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, and in 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 C_{1}, C_{2}, C_{3}, C_{4}, 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 C_{1} and C_{3} 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 C_{2} and C_{4} 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 (1) easily for regions of type I, and (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, by
The integral over C_{3} is negated because it goes in the negative direction from b to a, as C is oriented positively (anticlockwise). On C_{2} and C_{4}, x remains constant, meaning
Combining (3) with (4), we get (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 $\Gamma$ be a rectifiable, positively oriented Jordan curve in $\mathbb {R} ^{2}$ and let $R$ denote its inner region. Suppose that $A,B:{\overline {R}}\to \mathbb {R}$ are continuous functions with the property that $A$ has second partial derivative at every point of $R$, $B$ has first partial derivative at every point of $R$ and that the functions $D_{1}B,D_{2}A:R\to \mathbb {R}$ are Riemann-integrable over $R$. Then
We need the following lemmas whose proofs can be found in:^{[3]}
Lemma 1 (Decomposition Lemma). Assume $\Gamma$ is a rectifiable, positively oriented Jordan curve in the plane and let $R$ be its inner region. For every positive real $\delta$, let ${\mathcal {F}}(\delta )$ denote the collection of squares in the plane bounded by the lines $x=m\delta ,y=m\delta$, where $m$ runs through the set of integers. Then, for this $\delta$, there exists a decomposition of ${\overline {R}}$ into a finite number of non-overlapping subregions in such a manner that
Each one of the subregions contained in $R$, say $R_{1},R_{2},\ldots ,R_{k}$, is a square from ${\mathcal {F}}(\delta )$.
Each one of the remaining subregions, say $R_{k+1},\ldots ,R_{s}$, has as boundary a rectifiable Jordan curve formed by a finite number of arcs of $\Gamma$ and parts of the sides of some square from ${\mathcal {F}}(\delta )$.
Each one of the border regions $R_{k+1},\ldots ,R_{s}$ can be enclosed in a square of edge-length $2\delta$.
If $\Gamma _{i}$ is the positively oriented boundary curve of $R_{i}$, then $\Gamma =\Gamma _{1}+\Gamma _{2}+\cdots +\Gamma _{s}.$
The number $s-k$ of border regions is no greater than $4\!\left({\frac {\Lambda }{\delta }}+1\right)$, where $\Lambda$ is the length of $\Gamma$.
Lemma 2. Let $\Gamma$ be a rectifiable curve in the plane and let $\Delta _{\Gamma }(h)$ be the set of points in the plane whose distance from (the range of) $\Gamma$ is at most $h$. The outer Jordan content of this set satisfies ${\overline {c}}\,\,\Delta _{\Gamma }(h)\leq 2h\Lambda +\pi h^{2}$.
Lemma 3. Let $\Gamma$ be a rectifiable curve in $\mathbb {R} ^{2}$ and let $f:{\text{range of }}\Gamma \to \mathbb {R}$ be a continuous function. Then
$\left\vert \int _{\Gamma }f(x,y)\,dy\right\vert$ and
$\left\vert \int _{\Gamma }f(x,y)\,dx\right\vert$ are $\leq {\frac {1}{2}}\Lambda \Omega _{f},$where $\Omega _{f}$ is the oscillation of $f$ on the range of $\Gamma$.
Now we are in position to prove the Theorem:
Proof of Theorem. Let $\varepsilon$ be an arbitrary positive real number. By continuity of $A$, $B$ and compactness of ${\overline {R}}$, given $\varepsilon >0$, there exists $0<\delta <1$ such that whenever two points of ${\overline {R}}$ are less than $2{\sqrt {2}}\,\delta$ apart, their images under $A,B$ are less than $\varepsilon$ apart. For this $\delta$, consider the decomposition given by the previous Lemma. We have
Every point of a border region is at a distance no greater than $2{\sqrt {2}}\,\delta$ from $\Gamma$. Thus, if $K$ is the union of all border regions, then $K\subset \Delta _{\Gamma }(2{\sqrt {2}}\,\delta )$; hence $c(K)\leq {\overline {c}}\,\Delta _{\Gamma }(2{\sqrt {2}}\,\delta )\leq 4{\sqrt {2}}\,\delta +8\pi \delta ^{2}$, by Lemma 2. Notice that
for some $C>0$. Since this is true for every $\varepsilon >0$, 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:
The functions $A,B:{\overline {R}}\to \mathbb {R}$ are still assumed to be continuous. However, we now require them to be Fréchet-differentiable at every point of $R$. This implies the existence of all directional derivatives, in particular $D_{e_{i}}A=:D_{i}A,D_{e_{i}}B=:D_{i}B,\,i=1,2$, where, as usual, $(e_{1},e_{2})$ is the canonical ordered basis of $\mathbb {R} ^{2}$. In addition, we require the function $D_{1}B-D_{2}A$ to be Riemann-integrable over $R$.
As a corollary of this, we get the Cauchy Integral Theorem for rectifiable Jordan curves:
Theorem (Cauchy). If $\Gamma$ is a rectifiable Jordan curve in $\mathbb {C}$ and if $f:{\text{closure of inner region of }}\Gamma \to \mathbb {C}$ is a continuous mapping holomorphic throughout the inner region of $\Gamma$, then
$\int _{\Gamma }f=0,$
the integral being a complex contour integral.
Proof. We regard the complex plane as $\mathbb {R} ^{2}$. Now, define $u,v:{\overline {R}}\to \mathbb {R}$ to be such that $f(x+iy)=u(x,y)+iv(x,y).$ These functions are clearly continuous. It is well known that $u$ and $v$ are Fréchet-differentiable and that they satisfy the Cauchy-Riemann equations: $D_{1}v+D_{2}u=D_{1}u-D_{2}v={\text{zero function}}$.
Now, analyzing the sums used to define the complex contour integral in question, it is easy to realize that
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. Let $\Gamma _{0},\Gamma _{1},\ldots ,\Gamma _{n}$ be positively oriented rectifiable Jordan curves in $\mathbb {R} ^{2}$ satisfying
Suppose $p:{\overline {D}}\to \mathbb {R}$ and $q:{\overline {D}}\to \mathbb {R}$ are continuous functions whose restriction to $D$ is Fréchet-differentiable. If the function
Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the $xy$-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 $\mathbf {F} =(L,M,0)$. Start with the left side of Green's theorem:
The surface $S$ is just the region in the plane $D$, with the unit normal $\mathbf {\hat {n}}$ defined (by convention) to have a positive z component in order to match the "positive orientation" definitions for both theorems.
where $\nabla \cdot \mathbf {F}$ is the divergence on the two-dimensional vector field $\mathbf {F}$, and $\mathbf {\hat {n}}$ is the outward-pointing unit normal vector on the boundary.
To see this, consider the unit normal $\mathbf {\hat {n}}$ in the right side of the equation. Since in Green's theorem $d\mathbf {r} =(dx,dy)$ 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 $(dy,-dx)$. The length of this vector is ${\textstyle {\sqrt {dx^{2}+dy^{2}}}=ds.}$ So $(dy,-dx)=\mathbf {\hat {n}} \,ds.$
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.
^Apostol, Tom (1960). Mathematical Analysis (1 ed.). Reading, Massachusetts, U.S.A.: Addison-Wesley Publishing Company, INC.
^ ^{a}^{b}Stewart, James (1999). Calculus (6th ed.). Thomson, Brooks/Cole.
^George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, England: T. Wheelhouse, 1828). 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.
^Katz, Victor (2009). "22.3.3: Complex Functions and Line Integrals". A History of Mathematics: An Introduction. Addison-Wesley. pp. 801–5. ISBN 0-321-38700-7.
Further reading
Marsden, Jerrold E.; Tromba, Anthony J. (2003). "The Integral Theorems of Vector Analysis". Vector Calculus (Fifth ed.). New York: Freeman. pp. 518–608. ISBN 0-7167-4992-0.