Scalar line integrals with respect to arc length and vector-field line integrals

Depends on

• Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations
• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
• A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion

Used by

• Green's theorem is the curl statement for a planar field lifted to ℝ³ Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• The planar divergence theorem: the flux form of Green's theorem Corollary
• The scalar line integral of one is the arc length Corollary
• A curl-free C¹ field on the complement of a line that is not conservative Counterexample
• The vector field (y,0) gives different integrals along two paths with the same endpoints Counterexample
• The vortex field is closed but not exact on the punctured plane Counterexample
• Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence Definition
• Positive orientation of elementary-region boundaries Definition
• The induced boundary chain and circulation of a C² patch over a finite elementary Green region Definition
• A polynomial potential evaluates work along every path by endpoints Example
• A vector line integral around the vortex counts repeated traversals Example
• Scalar and vector line integrals along an affine line segment Example
• Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
• The planar divergence theorem on a rectangle, checked against a direct boundary computation Example
• The scalar line integral of x over the right unit semicircle equals two Example
• False: every closed C1 field on a connected open set is exact False statement
• FALSE: Stokes' theorem requires the surface to be a graph over a coordinate plane False statement
• False: vector line integrals are invariant under reversing a path False statement
• A vector line integral along an image arc is the parameter line integral of the pulled-back field Lemma
• The piecewise-C1 line-integral sums do not depend on the admissible partition Lemma
• The Type I boundary identity for the P dx term Lemma
• The Type II boundary identity for the Q dy term Lemma
• The winding number is the circulation of the planar vortex field divided by 2π Remark
• A continuous path-independent field has a potential constructed by line integrals Theorem
• For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals Theorem
• Green's theorem for finite unions of elementary regions Theorem
• Line integrals under reversal and concatenation Theorem
• Line-integral estimates by arc length and the supremum of the field Theorem
• Path independence is equivalent to zero integral around every closed piecewise-C1 path Theorem
• Scalar line integrals are parametrization-independent; vector line integrals retain orientation and change sign when it reverses Theorem
• The classical Stokes theorem for a C² patch over a finite elementary Green region Theorem
• The gradient theorem: the line integral of a gradient is the endpoint increment Theorem

Dependency tree · two levels

40 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 9.2 (standard reference, not scraped)