False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-13

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False: vector line integrals are invariant under reversing a path

Statement

For every continuous vector field $F$ and piecewise- $C^{1}$ path $γ$ on whose trace it is defined, $\int_{γ^{-}} F \cdot d r = \int_{γ} F \cdot d r .$

Facts & Assumptions

Given: The constant field $F = (1, 0)$ and the segment $γ (t) = (t, 0)$ on $[0, 1]$ .

[L1]

On a $C^{1}$ path $η$ , the vector line integral is the integral of $⟨ F (η (t)), η^{'} (t) ⟩$ (Scalar line integrals with respect to arc length and vector-field line integrals).

[L2]

An orientation-reversing reparametrization negates a vector line integral but leaves a scalar $d s$ integral unchanged (Scalar line integrals are parametrization-independent; vector line integrals retain orientation and change sign when it reverses).

[L3]

The integral of a constant $q$ on $[a, b]$ is $q (b - a)$ (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

Refutation

technique · direct

1.1

Since $γ^{'} (t) = (1, 0)$ , [L1] and [L3] give $\int_{γ} F \cdot d r = \int_{0}^{1} 1 d t = 1$ .

givenL1L3algebra

2.1

The reversal is orientation-reversing, so [L2] gives $\int_{γ^{-}} F \cdot d r = - 1$ .

givenstep 1.1L2algebra

3.1

Since $- 1 \neq 1$ , the statement is false. The scalar analogue is true because [L2] says that $d s$ removes the orientation sign.

step 1.1step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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, Proposition 9.2.15 (standard reference, not scraped)