Proof that reversing a path changes the sign of a line integral

Builds on Line integral of a vector field Line integral of a scalar field Rectifiable curve Partition of an interval Parametrised curve Fundamental theorem of calculus Cauchy–Schwarz inequality Triangle inequality Extreme value theorem

Proof

We work directly with the sums in the definitions. Reversal maps each partition of $[a, b]$ to another partition of the same mesh.

Let $\rho(t) = a + b - t$ . It is a decreasing bijection of $[a, b]$ onto itself, and $\gamma^- = \gamma \circ \rho$ .
Since $\gamma$ is piecewise $C^1$ , there is a partition $a = p_0 < \dots < p_m = b$ such that $\gamma$ is a $C^1$ curve on each $[p_{l-1}, p_l]$ . By the extreme value theorem there is $M \ge 0$ with $\norm{\gamma'} \le M$ on each piece, using one-sided derivatives at the $p_l$ .
Let $s < t$ lie in one piece. By the fundamental theorem of calculus in each component, $\gamma(t) - \gamma(s) = \int_s^t \gamma'(u) \, du$ . With $w = \gamma(t) - \gamma(s)$ , the Cauchy–Schwarz inequality gives $\norm{w}^2 = \int_s^t w \cdot \gamma'(u) \, du \le \norm{w} M (t - s)$ , so $\norm{w} \le M (t - s)$ .
For any $s < t$ in $[a, b]$ , insert the points $p_l$ between them. The triangle inequality and step 3 give $\norm{\gamma(t) - \gamma(s)} \le M (t - s)$ .
Hence each inscribed polygon of $\gamma$ has length at most $M (b - a)$ . So $\gamma$ is rectifiable, and the line integrals over $\gamma$ exist.
Let $[c, d] \subseteq [a, b]$ , and let $P$ be a partition $c = t_0 < \dots < t_k = d$ . Put $u_j = \rho(t_{k-j})$ . Then $P^* = (u_0, \dots, u_k)$ is a partition of $[\rho(d), \rho(c)]$ , and $P \mapsto P^*$ is a bijection between the partitions of the two intervals.
The terms $\norm{\gamma^-(t_i) - \gamma^-(t_{i-1})} = \norm{\gamma(u_{k-i+1}) - \gamma(u_{k-i})}$ are the terms of the polygon length of $\gamma$ for $P^*$ , in reverse order. So the two polygon lengths are equal.
Taking suprema, the length of $\gamma^-$ on $[c, d]$ equals the length of $\gamma$ on $[\rho(d), \rho(c)]$ . In particular $\gamma^-$ is rectifiable.
Now let $a = t_0 < \dots < t_k = b$ with points $\tau_i \in [t_{i-1}, t_i]$ . Let $u_j = \rho(t_{k-j})$ and, for $j = k + 1 - i$ , let $\sigma_j = \rho(\tau_i)$ . Then $\sigma_j \in [u_{j-1}, u_j]$ , and the two partitions have the same mesh.
For $j = k + 1 - i$ we have $\gamma^-(t_i) = \gamma(u_{j-1})$ , $\gamma^-(t_{i-1}) = \gamma(u_j)$ and $\gamma^-(\tau_i) = \gamma(\sigma_j)$ .
So $\sum_i F(\gamma^-(\tau_i)) \cdot \big( \gamma^-(t_i) - \gamma^-(t_{i-1}) \big) = -\sum_j F(\gamma(\sigma_j)) \cdot \big( \gamma(u_j) - \gamma(u_{j-1}) \big)$ .
By step 8, the length of $\gamma^-$ on $[t_{i-1}, t_i]$ equals the length $\Delta s_j$ of $\gamma$ on $[u_{j-1}, u_j]$ . So $\sum_i f(\gamma^-(\tau_i)) \Delta s^-_i = \sum_j f(\gamma(\sigma_j)) \Delta s_j$ , where $\Delta s^-_i$ is the length of $\gamma^-$ on $[t_{i-1}, t_i]$ .
Let the mesh tend to $0$ . The right sides of steps 11 and 12 tend to $-\int_\gamma F \cdot d\mathbf{r}$ and $\int_\gamma f \, ds$ . This holds by the definitions of the vector line integral and the scalar line integral.
Hence the left sides converge to the same values, so $\int_{\gamma^-} F \cdot d\mathbf{r} = -\int_\gamma F \cdot d\mathbf{r}$ and $\int_{\gamma^-} f \, ds = \int_\gamma f \, ds$ .

Idea

A partition of the reversed curve is a partition of the original curve read backwards. Each chord $\gamma(t_i) - \gamma(t_{i-1})$ turns into its negative, so the vector sums change sign. Lengths of pieces do not change, so the scalar sums stay the same.

Reading path

To understand this page from the ground up, read these 358 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that open balls are open, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the fundamental theorem of calculus, Proof of the mean value theorem from Rolle's theorem, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

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Proof that every spanning set contains a basis, by Zorn's lemma proof
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Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
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Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Isomorphism with the coordinate space statement
Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Continuity via closed sets statement
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof that differentiable functions are continuous proof
Proof of the uniqueness of limits proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Intermediate value theorem statement
Proof of continuity via open sets in metric spaces proof
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that metric spaces are Hausdorff proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Differentiable functions are continuous statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Lebesgue number lemma statement
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Proof of the triangle inequality for norms proof
Uniqueness of limits statement
Linearity of the integral statement
Proof of the monotonicity of the integral proof
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Squeeze theorem statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Proof that Darboux and Riemann integrals agree proof
Subsequences of a convergent sequence converge to the same limit statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Continuity via open sets statement
Compactness and sequential compactness agree in metric spaces statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Proof of Riemann's criterion for integrability proof
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Proof of the linearity of differentiation proof
Bolzano–Weierstrass theorem statement
Proof of the additivity of the integral over intervals proof
Proof of the derivative of a map into a product proof
Additivity of the integral over intervals statement
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Heine–Borel theorem statement
Proof of the first fundamental theorem of calculus proof
Constant multiple rule statement
Derivative of a map into R^n by components statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Equivalence of norms in finite dimension statement
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the mean value theorem for integrals proof
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that continuous partial derivatives imply differentiability proof
Proof of the second fundamental theorem of calculus proof
Continuous partial derivatives imply differentiability statement
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Fundamental theorem of calculus statement
Parametrised curve definition
Vector field definition
Rectifiable curve definition
Line integral of a vector field definition
Line integral of a scalar field definition
Proof that reversing a path changes the sign of a line integral proof

Where this leads

5 pages build on this one.

Tag	`9SZXY65416`
URI	`m2://9SZXY65416`
Kind	proof
Writing	W0 stub
MSC	26B12
Proves	Reversing a path changes the sign of a line integral
Uses	Line integral of a vector field, Line integral of a scalar field, Rectifiable curve, Partition of an interval, Parametrised curve, Fundamental theorem of calculus, Cauchy–Schwarz inequality, Triangle inequality, Extreme value theorem
Changed	4 November 2026

Proof that reversing a path changes the sign of a line integral

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