirement in exactness makes all mixed second partials of the potential available and continuous.
Depends on
- Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$
Used by
- A C¹ field on an open subset of ℝ³ is closed exactly when its curl vanishes Corollary
- On a star-shaped open domain, closed, exact, conservative, path-independent, and zero-loop are equivalent Corollary
- The vortex field is closed but not exact on the punctured plane Counterexample
- Vector potentials of a continuous field on an open subset of ℝ³ Definition
- Constructing a potential on a rectangle by coordinate-segment integrals Example
- False: every closed C1 field on a connected open set is exact False statement
- The winding number is the circulation of the planar vortex field divided by 2π Remark
- Every exact C1 vector field is closed Theorem
- Poincare's lemma on a star-shaped domain: every closed C1 field is exact Theorem
Dependency tree · two levels
24 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.-B. Campesato, Poincare Lemma, sections 1 and 2 (standard reference, not scraped)