# Il teorema di 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.

Contenuti 1 Teorema 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 Storia 10 Guarda anche 11 Riferimenti 12 Ulteriori letture 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, poi {stile di visualizzazione {scriptstyle C}} {stile di visualizzazione (l,dx+M,dio)= non _{D}sinistra({frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}Giusto)dx,dio} where the path of integration along C is anticlockwise.[1][2] 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, e in particolare, 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 (ancora, 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 unto_{C}l,dx=iint _{D}sinistra(-{frac {partial L}{parziale y}}Giusto)dA} (1) e {displaystyle unto_{C} M,dy=iint _{D}sinistra({frac {parziale M}{parziale x}}Giusto)dA} (2) are true, then Green's theorem follows immediately for the region D. We can prove (1) easily for regions of type I, e (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, di {stile di visualizzazione 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): {stile di visualizzazione {inizio{allineato}iint _{D}{frac {partial L}{parziale y}},dA&=int _{un}^{b},int _{g_{1}(X)}^{g_{2}(X)}{frac {partial L}{parziale y}}(X,y),dio,dx\&=int _{un}^{b}sinistra[l(X,g_{2}(X))-l(X,g_{1}(X))Giusto],dx.end{allineato}}} (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. Quindi {displaystyle int _{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. Quindi {displaystyle int _{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, significato {displaystyle int _{C_{4}}l(X,y),dx=int _{C_{2}}l(X,y),dx=0.} Perciò, {stile di visualizzazione {inizio{allineato}int _{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{allineato}}} (4) Combining (3) insieme a (4), noi abbiamo (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 {stile di visualizzazione Gamma } be a rectifiable, positively oriented Jordan curve in {displaystyle mathbb {R} ^{2}} e lascia {stile di visualizzazione R} denote its inner region. Supporre che {stile di visualizzazione A,B:{sopra {R}}a matematicabb {R} } are continuous functions with the property that {stile di visualizzazione A} has second partial derivative at every point of {stile di visualizzazione R} , {stile di visualizzazione B} has first partial derivative at every point of {stile di visualizzazione R} and that the functions {stile di visualizzazione D_{1}B,D_{2}UN:Rto mathbb {R} } are Riemann-integrable over {stile di visualizzazione R} . Quindi {displaystyle int _{Gamma }(UN,dx+B,dio)=int _{R}sinistra(D_{1}B(X,y)-D_{2}UN(X,y)Giusto),d(X,y).} We need the following lemmas whose proofs can be found in:[3] Lemma 1 (Decomposition Lemma) — Assume {stile di visualizzazione Gamma } is a rectifiable, positively oriented Jordan curve in the plane and let {stile di visualizzazione R} be its inner region. For every positive real {delta dello stile di visualizzazione } , permettere {stile di visualizzazione {matematico {F}}(delta )} denote the collection of squares in the plane bounded by the lines {displaystyle x=mdelta ,y=mdelta } , dove {stile di visualizzazione m} runs through the set of integers. Quindi, for this {delta dello stile di visualizzazione } , there exists a decomposition of {stile di visualizzazione {sopra {R}}} into a finite number of non-overlapping subregions in such a manner that Each one of the subregions contained in {stile di visualizzazione R} , dire {stile di visualizzazione R_{1},R_{2},ldot ,R_{K}} , is a square from {stile di visualizzazione {matematico {F}}(delta )} . Each one of the remaining subregions, dire {stile di visualizzazione R_{k+1},ldot ,R_{S}} , has as boundary a rectifiable Jordan curve formed by a finite number of arcs of {stile di visualizzazione Gamma } and parts of the sides of some square from {stile di visualizzazione {matematico {F}}(delta )} . Each one of the border regions {stile di visualizzazione R_{k+1},ldot ,R_{S}} can be enclosed in a square of edge-length {displaystyle 2delta } . Se {stile di visualizzazione Gamma _{io}} is the positively oriented boundary curve of {stile di visualizzazione R_{io}} , poi {displaystyle Gamma =Gamma _{1}+gamma _{2}+cdots +Gamma _{S}.} Il numero {displaystyle s-k} of border regions is no greater than {stile di testo 4!sinistra({frac {Lambda }{delta }}+1Giusto)} , dove {displaystyle Lambda } is the length of {stile di visualizzazione Gamma } .

Lemma 2 - Permettere {stile di visualizzazione Gamma } be a rectifiable curve in the plane and let {stile di visualizzazione Delta _{Gamma }(h)} be the set of points in the plane whose distance from (the range of) {stile di visualizzazione Gamma } è al massimo {stile di visualizzazione h} . The outer Jordan content of this set satisfies {stile di visualizzazione {sopra {c}},,Delta _{Gamma }(h)leq 2hLambda +pi h^{2}} .

Lemma 3 - Permettere {stile di visualizzazione Gamma } be a rectifiable curve in {displaystyle mathbb {R} ^{2}} e lascia {stile di visualizzazione f:{testo{range of }}Gamma to mathbb {R} } be a continuous function. Quindi {displaystyle leftvert int _{Gamma }f(X,y),dyrightvert leq {frac {1}{2}}Lambda Omega _{f},} e {displaystyle leftvert int _{Gamma }f(X,y),dxrightvert leq {frac {1}{2}}Lambda Omega _{f},} dove {stile di visualizzazione Omega _{f}} is the oscillation of {stile di visualizzazione f} on the range of {stile di visualizzazione Gamma } .

Now we are in position to prove the theorem: Proof of Theorem. Permettere {displaystyle varepsilon } be an arbitrary positive real number. By continuity of {stile di visualizzazione A} , {stile di visualizzazione B} and compactness of {stile di visualizzazione {sopra {R}}} , dato {displaystyle varepsilon >0} , lì esiste {stile di visualizzazione 00.} We may as well choose {delta dello stile di visualizzazione } so that the RHS of the last inequality is {stile di visualizzazione 0} . Since this is true for every {displaystyle varepsilon >0} , abbiamo chiuso.

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: Le funzioni {stile di visualizzazione A,B:{sopra {R}}a matematicabb {R} } are still assumed to be continuous. Tuttavia, we now require them to be Fréchet-differentiable at every point of {stile di visualizzazione R} . This implies the existence of all directional derivatives, in particolare {stile di visualizzazione D_{e_{io}}A=:D_{io}UN,D_{e_{io}}B=:D_{io}B,,i=1,2} , dove, as usual, {stile di visualizzazione (e_{1},e_{2})} is the canonical ordered basis of {displaystyle mathbb {R} ^{2}} . Inoltre, we require the function {stile di visualizzazione D_{1}B-D_{2}UN} to be Riemann-integrable over {stile di visualizzazione R} .

As a corollary of this, we get the Cauchy Integral Theorem for rectifiable Jordan curves: Teorema (Cauchy) — If {stile di visualizzazione Gamma } is a rectifiable Jordan curve in {displaystyle mathbb {C} } e se {stile di visualizzazione f:{testo{closure of inner region of }}Gamma to mathbb {C} } is a continuous mapping holomorphic throughout the inner region of {stile di visualizzazione Gamma } , poi {displaystyle int _{Gamma }f=0,} the integral being a complex contour integral.

Proof We regard the complex plane as {displaystyle mathbb {R} ^{2}} . Adesso, definire {stile di visualizzazione u,v:{sopra {R}}a matematicabb {R} } to be such that {stile di visualizzazione f(x+iy)= tu(X,y)+iv(X,y).} These functions are clearly continuous. It is well known that {stile di visualizzazione u} e {stile di visualizzazione v} are Fréchet-differentiable and that they satisfy the Cauchy-Riemann equations: {stile di visualizzazione D_{1}v+D_{2}u=D_{1}u-D_{2}v={testo{zero function}}} .

Adesso, analyzing the sums used to define the complex contour integral in question, it is easy to realize that {displaystyle int _{Gamma }f=int _{Gamma }tu,dx-v,dyquad +iint _{Gamma }v,dx+u,dio,} 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. Permettere {stile di visualizzazione Gamma _{0},gamma _{1},ldot ,gamma _{n}} be positively oriented rectifiable Jordan curves in {displaystyle mathbb {R} ^{2}} soddisfacente {stile di visualizzazione {inizio{allineato}gamma _{io}subset R_{0},&&{testo{Se }}1leq ileq n\Gamma _{io}sottoinsieme mathbb {R} ^{2}set meno {sopra {R}}_{j},&&{testo{Se }}1leq i,jleq n{testo{ e }}ineq j,fine{allineato}}} dove {stile di visualizzazione R_{io}} is the inner region of {stile di visualizzazione Gamma _{io}} . Permettere {displaystyle D=R_{0}set meno ({sopra {R}}_{1}tazza {sopra {R}}_{2}cup cdots cup {sopra {R}}_{n}).} Supponiamo {stile di visualizzazione p:{sopra {D}}a matematicabb {R} } e {stile di visualizzazione q:{sopra {D}}a matematicabb {R} } are continuous functions whose restriction to {stile di visualizzazione D} is Fréchet-differentiable. Se la funzione {stile di visualizzazione (X,y)longmapsto {frac {partial q}{partial e_{1}}}(X,y)-{frac {partial p}{partial e_{2}}}(X,y)} is Riemann-integrable over {stile di visualizzazione D} , poi {stile di visualizzazione {inizio{allineato}&int _{gamma _{0}}p(X,y),dx+q(X,y),dy-sum _{io=1}^{n}int _{gamma _{io}}p(X,y),dx+q(X,y),dy\[5pt]={}&int _{D}sinistra{{frac {partial q}{partial e_{1}}}(X,y)-{frac {partial p}{partial e_{2}}}(X,y)Giusto},d(X,y).fine{allineato}}} 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 {displaystyle mathbf {F} =(l,M,0)} . Start with the left side of Green's theorem: {displaystyle unto_{C}(l,dx+M,dio)=unto _{C}(l,M,0)cdot (dx,dio,dz)=unto _{C}mathbf {F} cdot dmathbf {r} .} The Kelvin–Stokes theorem: {displaystyle unto_{C}mathbf {F} cdot dmathbf {r} = non _{S}nabla volte mathbf {F} cdot mathbf {cappello {n}} ,dS.} The surface {stile di visualizzazione S} is just the region in the plane {stile di visualizzazione D} , with the unit normal {displaystyle mathbf {cappello {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 {cappello {n}} = sinistra[sinistra({frac {parziale 0}{parziale y}}-{frac {parziale M}{parziale z}}Giusto)mathbf {io} +sinistra({frac {partial L}{parziale z}}-{frac {parziale 0}{parziale x}}Giusto)mathbf {j} +sinistra({frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}Giusto)mathbf {K} Giusto]cdot mathbf {K} = sinistra({frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}Giusto).} Thus we get the right side of Green's theorem {displaystyle iint _{S}nabla volte mathbf {F} cdot mathbf {cappello {n}} ,dS=iint _{D}sinistra({frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}Giusto),dA.} Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: {displaystyle unto_{C}l,dx+M,dy = punto _{partial D}!omega =int _{D}domega =int _{D}{frac {partial L}{parziale y}},dywedge ,dx+{frac {parziale M}{parziale x}},dxwedge ,dy=iint _{D}sinistra({frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}Giusto),dx,dio.} 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}sinistra(mathbf {nabla } cdot mathbf {F} Giusto),dV=} {displaystyle partial scriptstyle V} {stile di visualizzazione (mathbf {F} cdot mathbf {cappello {n}} ),dS.} dove {displaystyle nabla cdot mathbf {F} } is the divergence on the two-dimensional vector field {displaystyle mathbf {F} } , e {displaystyle mathbf {cappello {n}} } is the outward-pointing unit normal vector on the boundary.

Per vedere questo, consider the unit normal {displaystyle mathbf {cappello {n}} } in the right side of the equation. Since in Green's theorem {displaystyle dmathbf {r} =(dx,dio)} is a vector pointing tangential along the curve, and the curve C is the positively oriented (cioè. anticlockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right of this; one choice would be {stile di visualizzazione (dio,-dx)} . The length of this vector is {stile di testo {mq {dx^{2}+dy^{2}}}=ds.} Così {stile di visualizzazione (dio,-dx)=mathbf {cappello {n}} ,ds.} Start with the left side of Green's theorem: {displaystyle unto_{C}(l,dx+M,dio)=unto _{C}(M,-l)cdot (dio,-dx)=unto _{C}(M,-l)cdot mathbf {cappello {n}} ,ds.} Applying the two-dimensional divergence theorem with {displaystyle mathbf {F} =(M,-l)} , we get the right side of Green's theorem: {displaystyle unto_{C}(M,-l)cdot mathbf {cappello {n}} ,ds=iint _{D}sinistra(nabla cdot (M,-l)Giusto),dA=iint _{D}sinistra({frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}Giusto),dA.} Area calculation Green's theorem can be used to compute area by line integral.[4] The area of a planar region {stile di visualizzazione D} è dato da {displaystyle A=iint _{D}dA.} Choose {stile di visualizzazione L} e {stile di visualizzazione M} tale che {stile di visualizzazione {frac {parziale M}{parziale x}}-{frac {partial L}{parziale y}}=1} , the area is given by {displaystyle A=oint _{C}(l,dx+M,dio).} Possible formulas for the area of {stile di visualizzazione D} include[4] {displaystyle A=oint _{C}X,dy=-oint _{C}y,dx={tfrac {1}{2}}unguento _{C}(-y,dx+x,dio).} 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. In 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.[5][6] 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. Cambridge University Press. ISBN 978-0-521-86153-3. ^ Spiegel, M. R.; Lipschutz, S.; Spellman, D. (2009). Vector Analysis. Schaum’s Outlines (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7. ^ Apostolo, Tom (1960). Mathematical Analysis (1 ed.). Lettura, Massachusetts, STATI UNITI D'AMERICA.: Addison-Wesley Publishing Company, INC. ^ Salta su: a b Stewart, Giacomo (1999). Calcolo (6th ed.). Thomson, Brooks/Cole. ISBN 9780534359492. ^ George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, Inghilterra: T. Wheelhouse, 1828). Green did not actually derive the form of "Il teorema di Green" which appears in this article; rather, he derived a form of the "divergence theorem", which appears on pages 10–12 of his Essay.

In 1846, the form of "Il teorema di Green" which appears in this article was first published, senza prove, 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, dove (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: Bernhard 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), (Gottinga, (Germania): Adalbert Rente, 1867); see pages 8–9. ^ Katz, Vincitore (2009). "22.3.3: Complex Functions and Line Integrals". A History of Mathematics: Un introduzione. 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 (Quinta ed.). New York: Libero. 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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