# Théorème de Green Green's theorem This article is about the theorem in the plane relating double integrals and line integrals. For Green's theorems relating volume integrals involving the Laplacian to surface integrals, see Green's identities. Not to be confused with Green's law for waves approaching a shoreline. Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem show Differential show Integral show Series hide Vector GradientDivergenceCurlLaplacianDirectional derivativeIdentities Theorems GradientGreen'sStokes'Divergencegeneralized Stokes show Multivariable show Advanced show Specialized show Miscellaneous vte In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.

Contenu 1 Théorème 2 Proof when D is a simple region 3 Proof for rectifiable Jordan curves 4 Validity under different hypotheses 5 Multiply-connected regions 6 Relationship to Stokes' theorem 7 Relationship to the divergence theorem 8 Area calculation 9 Histoire 10 Voir également 11 Références 12 Lectures complémentaires 13 External links Theorem Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (X, y) defined on an open region containing D and having continuous partial derivatives there, alors {style d'affichage {scriptstyle C}} {style d'affichage (L,dx+M,mourir)=iint _{ré}la gauche({frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}droit)dx,mourir} where the path of integration along C is anticlockwise. In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, et en particulier, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.

Proof when D is a simple region If D is a simple type of region with its boundary consisting of the curves C1, C2, C3, C4, half of Green's theorem can be demonstrated.

The following is a proof of half of the theorem for the simplified area D, a type I region where C1 and C3 are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the theorem when D is a type II region where C2 and C4 are curves connected by horizontal lines (encore, possibly of zero length). Putting these two parts together, the theorem is thus proven for regions of type III (defined as regions which are both type I and type II). The general case can then be deduced from this special case by decomposing D into a set of type III regions.

If it can be shown that {displaystyle oint _{C}L,dx=iint _{ré}la gauche(-{frac {partial L}{y partiel}}droit)dA} (1) et {displaystyle oint _{C} M,dy=iint _{ré}la gauche({frac {partiel M}{partiel x}}droit)dA} (2) are true, then Green's theorem follows immediately for the region D. We can prove (1) easily for regions of type I, et (2) for regions of type II. Green's theorem then follows for regions of type III.

Assume region D is a type I region and can thus be characterized, as pictured on the right, par {displaystyle D={(X,y)mid aleq xleq b,g_{1}(X)leq yleq g_{2}(X)}} where g1 and g2 are continuous functions on [un, b]. Compute the double integral in (1): {style d'affichage {commencer{aligné}iint _{ré}{frac {partial L}{y partiel}},dA&=int _{un}^{b},entier _{g_{1}(X)}^{g_{2}(X)}{frac {partial L}{y partiel}}(X,y),mourir,dx\&=int _{un}^{b}la gauche[L(X,g_{2}(X))-L(X,g_{1}(X))droit],dx.end{aligné}}} (3) Now compute the line integral in (1). C can be rewritten as the union of four curves: C1, C2, C3, C4.

With C1, use the parametric equations: x = x, y = g1(X), a ≤ x ≤ b. Alors {style d'affichage entier _{C_{1}}L(X,y),dx=int _{un}^{b}L(X,g_{1}(X)),dx.} With C3, use the parametric equations: x = x, y = g2(X), a ≤ x ≤ b. Alors {style d'affichage entier _{C_{3}}L(X,y),dx=-int _{-C_{3}}L(X,y),dx=-int _{un}^{b}L(X,g_{2}(X)),dx.} The integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively (anticlockwise). On C2 and C4, x remains constant, meaning {style d'affichage entier _{C_{4}}L(X,y),dx=int _{C_{2}}L(X,y),dx=0.} Par conséquent, {style d'affichage {commencer{aligné}entier _{C}L,dx&=int _{C_{1}}L(X,y),dx+int _{C_{2}}L(X,y),dx+int _{C_{3}}L(X,y),dx+int _{C_{4}}L(X,y),dx\&=int _{un}^{b}L(X,g_{1}(X)),dx-int _{un}^{b}L(X,g_{2}(X)),dx.end{aligné}}} (4) Combining (3) avec (4), on a (1) for regions of type I. A similar treatment yields (2) for regions of type II. Putting the two together, we get the result for regions of type III.

Proof for rectifiable Jordan curves We are going to prove the following Theorem — Let {style d'affichage Gamma } be a rectifiable, positively oriented Jordan curve in {style d'affichage mathbb {R} ^{2}} et laissez {style d'affichage R} denote its inner region. Supposer que {style d'affichage A,B:{surligner {R}}à mathbb {R} } are continuous functions with the property that {style d'affichage A} has second partial derivative at every point of {style d'affichage R} , {style d'affichage B} has first partial derivative at every point of {style d'affichage R} and that the functions {displaystyle D_{1}B,RÉ_{2}UN:Rto mathbb {R} } are Riemann-integrable over {style d'affichage R} . Alors {style d'affichage entier _{Gamma }(UN,dx+B,mourir)=int _{R}la gauche(RÉ_{1}B(X,y)-RÉ_{2}UN(X,y)droit),ré(X,y).} We need the following lemmas whose proofs can be found in: Lemme 1 (Decomposition Lemma) — Assume {style d'affichage Gamma } is a rectifiable, positively oriented Jordan curve in the plane and let {style d'affichage R} be its inner region. For every positive real {delta de style d'affichage } , laisser {style d'affichage {mathématique {F}}(delta )} denote the collection of squares in the plane bounded by the lines {displaystyle x=mdelta ,y=mdelta } , où {style d'affichage m} runs through the set of integers. Alors, for this {delta de style d'affichage } , there exists a decomposition of {style d'affichage {surligner {R}}} into a finite number of non-overlapping subregions in such a manner that Each one of the subregions contained in {style d'affichage R} , dire {style d'affichage R_{1},R_{2},ldots ,R_{k}} , is a square from {style d'affichage {mathématique {F}}(delta )} . Each one of the remaining subregions, dire {style d'affichage R_{k+1},ldots ,R_{s}} , has as boundary a rectifiable Jordan curve formed by a finite number of arcs of {style d'affichage Gamma } and parts of the sides of some square from {style d'affichage {mathématique {F}}(delta )} . Each one of the border regions {style d'affichage R_{k+1},ldots ,R_{s}} can be enclosed in a square of edge-length {displaystyle 2delta } . Si {style d'affichage Gamma _{je}} is the positively oriented boundary curve of {style d'affichage R_{je}} , alors {displaystyle Gamma =Gamma _{1}+Gamma _{2}+cdots +Gamma _{s}.} Le nombre {displaystyle s-k} of border regions is no greater than {style de texte 4!la gauche({frac {Lambda }{delta }}+1droit)} , où {style d'affichage Lambda } is the length of {style d'affichage Gamma } .

Lemme 2 — Let {style d'affichage Gamma } be a rectifiable curve in the plane and let {style d'affichage Delta _{Gamma }(h)} be the set of points in the plane whose distance from (la gamme de) {style d'affichage Gamma } est au plus {style d'affichage h} . The outer Jordan content of this set satisfies {style d'affichage {surligner {c}},,Delta _{Gamma }(h)leq 2hLambda +pi h^{2}} .

Lemme 3 — Let {style d'affichage Gamma } be a rectifiable curve in {style d'affichage mathbb {R} ^{2}} et laissez {style d'affichage f:{texte{range of }}Gamma to mathbb {R} } be a continuous function. Alors {displaystyle leftvert int _{Gamma }F(X,y),dyrightvert leq {frac {1}{2}}Lambda Omega _{F},} et {displaystyle leftvert int _{Gamma }F(X,y),dxrightvert leq {frac {1}{2}}Lambda Omega _{F},} où {style d'affichage Omega _{F}} is the oscillation of {style d'affichage f} on the range of {style d'affichage Gamma } .

Now we are in position to prove the theorem: Proof of Theorem. Laisser {displaystyle varepsilon } be an arbitrary positive real number. By continuity of {style d'affichage A} , {style d'affichage B} and compactness of {style d'affichage {surligner {R}}} , given {displaystyle varepsilon >0} , il existe {style d'affichage 00.} We may as well choose {delta de style d'affichage } so that the RHS of the last inequality is {style d'affichage 0} . Since this is true for every {displaystyle varepsilon >0} , nous avons fini.

Validity under different hypotheses The hypothesis of the last theorem are not the only ones under which Green's formula is true. Another common set of conditions is the following: Les fonctions {style d'affichage A,B:{surligner {R}}à mathbb {R} } are still assumed to be continuous. Cependant, we now require them to be Fréchet-differentiable at every point of {style d'affichage R} . This implies the existence of all directional derivatives, en particulier {displaystyle D_{e_{je}}A=:RÉ_{je}UN,RÉ_{e_{je}}B=:RÉ_{je}B,,je=1,2} , où, comme d'habitude, {style d'affichage (e_{1},e_{2})} is the canonical ordered basis of {style d'affichage mathbb {R} ^{2}} . en outre, we require the function {displaystyle D_{1}B-D_{2}UN} to be Riemann-integrable over {style d'affichage R} .

As a corollary of this, we get the Cauchy Integral Theorem for rectifiable Jordan curves: Théorème (Cauchy) — If {style d'affichage Gamma } is a rectifiable Jordan curve in {style d'affichage mathbb {C} } et si {style d'affichage f:{texte{closure of inner region of }}Gamma to mathbb {C} } is a continuous mapping holomorphic throughout the inner region of {style d'affichage Gamma } , alors {style d'affichage entier _{Gamma }f=0,} the integral being a complex contour integral.

Proof We regard the complex plane as {style d'affichage mathbb {R} ^{2}} . À présent, définir {style d'affichage u,v:{surligner {R}}à mathbb {R} } to be such that {style d'affichage f(x+iy)=u(X,y)+iv(X,y).} These functions are clearly continuous. It is well known that {style d'affichage u} et {style d'affichage v} are Fréchet-differentiable and that they satisfy the Cauchy-Riemann equations: {displaystyle D_{1}v+D_{2}u=D_{1}u-D_{2}v={texte{zero function}}} .

À présent, analyzing the sums used to define the complex contour integral in question, it is easy to realize that {style d'affichage entier _{Gamma }f=int _{Gamma }tu,dx-v,dyquad +iint _{Gamma }v,dx+u,mourir,} the integrals on the RHS being usual line integrals. These remarks allow us to apply Green's Theorem to each one of these line integrals, finishing the proof.

Multiply-connected regions Theorem. Laisser {style d'affichage Gamma _{0},Gamma _{1},ldots ,Gamma _{n}} be positively oriented rectifiable Jordan curves in {style d'affichage mathbb {R} ^{2}} satisfaisant {style d'affichage {commencer{aligné}Gamma _{je}subset R_{0},&&{texte{si }}1leq ileq n\Gamma _{je}sous-ensemble mathbb {R} ^{2}setmoins {surligner {R}}_{j},&&{texte{si }}1leq i,jleq n{texte{ et }}ineq j,fin{aligné}}} où {style d'affichage R_{je}} is the inner region of {style d'affichage Gamma _{je}} . Laisser {displaystyle D=R_{0}setmoins ({surligner {R}}_{1}Coupe {surligner {R}}_{2}cup cdots cup {surligner {R}}_{n}).} Supposer {style d'affichage p:{surligner {ré}}à mathbb {R} } et {style d'affichage q:{surligner {ré}}à mathbb {R} } are continuous functions whose restriction to {displaystyle D} is Fréchet-differentiable. If the function {style d'affichage (X,y)longmapsto {frac {partial q}{partial e_{1}}}(X,y)-{frac {partial p}{partial e_{2}}}(X,y)} is Riemann-integrable over {displaystyle D} , alors {style d'affichage {commencer{aligné}&int _{Gamma _{0}}p(X,y),dx+q(X,y),dy-sum _{je=1}^{n}entier _{Gamma _{je}}p(X,y),dx+q(X,y),dy\[5pt]={}&int _{ré}la gauche{{frac {partial q}{partial e_{1}}}(X,y)-{frac {partial p}{partial e_{2}}}(X,y)droit},ré(X,y).fin{aligné}}} Relationship to Stokes' theorem Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the {displaystyle xy} -plane.

We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0. Write F for the vector-valued function {style d'affichage mathbf {F} =(L,M,0)} . Start with the left side of Green's theorem: {displaystyle oint _{C}(L,dx+M,mourir)=point _{C}(L,M,0)cdot (dx,mourir,dz)=point _{C}mathbf {F} cdot dmathbf {r} .} The Kelvin–Stokes theorem: {displaystyle oint _{C}mathbf {F} cdot dmathbf {r} =iint _{S}nabla fois mathbf {F} cdot mathbf {chapeau {n}} ,dS.} The surface {style d'affichage S} is just the region in the plane {displaystyle D} , with the unit normal {style d'affichage mathbf {chapeau {n}} } defined (by convention) to have a positive z component in order to match the "positive orientation" definitions for both theorems.

The expression inside the integral becomes {displaystyle nabla times mathbf {F} cdot mathbf {chapeau {n}} =gauche[la gauche({frac {partiel 0}{y partiel}}-{frac {partiel M}{z partiel}}droit)mathbf {je} +la gauche({frac {partial L}{z partiel}}-{frac {partiel 0}{partiel x}}droit)mathbf {j} +la gauche({frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}droit)mathbf {k} droit]cdot mathbf {k} =gauche({frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}droit).} Thus we get the right side of Green's theorem {displaystyle iint _{S}nabla fois mathbf {F} cdot mathbf {chapeau {n}} ,dS=iint _{ré}la gauche({frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}droit),dA.} Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: {displaystyle oint _{C}L,dx+M,dy=point _{partial D}!oméga =int _{ré}domega = int _{ré}{frac {partial L}{y partiel}},dywedge ,dx+{frac {partiel M}{partiel x}},dxwedge ,dy=iint _{ré}la gauche({frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}droit),dx,mourir.} Relationship to the divergence theorem Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem: {displaystyle iiint _{V}la gauche(mathbf {nabla } cdot mathbf {F} droit),dV=} {displaystyle partial scriptstyle V} {style d'affichage (mathbf {F} cdot mathbf {chapeau {n}} ),dS.} où {displaystyle nabla cdot mathbf {F} } is the divergence on the two-dimensional vector field {style d'affichage mathbf {F} } , et {style d'affichage mathbf {chapeau {n}} } is the outward-pointing unit normal vector on the boundary.

Pour voir ça, consider the unit normal {style d'affichage mathbf {chapeau {n}} } in the right side of the equation. Since in Green's theorem {displaystyle dmathbf {r} =(dx,mourir)} is a vector pointing tangential along the curve, and the curve C is the positively oriented (c'est à dire. anticlockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right of this; one choice would be {style d'affichage (mourir,-dx)} . The length of this vector is {style de texte {sqrt {dx ^{2}+dy^{2}}}=ds.} Alors {style d'affichage (mourir,-dx)= mathbf {chapeau {n}} ,ds.} Start with the left side of Green's theorem: {displaystyle oint _{C}(L,dx+M,mourir)=point _{C}(M,-L)cdot (mourir,-dx)=point _{C}(M,-L)cdot mathbf {chapeau {n}} ,ds.} Applying the two-dimensional divergence theorem with {style d'affichage mathbf {F} =(M,-L)} , we get the right side of Green's theorem: {displaystyle oint _{C}(M,-L)cdot mathbf {chapeau {n}} ,ds=iint _{ré}la gauche(nabla cdot (M,-L)droit),dA=iint _{ré}la gauche({frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}droit),dA.} Area calculation Green's theorem can be used to compute area by line integral. The area of a planar region {displaystyle D} est donné par {displaystyle A=iint _{ré}dA.} Choose {displaystyle L} et {style d'affichage M} tel que {style d'affichage {frac {partiel M}{partiel x}}-{frac {partial L}{y partiel}}=1} , the area is given by {displaystyle A=oint _{C}(L,dx+M,mourir).} Possible formulas for the area of {displaystyle D} include {displaystyle A=oint _{C}X,dy=-oint _{C}y,dx={tfrac {1}{2}}point _{C}(-y,dx+x,mourir).} History It is named after George Green, who stated a similar result in an 1828 paper titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Dans 1846, Augustin-Louis Cauchy published a paper stating Green's theorem as the penultimate sentence. This is in fact the first printed version of Green's theorem in the form appearing in modern textbooks. Bernhard Riemann gave the first proof of Green's theorem in his doctoral dissertation on the theory of functions of a complex variable. See also Mathematics portal Planimeter – Tool for measuring area. Method of image charges – A method used in electrostatics that takes advantage of the uniqueness theorem (derived from Green's theorem) Shoelace formula – A special case of Green's theorem for simple polygons References ^ Riley, K. F.; Hobson, M. P; Bence, S. J. (2010). Mathematical Methods for Physics and Engineering. la presse de l'Universite de Cambridge. ISBN 978-0-521-86153-3. ^ Spiegel, M. R; Lipschutz, S; Spellman, ré. (2009). Vector Analysis. Schaum’s Outlines (2sd éd.). Colline McGraw. ISBN 978-0-07-161545-7. ^ Apôtre, À M (1960). Mathematical Analysis (1 éd.). En lisant, Massachusetts, ETATS-UNIS.: Addison-Wesley Publishing Company, INC. ^ Sauter à: a b Stewart, James (1999). Calcul (6e éd.). Thomson, Brooks/Cole. ISBN 9780534359492. ^ George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, England: J. Wheelhouse, 1828). Green did not actually derive the form of "Théorème de Green" which appears in this article; rather, he derived a form of the "divergence theorem", which appears on pages 10–12 of his Essay.

Dans 1846, the form of "Théorème de Green" which appears in this article was first published, sans preuve, in an article by Augustin Cauchy: UN. Cauchy (1846) "Sur les intégrales qui s'étendent à tous les points d'une courbe fermée" (On integrals that extend over all of the points of a closed curve), Comptes rendus, 23: 251–255. (The equation appears at the bottom of page 254, où (S) denotes the line integral of a function k along the curve s that encloses the area S.) A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his inaugural dissertation: Bernard Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Allemagne): Adalbert Rente, 1867); see pages 8–9. ^ Katz, Victor (2009). "22.3.3: Complex Functions and Line Integrals". A History of Mathematics: Une introduction. Addison-Wesley. pp. 801–5. ISBN 978-0-321-38700-4. Further reading Marsden, Jerrold E.; Tromba, Anthony J. (2003). "The Integral Theorems of Vector Analysis". Vector Calculus (Fifth ed.). New York: Homme libre. pp. 518–608. ISBN 0-7167-4992-0. External links Green's Theorem on MathWorld show vte Calculus Authority control: National libraries France (data)IsraelUnited States Categories: Theorems in calculus

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