Green's theorem

Theorem

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

∮

C

(

L

d

x

+

M

d

y

)

=

∬

D

(

∂

M

∂

x

−

∂

L

∂

y

)

d

A

{\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)dA}

where the path of integration along C is counterclockwise.

Application

Green's theorem in the plane relates line integrals around a simple closed curve to double integrals over the regions it encloses. It has two equivalent forms: the circulation form, which says the tangential line integral of a vector field

F

=

(

P

,

Q

)

{\displaystyle \mathbf {F} =(P,Q)}

around a positively oriented simple closed curve

C

{\displaystyle C}

equals the double integral of the scalar curl

∂

Q

∂

x

−

∂

P

∂

y

{\displaystyle {\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}}

over the region

D

{\displaystyle D}

bounded by

C

Proof when D is a simple region

The following is a proof of half of the theorem for the simplified area

D

{\displaystyle D}

, a type I region where

C

1

{\displaystyle C_{1}}

and

C

3

{\displaystyle 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

{\displaystyle D}

is a type II region where

C

2

{\displaystyle C_{2}}

and

C

4

{\displaystyle 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

{\displaystyle D}

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

ε

{\displaystyle \varepsilon }

be an arbitrary positive real number. By continuity of

A

{\displaystyle A}

,

B

{\displaystyle B}

and compactness of

R

¯

{\displaystyle {\overline {R}}}

, given

ε

>

0

{\displaystyle \varepsilon >0}

, there exists

0

<

δ

<

1

{\displaystyle 0<\delta <1}

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

:

R

¯

→

R

{\displaystyle 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

{\displaystyle R}

. This implies the existence of all directional derivatives, in particular

D

e

i

A

=:

D

i

A

,

D

e

i

B

=:

Multiply-connected regions

Theorem. Let

Γ

0

,

Γ

1

,

…

,

Γ

n

{\displaystyle \Gamma _{0},\Gamma _{1},\ldots ,\Gamma _{n}}

be positively oriented rectifiable Jordan curves in

R

2

{\displaystyle \mathbb {R} ^{2}}

satisfying

Γ

i

⊂

R

0

,

if

1

≤

i

≤

n

Γ

i

⊂

R

2

Relationship to Stokes' theorem

Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the

x

y

{\displaystyle 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

)

{\displaystyle \mathbf {F} =(L,M,0)}

. Start with the left side of Green's theorem:

∮

C

(

L

d

x

+

M

d

y

)

=

∮

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:

∬

D

(

∇

⋅

F

)

d

A

=

∮

C

F

⋅

n

^

d

s

,

{\displaystyle \iint _{D}\left(\nabla \cdot \mathbf {F} \right)dA=\oint _{C}\mathbf {F} \cdot \mathbf {\hat {n}} \,ds,}

where

∇

⋅

F

{\displaystyle \nabla \cdot \mathbf {F} }

is the divergence on the two-dimensional vector field

F

{\displaystyle \mathbf {F} }

Area calculation

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

D

{\displaystyle D}

is given by

A

=

∬

D

d

A

.

{\displaystyle A=\iint _{D}dA.}

Choose

L

{\displaystyle L}

and

M

{\displaystyle M}

such that

∂

M

∂

x

−

∂

L

∂

y

=

1

{\displaystyle {\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}=1}

. The area is given by

History

William Thomson (Lord Kelvin) named this theorem 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, which was almost forgotten until Thomson rediscovered it in 1845. In 1846, Augustin-Louis Cauchy published a paper stating Green's theorem as the penultimate sentence. A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation.