Green's theorem

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In vector calculus, Green's theorem relates a line integral around a simple closed curve Template:Mvar to a double integral over the plane region Template:Mvar (surface in <math>\R^2</math>) bounded by Template:Mvar. It is the two-dimensional special case of Stokes' theorem (surface in <math>\R^3</math>). In one dimension, it is equivalent to the fundamental theorem of calculus. In three dimensions, it is equivalent to the divergence theorem.

Theorem

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

<math display="block">\oint_C (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA</math>

where the path of integration along Template:Mvar is counterclockwise.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>

Application

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

File:Green's-theorem-simple-region.svg

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

and

are true, then Green's theorem follows immediately for the region D. We can prove (Template:EquationNote) easily for regions of type I, and (Template:EquationNote) 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 <math display="block">D = \{(x,y)\mid a\le x\le b, g_1(x) \le y \le g_2(x)\}</math> where g₁ and g₂ are continuous functions on Template:Closed-closed. Compute the double integral in (Template:EquationNote):

Now compute the line integral in (Template:EquationNote). 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 <math display="block">\int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x))\, dx.</math>

With C₃, use the parametric equations: x = x, y = g₂(x), a ≤ x ≤ b. Then <math display="block"> \int_{C_3} L(x,y)\, dx = -\int_{-C_3} L(x,y)\, dx = - \int_a^b L(x,g_2(x))\, dx.</math>

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 <math display="block"> \int_{C_4} L(x,y)\, dx = \int_{C_2} L(x,y)\, dx = 0.</math>

Therefore,

Combining (Template:EquationNote) with (Template:EquationNote), we get (Template:EquationNote) for regions of type I. A similar treatment yields (Template:EquationNote) 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:<ref>Template:Cite book</ref>

Now we are in position to prove the theorem:

Proof of Theorem. Let <math>\varepsilon</math> be an arbitrary positive real number. By continuity of <math>A</math>, <math>B</math> and compactness of <math>\overline{R}</math>, given <math>\varepsilon>0</math>, there exists <math>0<\delta<1</math> such that whenever two points of <math>\overline{R}</math> are less than <math>2\sqrt{2}\,\delta</math> apart, their images under <math>A, B</math> are less than <math>\varepsilon</math> apart. For this <math>\delta</math>, consider the decomposition given by the previous Lemma. We have <math display="block">\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.</math>

Put <math>\varphi := D_1 B - D_2 A</math>.

For each <math>i\in\{1,\ldots,k\}</math>, the curve <math>\Gamma_i</math> is a positively oriented square, for which Green's formula holds. Hence <math display="block">\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.</math>

Every point of a border region is at a distance no greater than <math>2\sqrt{2}\,\delta</math> from <math>\Gamma</math>. Thus, if <math>K</math> is the union of all border regions, then <math>K\subset \Delta_{\Gamma}(2\sqrt{2}\,\delta)</math>; hence <math>c(K)\le\overline{c}\,\Delta_{\Gamma}(2\sqrt{2}\,\delta)\le 4\sqrt{2}\,\delta+8\pi\delta^2</math>, by Lemma 2. Notice that <math display="block">\int_R \varphi\,\,-\int_{\bigcup_{i=1}^k R_i} \varphi=\int_K \varphi.</math> This yields <math display="block">\left\vert\sum_{i=1}^k \int_{\Gamma_i} A\,dx+B\,dy\quad-\int_R\varphi \right\vert \le M \delta(1+\pi\sqrt{2}\,\delta) \text{ for some } M > 0.</math>

We may as well choose <math>\delta</math> so that the RHS of the last inequality is <math><\varepsilon.</math>

The remark in the beginning of this proof implies that the oscillations of <math>A</math> and <math>B</math> on every border region is at most <math>\varepsilon</math>. We have <math display="block">\left\vert\sum_{i=k+1}^s \int_{\Gamma_i}A\,dx+B\,dy\right\vert\le\frac{1}{2} \varepsilon\sum_{i=k+1}^s \Lambda_i.</math>

By Lemma 1(iii), <math display="block">\sum_{i=k+1}^s \Lambda_i \le\Lambda + (4\delta)\,4\!\left(\frac{\Lambda}{\delta}+1\right)\le17\Lambda+16.</math>

Combining these, we finally get <math display="block">\left\vert\int_\Gamma A\,dx+B\,dy\quad-\int_R \varphi\right\vert< C \varepsilon,</math> for some <math>C > 0</math>. Since this is true for every <math>\varepsilon > 0</math>, 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 <math>A, B:\overline{R} \to \R</math> are still assumed to be continuous. However, we now require them to be Fréchet-differentiable at every point of <math>R</math>. This implies the existence of all directional derivatives, in particular <math>D_{e_i}A=:D_i A, D_{e_i}B=:D_i B, \,i=1,2</math>, where, as usual, <math>(e_1,e_2)</math> is the canonical ordered basis of <math>\R^2</math>. In addition, we require the function <math>D_1 B-D_2 A</math> to be Riemann-integrable over <math>R</math>.

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

Multiply-connected regions

Theorem. Let <math>\Gamma_0,\Gamma_1,\ldots,\Gamma_n</math> be positively oriented rectifiable Jordan curves in <math>\R^{2}</math> satisfying <math display="block">\begin{aligned} \Gamma_i \subset R_0,&&\text{if } 1\le i\le n\\ \Gamma_i \subset \R^2 \setminus \overline{R}_j,&&\text{if }1\le i,j \le n\text{ and }i\ne j, \end{aligned}</math> where <math>R_i</math> is the inner region of <math>\Gamma_i</math>. Let <math display="block">D = R_0 \setminus (\overline{R}_1 \cup \overline{R}_2 \cup \cdots \cup \overline{R}_n).</math>

Suppose <math>p: \overline{D} \to \R</math> and <math>q: \overline{D} \to \R</math> are continuous functions whose restriction to <math>D</math> is Fréchet-differentiable. If the function <math display="block">(x,y)\longmapsto\frac{\partial q}{\partial e_1}(x,y)-\frac{\partial p}{\partial e_2}(x,y)</math> is Riemann-integrable over <math>D</math>, then <math display="block">\begin{align} & \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{align}</math>

Relationship to Stokes' theorem

Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the <math>xy</math>-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 <math>\mathbf{F}=(L,M,0)</math>. Start with the left side of Green's theorem: <math display="block">\oint_C (L\, dx + M\, dy) = \oint_C (L, M, 0) \cdot (dx, dy, dz) = \oint_C \mathbf{F} \cdot d\mathbf{r}. </math>

The Kelvin–Stokes theorem: <math display="block">\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \nabla \times \mathbf{F} \cdot \mathbf{\hat n} \, dS. </math>

The surface <math>S</math> is just the region in the plane <math>D</math>, with the unit normal <math>\mathbf{\hat n}</math> 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 <math display="block">\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). </math>

Thus we get the right side of Green's theorem <math display="block">\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. </math>

Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: <math display="block">\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.</math>

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:

<math>\iint_D\left(\nabla\cdot\mathbf{F}\right)dA=\oint_C \mathbf{F} \cdot \mathbf{\hat n} \, ds,</math>

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

To see this, consider the unit normal <math>\mathbf{\hat n}</math> in the right side of the equation. Since in Green's theorem <math>d\mathbf{r} = (dx, dy)</math> 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 <math>(dy, -dx)</math>. The length of this vector is <math display="inline">\sqrt{dx^2 + dy^2} = ds.</math> So <math>(dy, -dx) = \mathbf{\hat n}\,ds.</math>

Start with the left side of Green's theorem: <math display="block">\oint_C (L\, dx + M\, dy) = \oint_C (M, -L) \cdot (dy, -dx) = \oint_C (M, -L) \cdot \mathbf{\hat n}\,ds.</math> Applying the two-dimensional divergence theorem with <math>\mathbf{F} = (M, -L)</math>, we get the right side of Green's theorem: <math display="block">\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.</math>

Area calculation

Green's theorem can be used to compute area by line integral.<ref name="stuart">Template:Cite book</ref> The area of a planar region <math>D</math> is given by <math display="block">A = \iint_D dA.</math>

Choose <math>L</math> and <math>M</math> such that <math>\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1</math>, the area is given by <math display="block">A = \oint_{C} (L\, dx + M\, dy).</math>

Possible formulas for the area of <math>D</math> include<ref name="stuart" /> <math display="block">A=\oint_C x\, dy = -\oint_C y\, dx = \tfrac 12 \oint_C (-y\, dx + x\, dy).</math>

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. 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.<ref>Template:Cite book</ref>

See also

• Template:Annotated link
• 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
• Desmos - A web based graphing calculator

Further reading

• Template:Cite book

External links

• Green's Theorem on MathWorld