In vector calculus, Green's theorem relates a
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
around a
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
to a
double integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
over the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
region bounded by . It is the two-dimensional special case of
Stokes' theorem.
Theorem
Let be a positively
oriented,
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
,
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
in a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, and let be the region bounded by . If and are functions of defined on an
open region
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
containing and have
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
there, then
where the path of integration along is
anticlockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
.
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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, and in particular, area
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
, 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 ''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.
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
where ''g''
1 and ''g''
2 are
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s on . Compute the double integral in ():
Now compute the line integral in (). ''C'' can be rewritten as the union of four curves: ''C''
1, ''C''
2, ''C''
3, ''C''
4.
With ''C''
1, use the
parametric equations: ''x'' = ''x'', ''y'' = ''g''
1(''x''), ''a'' ≤ ''x'' ≤ ''b''. Then
With ''C''
3, use the parametric equations: ''x'' = ''x'', ''y'' = ''g''
2(''x''), ''a'' ≤ ''x'' ≤ ''b''. Then
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
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:
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.
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
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:
Multiply-connected regions
Theorem. Let
be positively oriented rectifiable Jordan curves in
satisfying
where
is the inner region of
. Let
Suppose
and
are continuous functions whose restriction to
is Fréchet-differentiable. If the function
is Riemann-integrable over
, then
Relationship to Stokes' theorem
Green's theorem is a special case of the
Kelvin–Stokes theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
, when applied to a region in the
-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
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
-valued function
. Start with the left side of Green's theorem:
The Kelvin–Stokes theorem:
The surface
is just the region in the plane
, with the unit normal
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
Thus we get the right side of Green's theorem
Green's theorem is also a straightforward result of the general Stokes' theorem using
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s and
exterior derivatives:
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
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
:
:
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:
Area calculation
Green's theorem can be used to compute area by line integral.
The area of a planar region
is given by
Choose
and
such that
, the area is given by
Possible formulas for the area of
include
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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
gave the first proof of Green's theorem in his doctoral dissertation on the theory of functions of a complex variable.
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
The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian c ...
– A special case of Green's theorem for simple polygons
References
Further reading
*
External links
Green's Theorem on MathWorld
{{Authority control
Theorems in calculus
Articles containing proofs