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Scalar line integrals with respect to arc length and vector-field line integrals
Definition
Let be piecewise-. If , define both line integrals below to be . If , choose an admissible partition and a continuous derivative extension on each piece. Let be a continuous scalar field and a continuous vector field on a set containing the trace of . The scalar line integral with respect to arc length and the vector-field line integral are
where the inner product is The Euclidean inner product on . The summands exist by A continuous function on is Riemann integrable, by Heine-Cantor and Riemann's criterion. Independence of the admissible partition is proved in The piecewise-C1 line-integral sums do not depend on the admissible partition ↗.
Depends on
Used by
- The scalar line integral of one is the arc length Corollary
- 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
- 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
- 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: vector line integrals are invariant under reversing a path False statement
- 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
- A continuous path-independent field has a potential constructed by 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 gradient theorem: the line integral of a gradient is the endpoint increment Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 129 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis II, section 9.2 (standard reference, not scraped)