# Teorema 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.

Conteúdo 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 História 10 Veja também 11 Referências 12 Leitura adicional 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, então {estilo de exibição {scriptstyle C}} {estilo de exibição (eu,dx+M,dy)=iint_{D}deixei({fratura {parcial M}{x parcial}}-{fratura {partial L}{y parcial}}certo)dx,dy} 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, e em particular, 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 (novamente, 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 ungido_{C}eu,dx=iint _{D}deixei(-{fratura {partial L}{y parcial}}certo)dA} (1) e {displaystyle ungido_{C} M,dy=iint _{D}deixei({fratura {parcial M}{x parcial}}certo)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, por {displaystyle D={(x,y)mid aleq xleq b,g_{1}(x)leq yleq g_{2}(x)}} where g1 and g2 are continuous functions on [uma, b]. Compute the double integral in (1): {estilo de exibição {começar{alinhado}iint _{D}{fratura {partial L}{y parcial}},dA&=int _{uma}^{b},int_{g_{1}(x)}^{g_{2}(x)}{fratura {partial L}{y parcial}}(x,y),dy,dx\&=int _{uma}^{b}deixei[eu(x,g_{2}(x))-eu(x,g_{1}(x))certo],dx.end{alinhado}}} (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. Então {estilo de exibição int _{C_{1}}eu(x,y),dx=int_{uma}^{b}eu(x,g_{1}(x)),dx.} With C3, use the parametric equations: x = x, y = g2(x), a ≤ x ≤ b. Então {estilo de exibição int _{C_{3}}eu(x,y),dx=-int _{-C_{3}}eu(x,y),dx=-int _{uma}^{b}eu(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, significado {estilo de exibição int _{C_{4}}eu(x,y),dx=int_{C_{2}}eu(x,y),dx=0.} Portanto, {estilo de exibição {começar{alinhado}int_{C}eu,dx&=int _{C_{1}}eu(x,y),dx+int _{C_{2}}eu(x,y),dx+int _{C_{3}}eu(x,y),dx+int _{C_{4}}eu(x,y),dx\&=int _{uma}^{b}eu(x,g_{1}(x)),dx-int _{uma}^{b}eu(x,g_{2}(x)),dx.end{alinhado}}} (4) Combining (3) com (4), Nós temos (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 {displaystyle Gama } be a rectifiable, positively oriented Jordan curve in {estilo de exibição mathbb {R} ^{2}} e deixar {estilo de exibição R} denote its inner region. Suponha que {estilo de exibição A,B:{overline {R}}para mathbb {R} } are continuous functions with the property that {estilo de exibição A} has second partial derivative at every point of {estilo de exibição R} , {estilo de exibição B} has first partial derivative at every point of {estilo de exibição R} and that the functions {displaystyle D_{1}B,D_{2}UMA:Rto mathbb {R} } are Riemann-integrable over {estilo de exibição R} . Então {estilo de exibição int _{Gama }(UMA,dx+B,dy)=int_{R}deixei(D_{1}B(x,y)-D_{2}UMA(x,y)certo),d(x,y).} We need the following lemmas whose proofs can be found in: Lema 1 (Decomposition Lemma) — Assume {displaystyle Gama } is a rectifiable, positively oriented Jordan curve in the plane and let {estilo de exibição R} be its inner region. For every positive real {delta de estilo de exibição } , deixar {estilo de exibição {matemática {F}}(delta )} denote the collection of squares in the plane bounded by the lines {displaystyle x=mdelta ,y=mdelta } , Onde {estilo de exibição m} runs through the set of integers. Então, for this {delta de estilo de exibição } , there exists a decomposition of {estilo de exibição {overline {R}}} into a finite number of non-overlapping subregions in such a manner that Each one of the subregions contained in {estilo de exibição R} , dizer {estilo de exibição R_{1},R_{2},ldots ,R_{k}} , is a square from {estilo de exibição {matemática {F}}(delta )} . Each one of the remaining subregions, dizer {estilo de exibição R_{k+1},ldots ,R_{s}} , has as boundary a rectifiable Jordan curve formed by a finite number of arcs of {displaystyle Gama } and parts of the sides of some square from {estilo de exibição {matemática {F}}(delta )} . Each one of the border regions {estilo de exibição R_{k+1},ldots ,R_{s}} can be enclosed in a square of edge-length {displaystyle 2delta } . Se {displaystyle Gama _{eu}} is the positively oriented boundary curve of {estilo de exibição R_{eu}} , então {displaystyle Gamma =Gamma _{1}+Gamma _{2}+cdots +Gamma _{s}.} O número {displaystyle s-k} of border regions is no greater than {estilo de texto 4!deixei({fratura {Lambda }{delta }}+1certo)} , Onde {estilo de exibição Lambda } is the length of {displaystyle Gama } .

Lema 2 — Let {displaystyle Gama } be a rectifiable curve in the plane and let {estilo de exibição Delta _{Gama }(h)} be the set of points in the plane whose distance from (the range of) {displaystyle Gama } é no máximo {estilo de exibição h} . The outer Jordan content of this set satisfies {estilo de exibição {overline {c}},,Delta_{Gama }(h)leq 2hLambda +pi h^{2}} .

Lema 3 — Let {displaystyle Gama } be a rectifiable curve in {estilo de exibição mathbb {R} ^{2}} e deixar {estilo de exibição f:{texto{range of }}Gamma to mathbb {R} } be a continuous function. Então {displaystyle leftvert int _{Gama }f(x,y),dyrightvert leq {fratura {1}{2}}Lambda Omega _{f},} e {displaystyle leftvert int _{Gama }f(x,y),dxrightvert leq {fratura {1}{2}}Lambda Omega _{f},} Onde {displaystyle Omega _{f}} is the oscillation of {estilo de exibição f} on the range of {displaystyle Gama } .

Now we are in position to prove the theorem: Proof of Theorem. Deixar {displaystyle varepsilon } be an arbitrary positive real number. By continuity of {estilo de exibição A} , {estilo de exibição B} and compactness of {estilo de exibição {overline {R}}} , given {displaystyle varepsilon >0} , existe {estilo de exibição 00.} We may as well choose {delta de estilo de exibição } so that the RHS of the last inequality is {estilo de exibição 0} . Since this is true for every {displaystyle varepsilon >0} , we are done.

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: As funções {estilo de exibição A,B:{overline {R}}para mathbb {R} } are still assumed to be continuous. No entanto, we now require them to be Fréchet-differentiable at every point of {estilo de exibição R} . This implies the existence of all directional derivatives, em particular {displaystyle D_{e_{eu}}A=:D_{eu}UMA,D_{e_{eu}}B=:D_{eu}B,,i=1,2} , Onde, as usual, {estilo de exibição (e_{1},e_{2})} is the canonical ordered basis of {estilo de exibição mathbb {R} ^{2}} . Além disso, we require the function {displaystyle D_{1}B-D_{2}UMA} to be Riemann-integrable over {estilo de exibição R} .

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

Proof We regard the complex plane as {estilo de exibição mathbb {R} ^{2}} . Agora, definir {estilo de exibição você,v:{overline {R}}para mathbb {R} } to be such that {estilo de exibição f(x+iy)=u(x,y)+iv(x,y).} These functions are clearly continuous. It is well known that {estilo de exibição você} e {estilo de exibição 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={texto{zero function}}} .

Agora, analyzing the sums used to define the complex contour integral in question, it is easy to realize that {estilo de exibição int _{Gama }f=int _{Gama }você,dx-v,dyquad +iint _{Gama }v,dx+u,dy,} 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. Deixar {displaystyle Gama _{0},Gamma _{1},ldots ,Gamma _{n}} be positively oriented rectifiable Jordan curves in {estilo de exibição mathbb {R} ^{2}} satisfatório {estilo de exibição {começar{alinhado}Gamma _{eu}subset R_{0},&&{texto{E se }}1leq ileq n\Gamma _{eu}subconjunto mathbb {R} ^{2}setminus {overline {R}}_{j},&&{texto{E se }}1leq i,jleq n{texto{ e }}ineq j,fim{alinhado}}} Onde {estilo de exibição R_{eu}} is the inner region of {displaystyle Gama _{eu}} . Deixar {displaystyle D=R_{0}setminus ({overline {R}}_{1}copo {overline {R}}_{2}cup cdots cup {overline {R}}_{n}).} Suponha {estilo de exibição p:{overline {D}}para mathbb {R} } e {estilo de exibição q:{overline {D}}para mathbb {R} } are continuous functions whose restriction to {estilo de exibição D} is Fréchet-differentiable. If the function {estilo de exibição (x,y)longmapsto {fratura {partial q}{partial e_{1}}}(x,y)-{fratura {partial p}{partial e_{2}}}(x,y)} is Riemann-integrable over {estilo de exibição D} , então {estilo de exibição {começar{alinhado}&int _{Gamma _{0}}p(x,y),dx+q(x,y),dy-sum _{i=1}^{n}int_{Gamma _{eu}}p(x,y),dx+q(x,y),dy\[5pt]={}&int _{D}deixei{{fratura {partial q}{partial e_{1}}}(x,y)-{fratura {partial p}{partial e_{2}}}(x,y)certo},d(x,y).fim{alinhado}}} 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 {estilo de exibição mathbf {F} =(eu,M,0)} . Start with the left side of Green's theorem: {displaystyle ungido_{C}(eu,dx+M,dy)= unção _{C}(eu,M,0)cdot (dx,dy,dz)= unção _{C}mathbf {F} cdot dmathbf {r} .} The Kelvin–Stokes theorem: {displaystyle ungido_{C}mathbf {F} cdot dmathbf {r} =iint_{S}nabla vezes mathbf {F} cdot mathbf {chapéu {n}} ,dS.} The surface {estilo de exibição S} is just the region in the plane {estilo de exibição D} , with the unit normal {estilo de exibição mathbf {chapéu {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 {chapéu {n}} = esquerda[deixei({fratura {parcial 0}{y parcial}}-{fratura {parcial M}{z parcial}}certo)mathbf {eu} +deixei({fratura {partial L}{z parcial}}-{fratura {parcial 0}{x parcial}}certo)mathbf {j} +deixei({fratura {parcial M}{x parcial}}-{fratura {partial L}{y parcial}}certo)mathbf {k} certo]cdot mathbf {k} = esquerda({fratura {parcial M}{x parcial}}-{fratura {partial L}{y parcial}}certo).} Thus we get the right side of Green's theorem {displaystyle iint _{S}nabla vezes mathbf {F} cdot mathbf {chapéu {n}} ,dS=iint _{D}deixei({fratura {parcial M}{x parcial}}-{fratura {partial L}{y parcial}}certo),dA.} Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: {displaystyle ungido_{C}eu,dx+M,dy=oint _{partial D}!ômega =int_{D}domega =int_{D}{fratura {partial L}{y parcial}},dywedge ,dx+{fratura {parcial M}{x parcial}},dxwedge ,dy=iint _{D}deixei({fratura {parcial M}{x parcial}}-{fratura {partial L}{y parcial}}certo),dx,dy.} 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}deixei(mathbf {nabla } cdot mathbf {F} certo),dV=} {displaystyle partial scriptstyle V} {estilo de exibição (mathbf {F} cdot mathbf {chapéu {n}} ),dS.} Onde {displaystyle nabla cdot mathbf {F} } is the divergence on the two-dimensional vector field {estilo de exibição mathbf {F} } , e {estilo de exibição mathbf {chapéu {n}} } is the outward-pointing unit normal vector on the boundary.