Green's theorem explained

In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region (surface in

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

Theorem

Let be a positively oriented, piecewise smooth, simple closed curve in a plane, and let be the region bounded by . If and are functions of defined on an open region containing and have continuous partial derivatives there, then

$$\backslash oint\_C\; (L\backslash ,\; dx\; +\; M\backslash ,\; dy)\; =\; \backslash iint\_\; \backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; dA$$

where the path of integration along 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

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.

If it can be shown that

and

are true, then Green's theorem follows immediately for the region D. We can prove easily for regions of type I, and 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$$D\; =\; \backslash$$where g₁ and g₂ are continuous functions on . Compute the double integral in :

Now compute the line integral in . 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$$\backslash int\_\; L(x,y)\backslash ,\; dx\; =\; \backslash int\_a^b\; L(x,g\_1(x))\backslash ,\; dx.$$

With C₃, use the parametric equations: x = x, y = g₂(x), a ≤ x ≤ b. Then$$\backslash int\_\; L(x,y)\backslash ,\; dx\; =\; -\backslash int\_\; L(x,y)\backslash ,\; dx\; =\; -\; \backslash int\_a^b\; L(x,g\_2(x))\backslash ,\; dx.$$

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$$\backslash int\_\; L(x,y)\backslash ,\; dx\; =\; \backslash int\_\; L(x,y)\backslash ,\; dx\; =\; 0.$$

Therefore,

Combining with, we get for regions of type I. A similar treatment yields 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

We need the following lemmas whose proofs can be found in:^[3]

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$$\backslash int\_\backslash Gamma\; A\backslash ,dx+B\backslash ,dy=\backslash sum\_^k\; \backslash int\_\; A\backslash ,dx+B\backslash ,dy\backslash quad\; +\backslash sum\_^s\; \backslash int\_A\backslash ,dx+B\backslash ,dy.$$

Put

\varphi : = D _{1} B - D _{2} A .

For each

i\in\{ 1 ,\ldots,k\} , the curve \Gamma _{i} is a positively oriented square, for which Green's formula holds. Hence$$\backslash sum\_^k\; \backslash int\_A\backslash ,dx\; +\; B\backslash ,dy\; =\backslash sum\_^k\; \backslash int\_\; \backslash varphi\; =\; \backslash int\_\backslash ,\backslash varphi.$$

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)\le\overline{c}\Delta _{\Gamma} ( 2 \sqrt{ 2 }\delta)\le 4 \sqrt{ 2 }\delta +8 \pi\delta ^{2} , by Lemma 2. Notice that$$\backslash int\_R\; \backslash varphi\backslash ,\backslash ,-\backslash int\_\; \backslash varphi=\backslash int\_K\; \backslash varphi.$$This yields$$\backslash left\backslash vert\backslash sum\_^k\; \backslash int\_\; A\backslash ,dx+B\backslash ,dy\backslash quad-\backslash int\_R\backslash varphi\; \backslash right\backslash vert\; \backslash le\; M\; \backslash delta(1+\backslash pi\backslash sqrt\backslash ,\backslash delta)\; \backslash text\; M\; >\; 0.$$

We may as well choose

\delta so that the RHS of the last inequality is <\varepsilon.

The remark in the beginning of this proof implies that the oscillations of

A and B on every border region is at most \varepsilon . We have$$\backslash left\backslash vert\backslash sum\_^s\; \backslash int\_A\backslash ,dx+B\backslash ,dy\backslash right\backslash vert\backslash le\backslash frac\; \backslash varepsilon\backslash sum\_^s\; \backslash Lambda\_i.$$

By Lemma 1(iii),$$\backslash sum\_^s\; \backslash Lambda\_i\; \backslash le\backslash Lambda\; +\; (4\backslash delta)\backslash ,4\backslash !\backslash left(\backslash frac+1\backslash right)\backslash le17\backslash Lambda+16.$$

Combining these, we finally getfor 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 \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 \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:

Multiply-connected regions

Theorem. Let

\Gamma _{0,\Gamma} _{1,\ldots,\Gamma} _{n} be positively oriented rectifiable Jordan curves in \R ^{2} satisfyingwhere R _{i} is the inner region of \Gamma _{i} . Let$$D\; =\; R\_0\; \backslash setminus\; (\backslash overline\_1\; \backslash cup\; \backslash overline\_2\; \backslash cup\; \backslash cdots\; \backslash cup\; \backslash overline\_n).$$

Suppose

p: \overline{D} \to \R and q: \overline{D} \to \R are continuous functions whose restriction to D is Fréchet-differentiable. If the function$$(x,y)\backslash longmapsto\backslash frac(x,y)-\backslash frac(x,y)$$is Riemann-integrable over D , then

Relationship to Stokes' theorem

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

F = (L,M, 0 ) . Start with the left side of Green's theorem:$$\backslash oint\_C\; (L\backslash ,\; dx\; +\; M\backslash ,\; dy)\; =\; \backslash oint\_C\; (L,\; M,\; 0)\; \backslash cdot\; (dx,\; dy,\; dz)\; =\; \backslash oint\_C\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf.$$

The Kelvin–Stokes theorem:$$\backslash oint\_C\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; \backslash iint\_S\; \backslash nabla\; \backslash times\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash ,\; dS.$$

The surface

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

Thus we get the right side of Green's theorem$$\backslash iint\_S\; \backslash nabla\; \backslash times\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash ,\; dS\; =\; \backslash iint\_D\; \backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; \backslash ,\; dA.$$

Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives:$$\backslash oint\_C\; L\; \backslash ,dx\; +\; M\; \backslash ,dy=\; \backslash oint\_\; \backslash !\; \backslash omega=\; \backslash int\_D\; d\backslash omega=\; \backslash int\_D\; \backslash frac\; \backslash ,dy\; \backslash wedge\; \backslash ,dx\; +\; \backslash frac\; \backslash ,dx\; \backslash wedge\; \backslash ,dy=\; \backslash iint\_D\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right)\; \backslash ,dx\; \backslash ,dy.$$

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:

\iint _{D\left(\nabla ⋅ F\right)dA=\oint} _{C} F ⋅ \hatn ds,

where

\nabla ⋅ F is the divergence on the two-dimensional vector field F , and \hatn is the outward-pointing unit normal vector on the boundary.

To see this, consider the unit normal

\hatn in the right side of the equation. Since in Green's theorem d 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 $\backslash sqrt\; =\; ds.$ So (dy, - dx) = \hatn ds.

Start with the left side of Green's theorem:$$\backslash oint\_C\; (L\backslash ,\; dx\; +\; M\backslash ,\; dy)\; =\; \backslash oint\_C\; (M,\; -L)\; \backslash cdot\; (dy,\; -dx)\; =\; \backslash oint\_C\; (M,\; -L)\; \backslash cdot\; \backslash mathbf\backslash ,ds.$$Applying the two-dimensional divergence theorem with

F = (M, - L) , we get the right side of Green's theorem:$$\backslash oint\_C\; (M,\; -L)\; \backslash cdot\; \backslash mathbf\backslash ,ds\; =\; \backslash iint\_D\backslash left(\backslash nabla\; \backslash cdot\; (M,\; -L)\; \backslash right)\; \backslash ,\; dA\; =\; \backslash iint\_D\; \backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; \backslash ,\; dA.$$

Area calculation

Green's theorem can be used to compute area by line integral.^[4] The area of a planar region

D is given by$$A\; =\; \backslash iint\_D\; dA.$$

Choose

L and M such that (\partial M)/(\partial x) - (\partial L)/(\partial y) = 1 , the area is given by$$A\; =\; \backslash oint\_\; (L\backslash ,\; dx\; +\; M\backslash ,\; dy).$$

Possible formulas for the area of

D include$$A=\backslash oint\_C\; x\backslash ,\; dy\; =\; -\backslash oint\_C\; y\backslash ,\; dx\; =\; \backslash tfrac\; 12\; \backslash oint\_C\; (-y\backslash ,\; dx\; +\; x\backslash ,\; dy).$$

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

• • 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

Further reading

• Book: Jerrold E. . Marsden . Anthony J. . Tromba . The Integral Theorems of Vector Analysis . Vector Calculus . New York . Freeman . 2003 . Fifth . 0-7167-4992-0 . 518–608 . https://books.google.com/books?id=LiRLJf2m_dwC&pg=PA518 .

External links

• Green's Theorem on MathWorld

Notes and References

1. Book: Mathematical Methods for Physics and Engineering . registration . K. F. . Riley . M. P. . Hobson . S. J. . Bence . Cambridge University Press . 2010 . 978-0-521-86153-3 .
2. Book: Vector Analysis . 2nd . M. R. . Spiegel . S. . Lipschutz . D. . Spellman . Schaum’s Outlines . McGraw Hill . 2009 . 978-0-07-161545-7 .
3. Book: Apostol . Tom . Mathematical Analysis . 1960 . Addison-Wesley Publishing Company, INC . Reading, Massachusetts, U.S.A. . 1.
4. Book: Stewart, James . Calculus . 1999. registration. Thomson, Brooks/Cole . 9780534359492 . 6th.
5. 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.
6. Book: Katz, Victor. A History of Mathematics: An Introduction. Addison-Wesley. 2009. 978-0-321-38700-4. 801–5. 22.3.3: Complex Functions and Line Integrals.