Green's theorem

Green's theorem

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 (surface in ${\displaystyle \mathbb {R} ^{2}}$) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in ${\displaystyle \mathbb {R} ^{3}}$). In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to the divergence theorem.

Theorem

Green's 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 have continuous partial derivatives there, then

${\displaystyle \oint {C}(L\,dx+M\,dy)=\iint {D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)dA}$

where the path of integration along C is counterclockwise.

Application

Green's theorem

Green's theorem in the plane relates line integrals around a simple closed curve to double integrals over the regions it encloses. It has two equivalent forms: the circulation form, which says the tangential line integral of a vector field ${\displaystyle \mathbf {F} =(P,Q)}$ around a positively oriented simple closed curve ${\displaystyle C}$ equals the double integral of the scalar curl ${\displaystyle {\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}}$ over the region ${\displaystyle D}$ bounded by ${\displaystyle C}$; and the flux (divergence) form, which says that the normal line integral of ${\displaystyle \mathbf {F} }$ around ${\displaystyle C}$ equals the double integral of the divergence ${\displaystyle \nabla \cdot \mathbf {F} }$ over ${\displaystyle D}$. In applications, the circulation form is used for two-dimensional circulation and rotational flow calculations, while the flux form measures the net outflow across a closed boundary. Green's theorem also yields practical boundary-integral formulas for the area and centroid of a plane region.

Proof when D is a simple region

Green's theorem

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 (again, 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.

are true, then Green's theorem follows immediately for the region D. We can prove ((/wiki/Green'stheorem#math1)) easily for regions of type I, and ((/wiki/Green'stheorem#math2)) 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, by ${\displaystyle D=\{(x,y)\mid a\leq x\leq b,g{1}(x)\leq y\leq g{2}(x)\}}$ where g1 and g2 are continuous functions on [a, b]. Compute the double integral in ((/wiki/Green'stheorem#math1)):

Now compute the line integral in ((/wiki/Green'stheorem#math1)). 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. Then ${\displaystyle \int {C{1}}L(x,y)\,dx=\int {a}^{b}L(x,g{1}(x))\,dx.}$

With C3, use the parametric equations: x = x, y = g2(x), a ≤ x ≤ b. Then ${\displaystyle \int {C{3}}L(x,y)\,dx=\int {b}^{a}L(x,y)\,dx=-\int {a}^{b}L(x,g{2}(x))\,dx.}$

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 ${\displaystyle \int {C{4}}L(x,y)\,dx=\int {C{2}}L(x,y)\,dx=0.}$

Combining ((/wiki/Green'stheorem#math3)) with ((/wiki/Green'stheorem#math4)), we get ((/wiki/Green'stheorem#math1)) for regions of type I. A similar treatment using the same endpoints yields ((/wiki/Green'stheorem#math2)) for regions of type II. Putting the two together, we get the result for regions of type III.

Proof for rectifiable Jordan curves

Green's theorem

We need the following lemmas whose proofs can be found in:

Proof of Theorem. Let ${\displaystyle \varepsilon }$ be an arbitrary positive real number. By continuity of ${\displaystyle A}$, ${\displaystyle B}$ and compactness of ${\displaystyle {\overline {R}}}$, given ${\displaystyle \varepsilon >0}$, there exists ${\displaystyle 0<\delta <1}$ such that whenever two points of ${\displaystyle {\overline {R}}}$ are less than ${\displaystyle 2{\sqrt {2}}\,\delta }$ apart, their images under ${\displaystyle A,B}$ are less than ${\displaystyle \varepsilon }$ apart. For this ${\displaystyle \delta }$, consider the decomposition given by the previous Lemma. We have ${\displaystyle \int {\Gamma }A\,dx+B\,dy=\sum {i=1}^{k}\int {\Gamma {i}}A\,dx+B\,dy\quad +\sum {i=k+1}^{s}\int {\Gamma {i}}A\,dx+B\,dy.}$

For each ${\displaystyle i\in \{1,\ldots ,k\}}$, the curve ${\displaystyle \Gamma {i}}$ is a positively oriented square, for which Green's formula holds. Hence ${\displaystyle \sum {i=1}^{k}\int {\Gamma {i}}A\,dx+B\,dy=\sum {i=1}^{k}\int {R{i}}\varphi =\int {\bigcup {i=1}^{k}R{i}}\,\varphi .}$

Every point of a border region is at a distance no greater than ${\displaystyle 2{\sqrt {2}}\,\delta }$ from ${\displaystyle \Gamma }$. Thus, if ${\displaystyle K}$ is the union of all border regions, then ${\displaystyle K\subset \Delta {\Gamma }(2{\sqrt {2}}\,\delta )}$; hence ${\displaystyle c(K)\leq {\overline {c}}\,\Delta {\Gamma }(2{\sqrt {2}}\,\delta )\leq 4{\sqrt {2}}\,\delta +8\pi \delta ^{2}}$, by Lemma 2. Notice that ${\displaystyle \int {R}\varphi \,\,-\int {\bigcup {i=1}^{k}R{i}}\varphi =\int {K}\varphi .}$ This yields ${\displaystyle \left\vert \sum {i=1}^{k}\int {\Gamma {i}}A\,dx+B\,dy\quad -\int {R}\varphi \right\vert \leq M\delta (1+\pi {\sqrt {2}}\,\delta ){\text{ for some }}M>0.}$

We may as well choose ${\displaystyle \delta }$ so that the RHS of the last inequality is ${\displaystyle <\varepsilon .}$

The remark in the beginning of this proof implies that the oscillations of ${\displaystyle A}$ and ${\displaystyle B}$ on every border region is at most ${\displaystyle \varepsilon }$. We have ${\displaystyle \left\vert \sum {i=k+1}^{s}\int {\Gamma {i}}A\,dx+B\,dy\right\vert \leq {\frac {1}{2}}\varepsilon \sum {i=k+1}^{s}\Lambda {i}.}$

By Lemma 1(iii), ${\displaystyle \sum {i=k+1}^{s}\Lambda {i}\leq \Lambda +(4\delta )\,4\!\left({\frac {\Lambda }{\delta }}+1\right)\leq 17\Lambda +16.}$

Combining these, we finally get ${\displaystyle \left\vert \int {\Gamma }A\,dx+B\,dy\quad -\int {R}\varphi \right\vert <C\varepsilon ,}$ for some ${\displaystyle C>0}$. Since this is true for every ${\displaystyle \varepsilon >0}$, we are done.

Validity under different hypotheses

Green's theorem

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:

The functions ${\displaystyle A,B:{\overline {R}}\to \mathbb {R} }$ are still assumed to be continuous. However, we now require them to be Fréchet-differentiable at every point of ${\displaystyle R}$. This implies the existence of all directional derivatives, in particular ${\displaystyle D{e{i}}A=:D{i}A,D{e{i}}B=:D{i}B,\,i=1,2}$, where, as usual, ${\displaystyle (e{1},e{2})}$ is the canonical ordered basis of ${\displaystyle \mathbb {R} ^{2}}$. In addition, we require the function ${\displaystyle D{1}B-D{2}A}$ to be Riemann-integrable over ${\displaystyle R}$.

As a corollary of this, we get the Cauchy Integral Theorem for rectifiable Jordan curves:

Multiply-connected regions

Green's theorem

Theorem. Let ${\displaystyle \Gamma {0},\Gamma {1},\ldots ,\Gamma {n}}$ be positively oriented rectifiable Jordan curves in ${\displaystyle \mathbb {R} ^{2}}$ satisfying ${\displaystyle {\begin{aligned}\Gamma {i}\subset R{0},&&{\text{if }}1\leq i\leq n\\\Gamma {i}\subset \mathbb {R} ^{2}\setminus {\overline {R}}{j},&&{\text{if }}1\leq i,j\leq n{\text{ and }}i\neq j,\end{aligned}}}$ where ${\displaystyle R{i}}$ is the inner region of ${\displaystyle \Gamma {i}}$. Let ${\displaystyle D=R{0}\setminus ({\overline {R}}{1}\cup {\overline {R}}{2}\cup \cdots \cup {\overline {R}}{n}).}$

Suppose ${\displaystyle p:{\overline {D}}\to \mathbb {R} }$ and ${\displaystyle q:{\overline {D}}\to \mathbb {R} }$ are continuous functions whose restriction to ${\displaystyle D}$ is Fréchet-differentiable. If the function ${\displaystyle (x,y)\longmapsto {\frac {\partial q}{\partial e{1}}}(x,y)-{\frac {\partial p}{\partial e{2}}}(x,y)}$ is Riemann-integrable over ${\displaystyle D}$, then ${\displaystyle {\begin{aligned}&\int {\Gamma {0}}p(x,y)\,dx+q(x,y)\,dy-\sum {i=1}^{n}\int {\Gamma {i}}p(x,y)\,dx+q(x,y)\,dy\\[5pt]={}&\int {D}\left\{{\frac {\partial q}{\partial e{1}}}(x,y)-{\frac {\partial p}{\partial e{2}}}(x,y)\right\}\,d(x,y).\end{aligned}}}$

Relationship to Stokes' theorem

Green's 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: ${\displaystyle \oint {C}(L\,dx+M\,dy)=\oint {C}(L,M,0)\cdot (dx,dy,dz)=\oint {C}\mathbf {F} \cdot d\mathbf {r} .}$

The Kelvin–Stokes theorem: ${\displaystyle \oint {C}\mathbf {F} \cdot d\mathbf {r} =\iint {S}\nabla \times \mathbf {F} \cdot \mathbf {\hat {n}} \,dS.}$

The surface ${\displaystyle S}$ is just the region in the plane ${\displaystyle D}$, with the unit normal ${\displaystyle \mathbf {\hat {n}} }$ 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 ${\displaystyle \nabla \times \mathbf {F} \cdot \mathbf {\hat {n}} =\left[\left({\frac {\partial 0}{\partial y}}-{\frac {\partial M}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial L}{\partial z}}-{\frac {\partial 0}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\mathbf {k} \right]\cdot \mathbf {k} =\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right).}$

Thus we get the right side of Green's theorem ${\displaystyle \iint {S}\nabla \times \mathbf {F} \cdot \mathbf {\hat {n}} \,dS=\iint {D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dA.}$

Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: ${\displaystyle \oint {C}L\,dx+M\,dy=\oint {\partial D}\!\omega =\int {D}d\omega =\int {D}{\frac {\partial L}{\partial y}}\,dy\wedge \,dx+{\frac {\partial M}{\partial x}}\,dx\wedge \,dy=\iint {D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dx\,dy.}$

Relationship to the divergence theorem

Green's theorem

Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem:

${\displaystyle \iint {D}\left(\nabla \cdot \mathbf {F} \right)dA=\oint {C}\mathbf {F} \cdot \mathbf {\hat {n}} \,ds,}$

where ${\displaystyle \nabla \cdot \mathbf {F} }$ is the divergence on the two-dimensional vector field ${\displaystyle \mathbf {F} }$, and ${\displaystyle \mathbf {\hat {n}} }$ is the outward-pointing unit normal vector on the boundary.

To see this, consider the unit normal ${\displaystyle \mathbf {\hat {n}} }$ in the right side of the equation. Since in Green's theorem ${\displaystyle d\mathbf {r} =(dx,dy)}$ is a vector pointing tangential along the curve, and the curve C is the positively oriented (i.e. anticlockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right of this; one choice would be ${\displaystyle (dy,-dx)}$. The length of this vector is ${\textstyle {\sqrt {dx^{2}+dy^{2}}}=ds.}$ So ${\displaystyle (dy,-dx)=\mathbf {\hat {n}} \,ds.}$

Start with the left side of Green's theorem: ${\displaystyle \oint {C}(L\,dx+M\,dy)=\oint {C}(M,-L)\cdot (dy,-dx)=\oint {C}(M,-L)\cdot \mathbf {\hat {n}} \,ds.}$ Applying the two-dimensional divergence theorem with ${\displaystyle \mathbf {F} =(M,-L)}$, we get the right side of Green's theorem: ${\displaystyle \oint {C}(M,-L)\cdot \mathbf {\hat {n}} \,ds=\iint {D}\left(\nabla \cdot (M,-L)\right)\,dA=\iint {D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dA.}$

Area calculation

Green's theorem

Green's theorem can be used to compute area by line integral. The area of a planar region ${\displaystyle D}$ is given by ${\displaystyle A=\iint {D}dA.}$

Choose ${\displaystyle L}$ and ${\displaystyle M}$ such that ${\displaystyle {\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}=1}$, the area is given by ${\displaystyle A=\oint {C}(L\,dx+M\,dy).}$

Possible formulas for the area of ${\displaystyle D}$ include ${\displaystyle A=\oint {C}x\,dy=-\oint {C}y\,dx={\tfrac {1}{2}}\oint {C}(-y\,dx+x\,dy).}$

History

Green's theorem

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. Green did not actually derive the form of "Green's theorem" 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 "Green's theorem" which appears in this article was first published, without proof, in an article by Augustin Cauchy: A. 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, where (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), (Göttingen, (Germany): Adalbert Rente, 1867); see pages 8–9.