Theorem in calculus relating line and double integrals

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.

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.

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.

If it can be shown that

(1)

and

(2)

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

Theorem — Let be a rectifiable, positively oriented Jordan curve in and let denote its inner region. Suppose that are continuous functions with the property that has second partial derivative at every point of , has first partial derivative at every point of and that the functions are Riemann-integrable over . Then

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

Lemma 1 (Decomposition Lemma) — Assume is a rectifiable, positively oriented Jordan curve in the plane and let be its inner region. For every positive real , let denote the collection of squares in the plane bounded by the lines , where runs through the set of integers. Then, for this , there exists a decomposition of into a finite number of non-overlapping subregions in such a manner that

Each one of the subregions contained in , say , is a square from .

Each one of the remaining subregions, say , has as boundary a rectifiable Jordan curve formed by a finite number of arcs of and parts of the sides of some square from .

Each one of the border regions can be enclosed in a square of edge-length .

If is the positively oriented boundary curve of , then

The number of border regions is no greater than , where is the length of .

Lemma 2 — Let be a rectifiable curve in the plane and let be the set of points in the plane whose distance from (the range of) is at most . The outer Jordan content of this set satisfies .

Lemma 3 — Let be a rectifiable curve in and let be a continuous function. Then

and

where is the oscillation of on the range of .

Now we are in position to prove the theorem:

Proof of Theorem. Let be an arbitrary positive real number. By continuity of , and compactness of , given , there exists such that whenever two points of are less than apart, their images under are less than apart. For this , consider the decomposition given by the previous Lemma. We have

Put .

For each , the curve is a positively oriented square, for which Green’s formula holds. Hence

Every point of a border region is at a distance no greater than from . Thus, if is the union of all border regions, then ; hence , by Lemma 2. Notice that

This yields

We may as well choose so that the RHS of the last inequality is

The remark in the beginning of this proof implies that the oscillations of and on every border region is at most . We have

By Lemma 1(iii),

Combining these, we finally get

for some . Since this is true for every , we are done.

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 are still assumed to be continuous. However, we now require them to be Fréchet-differentiable at every point of . This implies the existence of all directional derivatives, in particular , where, as usual, is the canonical ordered basis of . In addition, we require the function to be Riemann-integrable over .

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

Theorem (Cauchy) — If is a rectifiable Jordan curve in and if is a continuous mapping holomorphic throughout the inner region of , then

the integral being a complex contour integral.

Proof

We regard the complex plane as . Now, define to be such that These functions are clearly continuous. It is well known that and are Fréchet-differentiable and that they satisfy the Cauchy-Riemann equations: .

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.

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

where is the divergence on the two-dimensional vector field , and is the outward-pointing unit normal vector on the boundary.

To see this, consider the unit normal in the right side of the equation. Since in Green’s theorem 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 . The length of this vector is So

Start with the left side of Green’s theorem:

Applying the two-dimensional divergence theorem with , we get the right side of Green’s theorem:

^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. ISBN978-0-321-38700-4.