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Generalized Stokes theorem
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<h2 id="introduction">Introduction</h2> <p>The <a href="/facts/Fundamental_theorem_of_calculus/pHeSGtUx">second fundamental theorem of calculus</a> states that the <a href="/facts/Integral/V7lyV997">integral</a> of a function f {\displaystyle f} over the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> [ a , b ] {\displaystyle [a,b]} can be calculated by finding an <a href="/facts/Antiderivative/5qDsudIv">antiderivative</a> F {\displaystyle F} of f {\displaystyle f} : ∫ a b f ( x ) d x = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a)\,.} </p><p>Stokes' theorem is a vast generalization of this theorem in the following sense. </p> <ul><li>By the choice of F {\displaystyle F} , d F d x = f ( x ) {\displaystyle {\frac {dF}{dx}}=f(x)} . In the parlance of <a href="/facts/Differential_form/T9gyHtMv">differential forms</a>, this is saying that f ( x ) d x {\displaystyle f(x)\,dx} is the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> of the 0-form, i.e. function, F {\displaystyle F} : in other words, that d F = f d x {\displaystyle dF=f\,dx} . The general Stokes theorem applies to higher <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> differential forms ω {\displaystyle \omega } instead of just 0-forms such as F {\displaystyle F} .</li> <li>A closed interval [ a , b ] {\displaystyle [a,b]} is a simple example of a one-dimensional <a href="/facts/Manifold_with_boundary/rXPERJtu">manifold with boundary</a>. Its boundary is the set consisting of the two points a {\displaystyle a} and b {\displaystyle b} . Integrating f {\displaystyle f} over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be <a href="/facts/Orientable/1SEQM2Pt">orientable</a>, and the form has to be <a href="/facts/Compact_support/BTuMGbsb">compactly supported</a> in order to give a well-defined integral.</li> <li>The two points a {\displaystyle a} and b {\displaystyle b} form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds M {\displaystyle M} with boundary. The boundary ∂ M {\displaystyle \partial M} of M {\displaystyle M} is itself a manifold and inherits a natural orientation from that of M {\displaystyle M} . For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, a {\displaystyle a} inherits the opposite orientation as b {\displaystyle b} , as they are at opposite ends of the interval. So, "integrating" F {\displaystyle F} over two boundary points a {\displaystyle a} , b {\displaystyle b} is taking the difference F ( b ) − F ( a ) {\displaystyle F(b)-F(a)} .</li></ul> <p>In even simpler terms, one can consider the points as boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral ( f d x = d F {\displaystyle f\,dx=dF} ) over a 1-dimensional manifold ( [ a , b ] {\displaystyle [a,b]} ) by considering the anti-derivative ( F {\displaystyle F} ) at the 0-dimensional boundaries ( { a , b } {\displaystyle \{a,b\}} ), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals ( d ω {\displaystyle d\omega } ) over n {\displaystyle n} -dimensional manifolds ( Ω {\displaystyle \Omega } ) by considering the antiderivative ( ω {\displaystyle \omega } ) at the ( n − 1 ) {\displaystyle (n-1)} -dimensional boundaries ( ∂ Ω {\displaystyle \partial \Omega } ) of the manifold. </p><p>So the fundamental theorem reads: ∫ [ a , b ] f ( x ) d x = ∫ [ a , b ] d F = ∫ ∂ [ a , b ] F = ∫ { a } − ∪ { b } + F = F ( b ) − F ( a ) . {\displaystyle \int _{[a,b]}f(x)\,dx=\int _{[a,b]}\,dF=\int _{\partial [a,b]}\,F=\int _{\{a\}^{-}\cup \{b\}^{+}}F=F(b)-F(a)\,.} </p> <h2 id="formulation-for-smooth-manifolds-with-boundary">Formulation for smooth manifolds with boundary</h2> <p>Let Ω {\displaystyle \Omega } be an <a href="/facts/Oriented_manifold/1SEQM2Pt">oriented</a> <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> of <a href="/facts/Dimension/0ktmUyac">dimension</a> n {\displaystyle n} with boundary and let α {\displaystyle \alpha } be a <a href="/facts/Smooth_function/mffL4ch2">smooth</a> n {\displaystyle n} -<a href="/facts/Differential_form/T9gyHtMv">differential form</a> that is <a href="/facts/Support_(mathematics)/BTuMGbsb">compactly supported</a> on Ω {\displaystyle \Omega } . First, suppose that α {\displaystyle \alpha } is compactly supported in the domain of a single, oriented <a href="/facts/Coordinate_chart/eLKXMzzR">coordinate chart</a> { U , φ } {\displaystyle \{U,\varphi \}} . In this case, we define the integral of α {\displaystyle \alpha } over Ω {\displaystyle \Omega } as ∫ Ω α = ∫ φ ( U ) ( φ − 1 ) ∗ α , {\displaystyle \int _{\Omega }\alpha =\int _{\varphi (U)}(\varphi ^{-1})^{*}\alpha \,,} i.e., via the <a href="/facts/Pullback_(differential_geometry)/EzMDBYfK">pullback</a> of α {\displaystyle \alpha } to R n {\displaystyle \mathbb {R} ^{n}} . </p><p>More generally, the integral of α {\displaystyle \alpha } over Ω {\displaystyle \Omega } is defined as follows: Let { ψ i } {\displaystyle \{\psi _{i}\}} be a <a href="/facts/Partition_of_unity/4byX17aR">partition of unity</a> associated with a <a href="/facts/Locally_finite_collection/WTjQrJ1S">locally finite</a> <a href="/facts/Cover_(topology)/2VUdelA4">cover</a> { U i , φ i } {\displaystyle \{U_{i},\varphi _{i}\}} of (consistently oriented) coordinate charts, then define the integral ∫ Ω α ≡ ∑ i ∫ U i ψ i α , {\displaystyle \int _{\Omega }\alpha \equiv \sum _{i}\int _{U_{i}}\psi _{i}\alpha \,,} where each term in the sum is evaluated by pulling back to R n {\displaystyle \mathbb {R} ^{n}} as described above. This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity. </p><p>The generalized Stokes theorem reads: </p> <p>Theorem (<i>Stokes–Cartan</i>)—Let ω {\displaystyle \omega } be a <a href="/facts/Infinitely_differentiable/mffL4ch2">smooth</a> ( n − 1 ) {\displaystyle (n-1)} -<a href="/facts/Differential_form/T9gyHtMv">form</a> with <a href="/facts/Compact_support/BTuMGbsb">compact support</a> on an <a href="/facts/Oriented/1SEQM2Pt">oriented</a>, n {\displaystyle n} -dimensional <a href="/facts/Manifold/rXPERJtu">manifold-with-boundary</a> Ω {\displaystyle \Omega } , where ∂ Ω {\displaystyle \partial \Omega } is given the induced orientation. Then ∫ Ω d ω = ∫ ∂ Ω ω . {\displaystyle \int _{\Omega }d\omega =\int _{\partial \Omega }\omega .} </p> <p>Here d {\displaystyle d} is the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a>, which is defined using the manifold structure only. The right-hand side is sometimes written as ∮ ∂ Ω ω {\textstyle \oint _{\partial \Omega }\omega } to stress the fact that the ( n − 1 ) {\displaystyle (n-1)} -manifold ∂ Ω {\displaystyle \partial \Omega } has no boundary.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> (This fact is also an implication of Stokes' theorem, since for a given smooth n {\displaystyle n} -dimensional manifold Ω {\displaystyle \Omega } , application of the theorem twice gives ∫ ∂ ( ∂ Ω ) ω = ∫ Ω d ( d ω ) = 0 {\textstyle \int _{\partial (\partial \Omega )}\omega =\int _{\Omega }d(d\omega )=0} for any ( n − 2 ) {\displaystyle (n-2)} -form ω {\displaystyle \omega } , which implies that ∂ ( ∂ Ω ) = ∅ {\displaystyle \partial (\partial \Omega )=\emptyset } .) The right-hand side of the equation is often used to formulate <i>integral</i> laws; the left-hand side then leads to equivalent <i>differential</i> formulations (see below). </p><p>The theorem is often used in situations where Ω {\displaystyle \Omega } is an embedded oriented submanifold of some bigger manifold, often R k {\displaystyle \mathbb {R} ^{k}} , on which the form ω {\displaystyle \omega } is defined. </p> <h2 id="topological-preliminaries-integration-over-chains">Topological preliminaries; integration over chains</h2> <p>Let M be a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a>. A (smooth) singular <a href="/facts/Simplex/0FCfNRN8">k-simplex</a> in M is defined as a <a href="/facts/Smooth_map/mffL4ch2">smooth map</a> from the standard simplex in R<i>k</i> to M. The group <i>C</i><i>k</i>(<i>M</i>, Z) of singular k-<a href="/facts/Chain_(algebraic_topology)/IDYbXM7u">chains</a> on M is defined to be the <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian group</a> on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a <a href="/facts/Chain_complex/YAlchtCT">chain complex</a>. The corresponding homology (resp. cohomology) group is isomorphic to the usual <a href="/facts/Singular_homology/6c2AHr85">singular homology</a> group <i>H</i><i>k</i>(<i>M</i>, Z) (resp. the <a href="/facts/Singular_cohomology/J7FJzTNh">singular cohomology</a> group <i>H</i><i>k</i>(<i>M</i>, Z)), defined using continuous rather than smooth simplices in M. </p><p>On the other hand, the differential forms, with exterior derivative, d, as the connecting map, form a cochain complex, which defines the <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a> groups H d R k ( M , R ) {\displaystyle H_{dR}^{k}(M,\mathbf {R} )} . </p><p>Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to R<i>k</i>. Extending by linearity allows one to integrate over chains. This gives a linear map from the space of k-forms to the kth group of singular cochains, <i>Ck</i>(<i>M</i>, Z), the linear functionals on <i>Ck</i>(<i>M</i>, Z). In other words, a k-form ω defines a functional I ( ω ) ( c ) = ∮ c ω . {\displaystyle I(\omega )(c)=\oint _{c}\omega .} on the k-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the <i>dual</i> of ∂ on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means: </p> <ol><li>closed forms, i.e., <i>dω</i> = 0, have zero integral over <i>boundaries</i>, i.e. over manifolds that can be written as ∂Σ<i>c</i> <i>Mc</i>, and</li> <li>exact forms, i.e., <i>ω</i> = <i>dσ</i>, have zero integral over <i>cycles</i>, i.e. if the boundaries sum up to the empty set: ∂Σ<i>c</i> <i>Mc</i> = ∅.</li></ol> <p><a href="/facts/De_Rham%27s_theorem/4qGKeJq0">De Rham's theorem</a> shows that this homomorphism is in fact an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>. So the converse to 1 and 2 above hold true. In other words, if {<i>ci</i>} are cycles generating the kth homology group, then for any corresponding real numbers, {<i>ai</i>} , there exist a closed form, ω, such that ∮ c i ω = a i , {\displaystyle \oint _{c_{i}}\omega =a_{i}\,,} and this form is unique up to exact forms. </p><p>Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Formally stated, the latter reads:<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <p>Theorem (<i>Stokes' theorem for chains</i>)—If c is a smooth k-chain in a smooth manifold M, and ω is a smooth (<i>k</i> − 1)-form on M, then ∫ ∂ c ω = ∫ c d ω . {\displaystyle \int _{\partial c}\omega =\int _{c}d\omega .} </p> <h2 id="underlying-principle">Underlying principle</h2> <p>To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an example for <i>d</i> = 2 dimensions. The essential idea can be understood by the diagram on the left, which shows that, in an oriented tiling of a manifold, the interior paths are traversed in opposite directions; their contributions to the path integral thus cancel each other pairwise. As a consequence, only the contribution from the boundary remains. It thus suffices to prove Stokes' theorem for sufficiently fine tilings (or, equivalently, <a href="/facts/Simplex/0FCfNRN8">simplices</a>), which usually is not difficult. </p> <h2 id="classical-vector-analysis-example">Classical vector analysis example</h2> <p>Let γ : [ a , b ] → R 2 {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}} be a <a href="/facts/Piecewise/nlQwptpe">piecewise</a> smooth <a href="/facts/Jordan_curve/zKpty4QT">Jordan plane curve</a>. The <a href="/facts/Jordan_curve_theorem/zKpty4QT">Jordan curve theorem</a> implies that γ {\displaystyle \gamma } divides R 2 {\displaystyle \mathbb {R} ^{2}} into two components, a <a href="/facts/Compact_space/d0cgXJH7">compact</a> one and another that is non-compact. Let D {\displaystyle D} denote the compact part that is bounded by γ {\displaystyle \gamma } and suppose ψ : D → R 3 {\displaystyle \psi :D\to \mathbb {R} ^{3}} is smooth, with S = ψ ( D ) {\displaystyle S=\psi (D)} . If Γ {\displaystyle \Gamma } is the <a href="/facts/Space_curve/xBQ0pTBW">space curve</a> defined by Γ ( t ) = ψ ( γ ( t ) ) {\displaystyle \Gamma (t)=\psi (\gamma (t))} <a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and F {\displaystyle {\textbf {F}}} is a smooth vector field on R 3 {\displaystyle \mathbb {R} ^{3}} , then:<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> ∮ Γ F ⋅ d Γ = ∬ S ( ∇ × F ) ⋅ d S {\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\left(\nabla \times \mathbf {F} \right)\cdot \,d\mathbf {S} } </p><p>This classical statement is a special case of the general formulation after making an identification of vector field with a 1-form and its curl with a two form through ( F x F y F z ) ⋅ d Γ → F x d x + F y d y + F z d z {\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\\\end{pmatrix}}\cdot d\Gamma \to F_{x}\,dx+F_{y}\,dy+F_{z}\,dz} ∇ × ( F x F y F z ) ⋅ d S = ( ∂ y F z − ∂ z F y ∂ z F x − ∂ x F z ∂ x F y − ∂ y F x ) ⋅ d S → d ( F x d x + F y d y + F z d z ) = ( ∂ y F z − ∂ z F y ) d y ∧ d z + ( ∂ z F x − ∂ x F z ) d z ∧ d x + ( ∂ x F y − ∂ y F x ) d x ∧ d y . {\displaystyle {\begin{aligned}&\nabla \times {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}\cdot d\mathbf {S} ={\begin{pmatrix}\partial _{y}F_{z}-\partial _{z}F_{y}\\\partial _{z}F_{x}-\partial _{x}F_{z}\\\partial _{x}F_{y}-\partial _{y}F_{x}\\\end{pmatrix}}\cdot d\mathbf {S} \to \\[1.4ex]&d(F_{x}\,dx+F_{y}\,dy+F_{z}\,dz)=\left(\partial _{y}F_{z}-\partial _{z}F_{y}\right)dy\wedge dz+\left(\partial _{z}F_{x}-\partial _{x}F_{z}\right)dz\wedge dx+\left(\partial _{x}F_{y}-\partial _{y}F_{x}\right)dx\wedge dy.\end{aligned}}} </p> <h2 id="generalization-to-rough-sets">Generalization to rough sets</h2> <p>The formulation above, in which Ω {\displaystyle \Omega } is a smooth manifold with boundary, does not suffice in many applications. For example, if the domain of integration is defined as the plane region between two x {\displaystyle x} -coordinates and the graphs of two functions, it will often happen that the domain has corners. In such a case, the corner points mean that Ω {\displaystyle \Omega } is not a smooth manifold with boundary, and so the statement of Stokes' theorem given above does not apply. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. This is because Ω {\displaystyle \Omega } and its boundary are well-behaved away from a small set of points (a <a href="/facts/Measure_zero/kMudL8Mn">measure zero</a> set). </p><p>A version of Stokes' theorem that allows for roughness was proved by Whitney.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Assume that D {\displaystyle D} is a connected bounded open subset of R n {\displaystyle \mathbb {R} ^{n}} . Call D {\displaystyle D} a <i>standard domain</i> if it satisfies the following property: there exists a subset P {\displaystyle P} of ∂ D {\displaystyle \partial D} , open in ∂ D {\displaystyle \partial D} , whose complement in ∂ D {\displaystyle \partial D} has <a href="/facts/Hausdorff_measure/Me6eTFVG">Hausdorff ( n − 1 ) {\displaystyle (n-1)} -measure</a> zero; and such that every point of P {\displaystyle P} has a <i>generalized normal vector</i>. This is a vector v ( x ) {\displaystyle {\textbf {v}}(x)} such that, if a coordinate system is chosen so that v ( x ) {\displaystyle {\textbf {v}}(x)} is the first basis vector, then, in an open neighborhood around x {\displaystyle x} , there exists a smooth function f ( x 2 , … , x n ) {\displaystyle f(x_{2},\dots ,x_{n})} such that P {\displaystyle P} is the graph { x 1 = f ( x 2 , … , x n ) } {\displaystyle \{x_{1}=f(x_{2},\dots ,x_{n})\}} and D {\displaystyle D} is the region { x 1 : x 1 < f ( x 2 , … , x n ) } {\displaystyle \{x_{1}:x_{1}<f(x_{2},\dots ,x_{n})\}} . Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff ( n − 1 ) {\displaystyle (n-1)} -measure and a finite or countable union of smooth ( n − 1 ) {\displaystyle (n-1)} -manifolds, each of which has the domain on only one side. He then proves that if D {\displaystyle D} is a standard domain in R n {\displaystyle \mathbb {R} ^{n}} , ω {\displaystyle \omega } is an ( n − 1 ) {\displaystyle (n-1)} -form which is defined, continuous, and bounded on D ∪ P {\displaystyle D\cup P} , smooth on D {\displaystyle D} , integrable on P {\displaystyle P} , and such that d ω {\displaystyle d\omega } is integrable on D {\displaystyle D} , then Stokes' theorem holds, that is, ∫ P ω = ∫ D d ω . {\displaystyle \int _{P}\omega =\int _{D}d\omega \,.} </p><p>The study of measure-theoretic properties of rough sets leads to <a href="/facts/Geometric_measure_theory/NLIQxYf9">geometric measure theory</a>. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> <h2 id="special-cases">Special cases</h2> <p>The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases. The traditional versions can be formulated using <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> without the machinery of differential geometry, and thus are more accessible. Further, they are older and their names are more familiar as a result. The traditional forms are often considered more convenient by practicing scientists and engineers but the non-naturalness of the traditional formulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindrical coordinates. There is potential for confusion in the way names are applied, and the use of dual formulations. </p> <h3>Classical (vector calculus) case</h3> <p class="note">Main article: <a href="/facts/Stokes%27_theorem/z0pAoj2x">Stokes' theorem</a></p> <p>This is a (dualized) (1 + 1)-dimensional case, for a 1-form (dualized because it is a statement about <a href="/facts/Vector_field/ccFNuPPp">vector fields</a>). This special case is often just referred to as <i>Stokes' theorem</i> in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as the <a href="/facts/Curl_(mathematics)/GuS8dc7o">curl</a> theorem. </p><p>The classical Stokes' theorem relates the <a href="/facts/Surface_integral/oPk5IRI1">surface integral</a> of the <a href="/facts/Curl_(mathematics)/GuS8dc7o">curl</a> of a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> over a surface Σ {\displaystyle \Sigma } in Euclidean three-space to the <a href="/facts/Line_integral/10s4B3Nn">line integral</a> of the vector field over its boundary. It is a special case of the general Stokes theorem (with n = 2 {\displaystyle n=2} ) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, ∂ Σ {\displaystyle \partial \Sigma } , must have positive <a href="/facts/Curve_orientation/HLpSX2Q2">orientation</a>, meaning that ∂ Σ {\displaystyle \partial \Sigma } points counterclockwise when the <a href="/facts/Normal_(geometry)/jhtLIhcu">surface normal</a>, n {\displaystyle n} , points toward the viewer. </p><p> One consequence of this theorem is that the <a href="/facts/Field_line/Nxr20d47">field lines</a> of a vector field with zero curl cannot be closed contours. The formula can be rewritten as:</p> <p>Theorem—Suppose F = ( P ( x , y , z ) , Q ( x , y , z ) , R ( x , y , z ) ) {\displaystyle {\textbf {F}}={\big (}P(x,y,z),Q(x,y,z),R(x,y,z){\big )}} is defined in a region with smooth surface Σ {\displaystyle \Sigma } and has continuous first-order <a href="/facts/Partial_derivatives/JWHsBkAu">partial derivatives</a>. Then ∬ Σ ( ( ∂ R ∂ y − ∂ Q ∂ z ) d y d z + ( ∂ P ∂ z − ∂ R ∂ x ) d z d x + ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y ) = ∮ ∂ Σ ( P d x + Q d y + R d z ) , {\displaystyle \iint _{\Sigma }{\Biggl (}\left({\frac {\partial R}{\partial y}}-{\frac {\partial Q}{\partial z}}\right)dy\,dz+\left({\frac {\partial P}{\partial z}}-{\frac {\partial R}{\partial x}}\right)dz\,dx+\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)dx\,dy{\Biggr )}=\oint _{\partial \Sigma }{\Big (}P\,dx+Q\,dy+R\,dz{\Big )}\,,} where P , Q {\displaystyle P,Q} and R {\displaystyle R} are the components of F {\displaystyle {\textbf {F}}} , and ∂ Σ {\displaystyle \partial \Sigma } is the boundary of the region Σ {\displaystyle \Sigma } . </p> <h3>Green's theorem</h3> <p><a href="/facts/Green%27s_theorem/WBDk654A">Green's theorem</a> is immediately recognizable as the third integrand of both sides in the integral in terms of P, Q, and R cited above. </p> <h4>In electromagnetism</h4> <p>Two of the four <a href="/facts/Maxwell_equations/LfJ1drHS">Maxwell equations</a> involve curls of 3-D vector fields, and their differential and integral forms are related by the special 3-dimensional (vector calculus) case of <a href="/facts/Stokes%27_theorem/z0pAoj2x">Stokes' theorem</a>. Caution must be taken to avoid cases with moving boundaries: the partial time derivatives are intended to exclude such cases. If moving boundaries are included, interchange of integration and differentiation introduces terms related to boundary motion not included in the results below (see <a href="/facts/Differentiation_under_the_integral_sign/bFKo8h0I">Differentiation under the integral sign</a>): </p> <table><tbody><tr><th>Name</th><th><a href="/facts/Partial_differential_equation/DyPsBlv7">Differential</a> form</th><th><a href="/facts/Integral/V7lyV997">Integral</a> form (using three-dimensional Stokes theorem plus relativistic invariance, ∫ ∂ ∂ t ⋯ → d d t ∫ ⋯ {\displaystyle \textstyle \int {\tfrac {\partial }{\partial t}}\dots \to {\tfrac {d}{dt}}\textstyle \int \cdots } )</th></tr><tr><td>Maxwell–Faraday equation <a href="/facts/Faraday%27s_law_of_induction/PFBreB0F">Faraday's law of induction</a>:</td><td> ∇ × E = − ∂ B ∂ t {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}} </td><td> ∮ C E ⋅ d l = ∬ S ∇ × E ⋅ d A = − ∬ S ∂ B ∂ t ⋅ d A {\displaystyle {\begin{aligned}\oint _{C}\mathbf {E} \cdot d\mathbf {l} &=\iint _{S}\nabla \times \mathbf {E} \cdot d\mathbf {A} \\&=-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot d\mathbf {A} \end{aligned}}} <p>(with C and S not necessarily stationary)</p></td></tr><tr><td><a href="/facts/Amp%C3%A8re%27s_law/B6WVCpc6">Ampère's law</a> (with Maxwell's extension):</td><td> ∇ × H = J + ∂ D ∂ t {\displaystyle \nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}} </td><td> ∮ C H ⋅ d l = ∬ S ∇ × H ⋅ d A = ∬ S J ⋅ d A + ∬ S ∂ D ∂ t ⋅ d A {\displaystyle {\begin{aligned}\oint _{C}\mathbf {H} \cdot d\mathbf {l} &=\iint _{S}\nabla \times \mathbf {H} \cdot d\mathbf {A} \\&=\iint _{S}\mathbf {J} \cdot d\mathbf {A} +\iint _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot d\mathbf {A} \end{aligned}}} <p>(with C and S not necessarily stationary)</p></td></tr></tbody></table> <p>The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in <a href="/facts/SI_units/S7YLglNz">SI units</a>. In other systems of units, such as <a href="/facts/Maxwell%27s_equations/LfJ1drHS">CGS</a> or <a href="/facts/Gaussian_units/1mtqI96c">Gaussian units</a>, the scaling factors for the terms differ. For example, in Gaussian units, Faraday's law of induction and Ampère's law take the forms:<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> ∇ × E = − 1 c ∂ B ∂ t , ∇ × H = 1 c ∂ D ∂ t + 4 π c J , {\displaystyle {\begin{aligned}\nabla \times \mathbf {E} &=-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}\,,\\\nabla \times \mathbf {H} &={\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {J} \,,\end{aligned}}} respectively, where c is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> in vacuum. </p> <h3>Divergence theorem</h3> <p>Likewise, the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a> ∫ V o l ∇ ⋅ F d V o l = ∮ ∂ Vol F ⋅ d Σ {\displaystyle \int _{\mathrm {Vol} }\nabla \cdot \mathbf {F} \,d_{\mathrm {Vol} }=\oint _{\partial \operatorname {Vol} }\mathbf {F} \cdot d{\boldsymbol {\Sigma }}} is a special case if we identify a vector field with the ( n − 1 ) {\displaystyle (n-1)} -form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case F = f c → {\displaystyle {\textbf {F}}=f{\vec {c}}} where c → {\displaystyle {\vec {c}}} is an arbitrary constant vector. Working out the divergence of the product gives c → ⋅ ∫ V o l ∇ f d V o l = c → ⋅ ∮ ∂ V o l f d Σ . {\displaystyle {\vec {c}}\cdot \int _{\mathrm {Vol} }\nabla f\,d_{\mathrm {Vol} }={\vec {c}}\cdot \oint _{\partial \mathrm {Vol} }f\,d{\boldsymbol {\Sigma }}\,.} Since this holds for all c → {\displaystyle {\vec {c}}} we find ∫ V o l ∇ f d V o l = ∮ ∂ V o l f d Σ . {\displaystyle \int _{\mathrm {Vol} }\nabla f\,d_{\mathrm {Vol} }=\oint _{\partial \mathrm {Vol} }f\,d{\boldsymbol {\Sigma }}\,.} </p> <h3>Volume integral of gradient of scalar field</h3> <p>Let f : Ω → R {\displaystyle f:\Omega \to \mathbb {R} } be a <a href="/facts/Scalar_field/8MtPTGob">scalar field</a>. Then ∫ Ω ∇ → f = ∫ ∂ Ω n → f {\displaystyle \int _{\Omega }{\vec {\nabla }}f=\int _{\partial \Omega }{\vec {n}}f} where n → {\displaystyle {\vec {n}}} is the <a href="/facts/Normal_vector/jhtLIhcu">normal vector</a> to the surface ∂ Ω {\displaystyle \partial \Omega } at a given point. </p><p>Proof: Let c → {\displaystyle {\vec {c}}} be a vector. Then 0 = ∫ Ω ∇ → ⋅ c → f − ∫ ∂ Ω n → ⋅ c → f by the divergence theorem = ∫ Ω c → ⋅ ∇ → f − ∫ ∂ Ω c → ⋅ n → f = c → ⋅ ∫ Ω ∇ → f − c → ⋅ ∫ ∂ Ω n → f = c → ⋅ ( ∫ Ω ∇ → f − ∫ ∂ Ω n → f ) {\displaystyle {\begin{aligned}0&=\int _{\Omega }{\vec {\nabla }}\cdot {\vec {c}}f-\int _{\partial \Omega }{\vec {n}}\cdot {\vec {c}}f&{\text{by the divergence theorem}}\\&=\int _{\Omega }{\vec {c}}\cdot {\vec {\nabla }}f-\int _{\partial \Omega }{\vec {c}}\cdot {\vec {n}}f\\&={\vec {c}}\cdot \int _{\Omega }{\vec {\nabla }}f-{\vec {c}}\cdot \int _{\partial \Omega }{\vec {n}}f\\&={\vec {c}}\cdot \left(\int _{\Omega }{\vec {\nabla }}f-\int _{\partial \Omega }{\vec {n}}f\right)\end{aligned}}} Since this holds for any c → {\displaystyle {\vec {c}}} (in particular, for every <a href="/facts/Basis_vector/89IPoN6c">basis vector</a>), the result follows. </p> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Chandrasekhar%E2%80%93Wentzel_lemma/1VJkhspk">Chandrasekhar–Wentzel lemma</a></li></ul> <h2 id="footnotes">Footnotes</h2> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Helmut_Grunsky/MVsm31Ir">Grunsky, Helmut</a> (1983). <a href="https://archive.org/details/generalstokesthe0000grun"><i>The General Stokes' Theorem</i></a>. Boston: Pitman. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-273-08510-7.</li> <li>Katz, Victor J. (May 1979). "The History of Stokes' Theorem". <i>Mathematics Magazine</i>. 52 (3): 146–156. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2690275">10.2307/2690275</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2690275">2690275</a>.</li> <li><a href="/facts/Lynn_Harold_Loomis/P3yIpycg">Loomis, Lynn Harold</a>; <a href="/facts/Shlomo_Sternberg/IsGM8VzY">Sternberg, Shlomo</a> (2014). <a href="https://archive.org/details/LoomisL.H.SternbergS.AdvancedCalculusRevisedEditionJonesAndBartlett"><i>Advanced Calculus</i></a>. Hackensack, New Jersey: World Scientific. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4583-93-0.</li> <li><a href="/facts/Ib_Madsen/8Nt7CoA3">Madsen, Ib</a>; Tornehave, Jørgen (1997). <a href="https://archive.org/details/MadsenI.H.TornehaveJ.FromCalculusToCohomologyDeRhamCohomologyAndCharacteristicClasses1996"><i>From Calculus to Cohomology: De Rham cohomology and characteristic classes</i></a>. Cambridge, UK: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-58956-8.</li> <li><a href="/facts/Jerrold_E._Marsden/I07WtXQe">Marsden, Jerrold E.</a>; Anthony, Tromba (2003). <i>Vector Calculus</i> (5th ed.). W. H. Freeman.</li> <li>Lee, John (2003). <a href="https://archive.org/details/GraduateTextsInMathematics218LeeJ.M.IntroductionToSmoothManifoldsSpringer2012"><i>Introduction to Smooth Manifolds</i></a>. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95448-6.</li> <li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1976). <a href="https://archive.org/details/1979RudinW"><i>Principles of Mathematical Analysis</i></a>. New York, NY: McGraw–Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-054235-X.</li> <li><a href="/facts/Michael_Spivak/C67tKX7d">Spivak, Michael</a> (1965). <a href="/facts/Calculus_on_Manifolds_(book)/JENnXtEC"><i>Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus</i></a>. San Francisco: Benjamin Cummings. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8053-9021-9.</li> <li>Stewart, James (2009). <a href="https://books.google.com/books?id=Vou3MZu_7tcC&pg=PA960"><i>Calculus: Concepts and Contexts</i></a>. Cengage Learning. pp. 960–967. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-495-55742-5.</li> <li>Stewart, James (2003). <i>Calculus: Early Transcendental Functions</i> (5th ed.). Brooks/Cole.</li> <li><a href="/facts/Loring_W._Tu/rPz68vXp">Tu, Loring W.</a> (2011). <i>An Introduction to Manifolds</i> (2nd ed.). New York: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-7399-3.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Stokes_formula">"Stokes formula"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="http://higheredbcs.wiley.com/legacy/college/hugheshallett/0471484822/theory/hh_focusontheory_sectionm.pdf">Proof of the Divergence Theorem and Stokes' Theorem</a></li> <li><a href="http://tutorial.math.lamar.edu/classes/calcIII/stokestheorem.aspx">Calculus 3 – Stokes Theorem from lamar.edu</a> – an expository explanation</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Michel Moisan; Jacques Pelletier. Physics of Collisional Plasmas – Introduction to. Springer. <a href="https://www.springer.com/gp/book/9789400745575" target="_blank">https://www.springer.com/gp/book/9789400745575</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Spivak, Michael (1965). Calculus on manifolds : a modern approach to classical theorems of advanced calculus. New York: Avalon Publishing. ISBN 0-8053-9021-9. OCLC 187146. <a href="0-8053-9021-9" target="_blank">0-8053-9021-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Cartan, Élie (1945). Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques. Paris: Hermann. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Katz, Victor J. (1979-01-01). "The History of Stokes' Theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.2307/2690275. JSTOR 2690275. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Katz, Victor J. (1999). "5. Differential Forms". In James, I. M. (ed.). History of Topology. Amsterdam: Elsevier. pp. 111–122. ISBN 9780444823755. <a href="9780444823755" target="_blank">9780444823755</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>See: Katz, Victor J. (May 1979). "The history of Stokes' theorem". Mathematics Magazine. 52 (3): 146–156. doi:10.1080/0025570x.1979.11976770. The letter from Thomson to Stokes appears in: Thomson, William; Stokes, George Gabriel (1990). Wilson, David B. (ed.). The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Volume 1: 1846–1869. Cambridge, England: Cambridge University Press. pp. 96–97. ISBN 9780521328319. Neither Thomson nor Stokes published a proof of the theorem. The first published proof appeared in 1861 in: Hankel, Hermann (1861). Zur allgemeinen Theorie der Bewegung der Flüssigkeiten [On the general theory of the movement of fluids]. Göttingen, Germany: Dieterische University Buchdruckerei. pp. 34–37. Hankel doesn't mention the author of the theorem. In a footnote, Larmor mentions earlier researchers who had integrated, over a surface, the curl of a vector field. See: Stokes, George Gabriel (1905). Larmor, Joseph; Strutt, John William (eds.). Mathematical and Physical Papers by the late Sir George Gabriel Stokes. Vol. 5. Cambridge, England: University of Cambridge Press. pp. 320–321. <a href="9780521328319" target="_blank">9780521328319</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, England: OUP Oxford. p. 146. ISBN 0198505930. <a href="0198505930" target="_blank">0198505930</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Spivak (1965), p. vii, Preface. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Spivak (1965), p. vii, Preface. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>See: The 1854 Smith's Prize Examination is available online at: Clerk Maxwell Foundation. Maxwell took this examination and tied for first place with Edward John Routh. See: Clerk Maxwell, James (1990). Harman, P. M. (ed.). The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862. Cambridge, England: Cambridge University Press. p. 237, footnote 2. ISBN 9780521256254. See also Smith's prize or the Clerk Maxwell Foundation. Clerk Maxwell, James (1873). A Treatise on Electricity and Magnetism. Vol. 1. Oxford, England: Clarendon Press. pp. 25–27. In a footnote on page 27, Maxwell mentions that Stokes used the theorem as question 8 in the Smith's Prize Examination of 1854. This footnote appears to have been the cause of the theorem's being known as "Stokes' theorem". <a href="9780521256254" target="_blank">9780521256254</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>For mathematicians this fact is known, therefore the circle is redundant and often omitted. However, one should keep in mind here that in thermodynamics, where frequently expressions as ∮ W { d total U } {\displaystyle \oint _{W}\{{\text{d}}_{\text{total}}U\}} appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path W {\displaystyle W} is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where U {\displaystyle U} is a function of the temperature α 1 = T {\displaystyle \alpha _{1}=T} , the volume α 2 = V {\displaystyle \alpha _{2}=V} , and the electrical polarization α 3 = P {\displaystyle \alpha _{3}=P} of the sample, one has { d total U } = ∑ i = 1 3 ∂ U ∂ α i d α i , {\displaystyle \{d_{\text{total}}U\}=\sum _{i=1}^{3}{\frac {\partial U}{\partial \alpha _{i}}}\,d\alpha _{i}\,,} and the circle is really necessary, e.g. if one considers the differential consequences of the integral postulate ∮ W { d total U } = ! 0 . {\displaystyle \oint _{W}\,\{d_{\text{total}}U\}\,{\stackrel {!}{=}}\,0\,.} <a href="/wiki/Thermodynamics" target="_blank">/wiki/Thermodynamics</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Renteln, Paul (2014). Manifolds, Tensors, and Forms. Cambridge, UK: Cambridge University Press. pp. 158–175. ISBN 9781107324893. <a href="9781107324893" target="_blank">9781107324893</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Lee, John M. (2013). Introduction to Smooth Manifolds. New York: Springer. p. 481. ISBN 9781441999818. <a href="9781441999818" target="_blank">9781441999818</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>γ {\displaystyle \gamma } and Γ {\displaystyle \Gamma } are both loops, however, Γ {\displaystyle \Gamma } is not necessarily a Jordan curve <a href="/wiki/Jordan_curve" target="_blank">/wiki/Jordan_curve</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Stewart, James (2010). Essential Calculus: Early Transcendentals. Cole. <a href="https://books.google.com/books?id=btIhvKZCkTsC&pg=PA786" target="_blank">https://books.google.com/books?id=btIhvKZCkTsC&pg=PA786</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>This proof is based on the Lecture Notes given by Prof. Robert Scheichl (University of Bath, U.K) [1], please refer the [2] <a href="/wiki/University_of_Bath" target="_blank">/wiki/University_of_Bath</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>"This proof is also same to the proof shown in". <a href="https://www.proofwiki.org/wiki/Classical_Stokes'_Theorem" target="_blank">https://www.proofwiki.org/wiki/Classical_Stokes'_Theorem</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Whitney, Geometric Integration Theory, III.14. <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Harrison, J. (October 1993). "Stokes' theorem for nonsmooth chains". Bulletin of the American Mathematical Society. New Series. 29 (2): 235–243. arXiv:math/9310231. Bibcode:1993math.....10231H. doi:10.1090/S0273-0979-1993-00429-4. S2CID 17436511. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Jackson, J. D. (1975). Classical Electrodynamics (2nd ed.). New York, NY: Wiley. ISBN 9780471431329. <a href="9780471431329" target="_blank">9780471431329</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Born, M.; Wolf, E. (1980). Principles of Optics (6th ed.). Cambridge, England: Cambridge University Press. <a href="/wiki/Principles_of_Optics" target="_blank">/wiki/Principles_of_Optics</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> </ol>