Proof of Stokes' theorem by pulling back to Green's theorem

Builds on Parametrised surface Surface integral Flux through a surface Curl Cross product Line integral of a vector field Green's theorem Chain rule in several variables Chain rule for a composition with a curve Product rule Schwarz's theorem for C^2 functions Line integral as an integral over the parameter interval Additivity of line integrals under concatenation Mean value theorem C^k function characterised by continuous derivatives Continuous partial derivatives imply differentiability

Proof

We pull both integrals back to $D$ and apply Green's theorem there.

Let $r$ be $C^2$ on an open set $W \supseteq D$ , and let $F = (F_1, F_2, F_3)$ be $C^1$ on an open set $V \supseteq S$ . The set $W' = W \cap r^{-1}(V)$ is open, since $r$ is continuous, and it contains $D$ .
Write $r_u, r_v$ for the partial derivatives of the parametrised surface $r$ , and $a = r_u$ , $b = r_v$ with components $a_j, b_j$ . Write $\partial_j F_i = \partial F_i / \partial x_j$ .
On $W'$ define $L(u, v) = F(r(u, v)) \cdot r_u(u, v)$ and $M(u, v) = F(r(u, v)) \cdot r_v(u, v)$ .
Each $F_i$ is differentiable, by continuous partial derivatives imply differentiability. By the chain rule, $\partial_v (F_i \circ r) = \sum_j \partial_j F_i(r) \, b_j$ and $\partial_u (F_i \circ r) = \sum_j \partial_j F_i(r) \, a_j$ .
By the product rule in each variable, $\partial_v L = \sum_{i, j} \partial_j F_i(r) \, b_j a_i + F(r) \cdot r_{uv}$ and $\partial_u M = \sum_{i, j} \partial_j F_i(r) \, a_j b_i + F(r) \cdot r_{vu}$ .
These partial derivatives are continuous on $W'$ , because $F$ is $C^1$ and $r$ is $C^2$ . By Schwarz's theorem, $r_{uv} = r_{vu}$ .
Hence $\partial_u M - \partial_v L = \sum_{i, j} \partial_j F_i(r) \, (a_j b_i - a_i b_j)$ . The terms with $i = j$ are $0$ . Grouping the terms $(i, j)$ and $(j, i)$ , the sum is $\sum_{i < j} (\partial_j F_i - \partial_i F_j)(r) \, (a_j b_i - a_i b_j)$ .
By the definitions of the curl and the cross product, the pair $(1, 2)$ gives $(\partial_1 F_2 - \partial_2 F_1)(a_1 b_2 - a_2 b_1) = (\nabla \times F)_3 \, (a \times b)_3$ .
The pair $(1, 3)$ gives $(\partial_3 F_1 - \partial_1 F_3)(a_3 b_1 - a_1 b_3) = (\nabla \times F)_2 \, (a \times b)_2$ .
The pair $(2, 3)$ gives $(\partial_2 F_3 - \partial_3 F_2)(a_2 b_3 - a_3 b_2) = (\nabla \times F)_1 \, (a \times b)_1$ .
By steps 7 to 10, $\partial_u M - \partial_v L = (\nabla \times F)(r) \cdot (r_u \times r_v)$ on $W'$ .
Let $\gamma = (\gamma_1, \gamma_2) \colon [\alpha, \beta] \to \R^2$ , with break points $\alpha = p_0 < \dots < p_m = \beta$ such that $\gamma$ is $C^1$ on each $[p_{l-1}, p_l]$ .
On $(p_{l-1}, p_l)$ , the chain rule along a curve gives $(r \circ \gamma)'(t) = r_u(\gamma(t)) \, \gamma_1'(t) + r_v(\gamma(t)) \, \gamma_2'(t)$ . The right side is continuous on $[p_{l-1}, p_l]$ .
Apply the mean value theorem in each component. The one-sided derivatives of $r \circ \gamma$ at $p_{l-1}$ and $p_l$ exist and equal the limits of the right side. So $r \circ \gamma$ is $C^1$ on each piece, and $\partial S = r \circ \gamma$ is piecewise $C^1$ .
By additivity and the parametric formula on each piece, $\oint_{\partial S} F \cdot d\mathbf{r} = \sum_l \int_{p_{l-1}}^{p_l} F(r(\gamma(t))) \cdot (r \circ \gamma)'(t) \, dt$ .
By step 13 and the definition of $L$ and $M$ , the integrand is $L(\gamma(t)) \, \gamma_1'(t) + M(\gamma(t)) \, \gamma_2'(t)$ . With the parametric formula and additivity again, $\oint_{\partial S} F \cdot d\mathbf{r} = \oint_\gamma (L \, du + M \, dv)$ , a line integral in the plane.
The functions $L$ and $M$ have continuous partial derivatives on the open set $W' \supseteq D$ , by step 6. By Green's theorem and step 11, $\oint_\gamma (L \, du + M \, dv) = \iint_D (\nabla \times F)(r) \cdot (r_u \times r_v) \, du \, dv$ .
Since $r$ is regular, $r_u \times r_v \ne 0$ on $D$ , and $n = (r_u \times r_v) / \norm{r_u \times r_v}$ . By the definitions of the flux and the surface integral, $\iint_S (\nabla \times F) \cdot n \, dS = \iint_D (\nabla \times F)(r) \cdot (r_u \times r_v) \, du \, dv$ .
Steps 16 to 18 give $\oint_{\partial S} F \cdot d\mathbf{r} = \iint_S (\nabla \times F) \cdot n \, dS$ .

Idea

Through the parametrisation, the circulation of $F$ around $\partial S$ becomes a line integral of the plane field $(L, M) = (F \cdot r_u, F \cdot r_v)$ around $\gamma$ . Green's theorem turns it into a double integral of $\partial_u M - \partial_v L$ . The second derivatives of $r$ cancel, and what remains is exactly the curl of $F$ dotted with the normal vector $r_u \times r_v$ .

Reading path

To understand this page from the ground up, read these 407 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 chain rule in several variables, 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 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 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.

Basis concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Element definition
Euclidean metric example
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Extended real number line object
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Integers object
Limit point of a set definition
Linear isomorphism concept
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Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Supremum concept
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Having a directional derivative definition
Total derivative concept
Cross product definition
Function of several variables concept
Vector-valued function concept
Ring of integers statement
Limit of a function definition
Jordan measure definition
Local minimum definition
Successor function definition
Archimedean ordered field definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
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Norm definition
p-norm on R^n example
Complex conjugate definition
Empty set object
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Ordered pair definition
Differentiable function of several variables concept
Partial derivative concept
Limit of a function of several variables concept
Parametrised curve concept
Intersection definition
Union definition
Continuity at a point definition
Derivative definition
Addition of natural numbers definition
Riemann integrable function definition
Proof of the derivative of a continuous linear map proof
Biconditional definition
Truth table concept
Modus ponens concept
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Inner product space definition
Linear independence definition
Composition of linear maps definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
Normed vector space definition
Completeness axiom statement
Cartesian product definition
Proof of the equality of complex numbers proof
Gradient concept
Jacobian matrix concept
Partial derivative definition
Continuity in R^n definition
Differentiable function of several variables definition
Derivative of a continuous linear map statement
Continuous function definition
Antiderivative definition
Differentiable function definition
Union of a family of sets definition
Subset definition
Riemann integral definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Mathematical proof concept
Proof of the derivative as a little-o approximation proof
Order on the natural numbers definition
Propositional logic concept
Uniqueness quantifier definition
The rationals form an ordered field statement
Convergent sequence definition
Sequential continuity definition
Upper and lower Darboux sums definition
Norm induced by an inner product definition
Spanning set definition
Equality of complex numbers statement
Higher partial derivatives definition
Double integral definition
Binary relation definition
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Universal set definition
Gradient definition
Proof of the derivative of a map into R^n by components proof
Topological space definition
Derivative as a little-o approximation statement
Order on the reals definition
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Upper and lower Darboux integrals definition
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Reflexive relation definition
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Strong induction statement
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Proof of the epsilon-delta form of the limit of a function proof
Proof of the epsilon-delta form of continuity proof
Choice function definition
Image of a set definition
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Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
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Epsilon-delta definition of a limit statement
Axiom of choice statement
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Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
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Open set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Closed set definition
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Proof that the derivative gives the directional derivatives proof
Equivalent metrics definition
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Open cover definition
Proof of the description of the open sets of a metric space proof
Proof that open sets are the sets that are neighbourhoods of all their points proof
Zorn's lemma statement
The reals form an ordered field statement
Proof that a set is open if and only if its complement is closed proof
Derivative gives the directional derivatives statement
Equivalent norms definition
Proof that the composition of continuous maps is continuous proof
Compact space definition
Open sets of a metric space statement
Open sets characterised by neighbourhoods statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
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
Composition of continuous maps is continuous statement
Continuous linear map definition
Compact set definition
Proof that closed sets contain their limits proof
Continuity at a point of a topological space definition
Proof of the triangle inequality for real numbers proof
Proof that open balls are open proof
Proof that the absolute value is multiplicative 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
Closed sets contain their limits statement
Proof of continuity via closed sets proof
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
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 of the uniqueness of limits proof
Proof of continuity via open sets in metric spaces proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof that differentiable functions are continuous proof
Proof that limits preserve non-strict inequalities proof
Proof of the squeeze theorem for sequences proof
Proof that metric spaces are Hausdorff proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Intermediate value theorem statement
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 a complex number times its conjugate is the square of its modulus proof
Proof that compact subsets of a Hausdorff space are closed proof
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
Proof that compactness and sequential compactness agree in metric spaces proof
Lebesgue number lemma statement
Differentiable functions are continuous statement
Proof of the sequence characterisation of uniform continuity proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Proof of the triangle inequality for norms proof
Limits preserve non-strict inequalities statement
Linearity of the integral statement
Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Squeeze theorem statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
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
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 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
Heine–Borel theorem statement
Continuous functions on a closed bounded interval are uniformly continuous 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 that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof of the second fundamental theorem of calculus proof
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
Parametrised surface 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
Line integral of a scalar field definition
Regular curve definition
Symmetry of second derivatives statement
Tangent plane and normal vector of a surface definition
Schwarz's theorem for C^2 functions statement
Green's theorem statement
Arc length parametrisation definition
Proof of the parametric formula for line integrals proof
Surface area definition
Proof of the additivity of line integrals under concatenation proof
Line integral as an integral over the parameter interval statement
Surface integral definition
Additivity of line integrals under concatenation statement
Orientable surface definition
Flux through a surface definition
Proof of Stokes' theorem by pulling back to Green's theorem proof

Where this leads

19 pages build on this one.

Tag	`BNRX1PNN39`
URI	`m2://BNRX1PNN39`
Kind	proof
Writing	W0 stub
MSC	26B20
Proves	Stokes' theorem
Uses	Parametrised surface, Surface integral, Flux through a surface, Curl, Cross product, Line integral of a vector field, Green's theorem, Chain rule in several variables, Chain rule for a composition with a curve, Product rule, Schwarz's theorem for C^2 functions, Line integral as an integral over the parameter interval, Additivity of line integrals under concatenation, Mean value theorem, C^k function characterised by continuous derivatives, Continuous partial derivatives imply differentiability
Changed	4 November 2026

Proof of Stokes' theorem by pulling back to Green's theorem

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