Proof of the homotopy invariance of line integrals of curl-free fields by local potentials

Builds on Curl Line integral of a vector field Closed curve Curl-free vector field on a convex set is conservative Conservative vector field Gradient theorem Additivity of line integrals under concatenation Lebesgue number lemma Heine–Borel theorem Diameter of a set Open ball Convex set Triangle inequality

Proof

We cut the square into small squares whose images lie in balls, use a potential on each ball, and add up. Only the continuity of $H$ is used.

Let $K = [0, 1]^2$ . It is compact by the Heine–Borel theorem, and $H \colon K \to U$ is continuous.
Each open ball $B = B(y, r)$ is convex: for $z, w \in B$ and $\lambda \in [0, 1]$ , the triangle inequality gives $\norm{(1 - \lambda) z + \lambda w - y} < r$ . So an intersection of two open balls is convex.
On each open ball $B \subseteq U$ , the field $F$ is $C^1$ with $\nabla \times F = 0$ . By the curl-free fields on convex sets result, there is a $C^1$ function $\phi_B$ on $B$ with $\nabla \phi_B = F$ .
Claim: let $B, B' \subseteq U$ be open balls and $y, z \in B \cap B'$ . Then $\phi_B(z) - \phi_B(y) = \phi_{B'}(z) - \phi_{B'}(y)$ .
Proof of the claim. By step 2, the segment $\sigma$ from $y$ to $z$ lies in $B \cap B'$ . The gradient theorem on $B$ and on $B'$ gives $\int_\sigma F \cdot d\mathbf{r} = \phi_B(z) - \phi_B(y)$ and $\int_\sigma F \cdot d\mathbf{r} = \phi_{B'}(z) - \phi_{B'}(y)$ .
The sets $H^{-1}(B)$ , for open balls $B \subseteq U$ , are open in $K$ and cover $K$ , since $U$ is open. By the Lebesgue number lemma there is $\eta > 0$ such that each subset of $K$ of diameter less than $\eta$ lies in one of them.
Choose an integer $N$ with $\sqrt{2} / N < \eta$ . For $j, k \in \set{1, \dots, N}$ , let $Q_{jk} = [\tfrac{j-1}{N}, \tfrac{j}{N}] \times [\tfrac{k-1}{N}, \tfrac{k}{N}]$ . Its diameter is $\sqrt{2} / N$ , so $H(Q_{jk}) \subseteq B_{jk}$ for some open ball $B_{jk} \subseteq U$ . Write $\phi_{jk} = \phi_{B_{jk}}$ .
For $j, k \in \set{0, \dots, N}$ , let $v_{jk} = (j/N, k/N)$ and $P_{jk} = H(v_{jk})$ . An edge is a side of some $Q_{jk}$ , from $v$ to $w$ , directed so that one coordinate increases.
For an edge from $v$ to $w$ and a square $Q_{jk}$ that contains it, let $E = \phi_{jk}(H(w)) - \phi_{jk}(H(v))$ . If two squares contain the edge, then $H(v)$ and $H(w)$ lie in both balls. By step 4 they give the same $E$ . So each edge $e$ has a well-defined value $E(e)$ .
Let $h_{jk}$ be the edge from $v_{j-1, k}$ to $v_{jk}$ , and $s_{jk}$ the edge from $v_{j, k-1}$ to $v_{jk}$ . The boundary of $Q_{jk}$ consists of $h_{j, k-1}$ , $s_{jk}$ , $h_{jk}$ and $s_{j-1, k}$ .
Compute all four values with $\phi_{jk}$ . Then $E(h_{j, k-1}) + E(s_{jk}) - E(h_{jk}) - E(s_{j-1, k})$ equals $[\phi_{jk}(P_{j, k-1}) - \phi_{jk}(P_{j-1, k-1})] + [\phi_{jk}(P_{jk}) - \phi_{jk}(P_{j, k-1})] - [\phi_{jk}(P_{jk}) - \phi_{jk}(P_{j-1, k})] - [\phi_{jk}(P_{j-1, k}) - \phi_{jk}(P_{j-1, k-1})]$ , which is $0$ .
Sum step 11 over all $j, k \in \set{1, \dots, N}$ . The terms with $h$ telescope in $k$ , and the terms with $s$ telescope in $j$ . This gives $\sum_{j=1}^N \big( E(h_{j0}) - E(h_{jN}) \big) + \sum_{k=1}^N \big( E(s_{Nk}) - E(s_{0k}) \big) = 0$ .
Since $H(0, s) = H(1, s)$ , we have $P_{0k} = P_{Nk}$ for all $k$ . The points $P_{0, k-1}$ and $P_{0, k}$ lie in $H(Q_{1k}) \subseteq B_{1k}$ , and as $P_{N, k-1}$ and $P_{N, k}$ they lie in $H(Q_{Nk}) \subseteq B_{Nk}$ .
By step 4, $E(s_{Nk}) = \phi_{Nk}(P_{0k}) - \phi_{Nk}(P_{0, k-1}) = \phi_{1k}(P_{0k}) - \phi_{1k}(P_{0, k-1}) = E(s_{0k})$ . With step 12, $\sum_j E(h_{j0}) = \sum_j E(h_{jN})$ .
Let $\beta_j$ be the restriction of $c_0$ to $[\tfrac{j-1}{N}, \tfrac{j}{N}]$ . It is piecewise $C^1$ , and its image $H(h_{j0})$ lies in $B_{j1}$ , since $H(t, 0) = c_0(t)$ .
By the gradient theorem on $B_{j1}$ , $\int_{\beta_j} F \cdot d\mathbf{r} = \phi_{j1}(P_{j0}) - \phi_{j1}(P_{j-1, 0}) = E(h_{j0})$ . By additivity, $\oint_{c_0} F \cdot d\mathbf{r} = \sum_j E(h_{j0})$ .
In the same way, using $H(t, 1) = c_1(t)$ and the balls $B_{jN}$ , $\oint_{c_1} F \cdot d\mathbf{r} = \sum_j E(h_{jN})$ .
By steps 14, 16 and 17, $\oint_{c_0} F \cdot d\mathbf{r} = \oint_{c_1} F \cdot d\mathbf{r}$ .

Idea

A curl-free field has a potential on every ball, though maybe not on all of $U$ . Cut the square into pieces so small that each piece of the homotopy lies in one ball. Around each small square the potential differences cancel, and adding all squares leaves only the two loops, because the two sides are glued by $H(0, s) = H(1, s)$ .

Reading path

To understand this page from the ground up, read these 406 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof that a curl-free vector field on a convex set is conservative, 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 chain rule in several variables, Proof of the gradient theorem, 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 of differentiation under the integral sign, 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 of the linearity of the derivative, Proof of the parametric formula for line integrals, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the derivative of a continuous linear map, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, 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 of the Heine–Cantor theorem by a finite cover of balls, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, 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 of Schwarz's theorem for C^2 functions by the mean value theorem, 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 additivity of line integrals under concatenation, 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 product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the symmetry of second derivatives by a second difference, 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 that the curl of a gradient is zero, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

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Closed sets contain their limits statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Triangle inequality statement
Absolute value is multiplicative statement
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Proof that the reals are Archimedean from the completeness axiom proof
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Continuity via closed sets statement
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof of the uniqueness of limits proof
Proof that differentiable functions are continuous proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
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Proof that metric spaces are Hausdorff proof
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The reals are Archimedean statement
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Proof that a complex number times its conjugate is the square of its modulus proof
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Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Differentiable functions are continuous statement
Proof that compactness and sequential compactness agree in metric spaces proof
Lebesgue number lemma statement
Proof of the sequence characterisation of uniform continuity proof
Limit of a sum of sequences statement
Proof of the triangle inequality for norms proof
Uniqueness of limits statement
Linearity of the integral statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that Darboux and Riemann integrals agree proof
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
Compact subsets of a metric space are bounded statement
Compactness and sequential compactness agree in metric spaces 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
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof of the Heine–Cantor theorem by a finite cover of balls proof
Proof that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits proof
Proof of the Cauchy–Schwarz inequality proof
Proof that continuity is equivalent to sequential continuity proof
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Proof of Riemann's criterion for integrability proof
Heine–Cantor theorem statement
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 product rule proof
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
Proof of the linearity of the derivative proof
Product rule statement
Additivity of the integral over intervals statement
Linearity of differentiation statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Operator norm definition
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Linearity of the derivative statement
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Heine–Borel theorem statement
Boundedness 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 that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem 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 of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
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 of the chain rule in several variables proof
Proof of differentiation under the integral sign proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof of the second fundamental theorem of calculus proof
Differentiation under the integral sign statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Fundamental theorem of calculus statement
Proof of the mean value inequality proof
Proof of Schwarz's theorem for C^2 functions by the mean value theorem proof
Parametrised curve definition
Vector field definition
Mean value inequality statement
Closed curve definition
Rectifiable curve definition
Conservative vector field definition
Curl definition
Tangent vector of a curve definition
Proof of the symmetry of second derivatives by a second difference proof
Line integral of a vector field definition
Proof that a curl-free vector field on a convex set is conservative proof
Line integral of a scalar field definition
Regular curve definition
Scalar potential definition
Symmetry of second derivatives statement
Schwarz's theorem for C^2 functions statement
Arc length parametrisation definition
Proof of the parametric formula for line integrals proof
Proof of the additivity of line integrals under concatenation proof
Proof that the curl of a gradient is zero proof
Line integral as an integral over the parameter interval statement
Additivity of line integrals under concatenation statement
Curl of a gradient is zero statement
Curl-free vector field on a convex set is conservative statement
Proof of the gradient theorem proof
Gradient theorem statement
Proof of the homotopy invariance of line integrals of curl-free fields by local potentials proof

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Homotopy invariance of line integrals of closed forms

Tag	`337Y3FBKTR`
URI	`m2://337Y3FBKTR`
Kind	proof
Writing	W0 stub
MSC	26B20
Proves	Homotopy invariance of line integrals of closed forms
Uses	Curl, Line integral of a vector field, Closed curve, Curl-free vector field on a convex set is conservative, Conservative vector field, Gradient theorem, Additivity of line integrals under concatenation, Lebesgue number lemma, Heine–Borel theorem, Diameter of a set, Open ball, Convex set, Triangle inequality
Changed	4 November 2026

Proof of the homotopy invariance of line integrals of curl-free fields by local potentials

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