CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Stokes agrees with the fundamental theorem of calculus

Statement

Assume ${A C}_{ω}$ . For $a < b$ , orient $[a, b]$ increasingly. Every smooth $f$ on this interval satisfies $\int_{[a, b]} d f = f (b) - f (a),$ where the boundary point signs are $- 1$ at $a$ and $+ 1$ at $b$ . This agrees with the Riemann fundamental theorem of calculus.

Facts & Assumptions

[F1]

The general Stokes theorem: Assume ${A C}_{ω}$ . Let $M$ be an oriented smooth $n$ -manifold with boundary, $n \geq 1$ , and let $η \in Ω_{c}^{n - 1} (M)$ . With $j : \partial M ↪ M$ and the outward-normal-first orientation, $\int_{M} d η = \int_{\partial M} j^{*} η .$ An empty boundary contributes zero; in dimension one its integral is a finite signed sum of point values.

[F2]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: Let $a < b$ . Suppose $G : [a, b] \to R$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . If $f : [a, b] \to R$ is Riemann integrable and $f (x) = G^{'} (x) (a < x < b),$ then $\int_{a}^{b} f = G (b) - G (a) .$ No derivative of $G$ at either endpoint is assumed, and the two endpoint values assigned to the integrable extension $f$ do not enter the conclusion.

Proof

Given: The objects and hypotheses in the statement above.

1.1F1

The interval is compact, and the outward directions are $- \partial_{t}$ at $a$ and $+ \partial_{t}$ at $b$ . Outward-first gives the point signs $- 1, + 1$ . Stokes therefore yields the difference $f (b) - f (a)$ , including constant and zero functions.

2.1F2step 1.1∎

In its increasing coordinate, $d f = f^{'} (t) d t$ , so the left side is the ordinary Riemann integral of $f^{'}$ . The published FTC applies: $f$ is continuous on the closed interval and differentiable inside, and its smooth derivative is Riemann integrable. It gives the same endpoint difference. The condition $a < b$ avoids treating a point as a one-manifold.

Depends on

Used by

Stokes on an interval with both endpoint chart signs Example
False: all exact forms integrate to zero everywhere False statement

Dependency tree · two levels

11 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

Lee Example 16.12, p.414 (standard reference, not scraped)