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5.5 Green’s Theorem Again

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Remark 5.5.1. Now I have explored all the ideas on differential length and

area stretching, I can prove Green’s Theorem for regions which are the images

of squares by maps which are smooth embeddings (except perhaps on sets

over which the integral of any function will be zero, where they only have to

be continuous). The ideas here will generalise considerably.

Exercise 5.5.1. Show that a disc can be obtained as the image of a square

by a map g which is differentiable and has a differentiable inverse at every

point except the top and bottom of the square.

90 CHAPTER 5. GREEN’S THEOREM

Remark 5.5.2. Suppose we have a differential 1-form on R2 and a curve in

R2 defined by a smooth embedding g : I −! R2.

Then we can pull back the differential 1-form in a similar way to the way we

pulled back a function:

Definition 5.13. If g : I −! U _ R2 is a smooth embedding and if ! is a

differential 1-form on U, g_! is the differential 1-form on I defined by

g_! , P _ g

dt + Q _ g

where

! , P dx + Q dy

and

g(t) =

x(t)

y(t)

Remark 5.5.3. If you draw a picture of this you will see that we are turning

the vector field on R2 into one along the curve by just looking at the

component tangent to the curve and pulling this back to I.

Remark 5.5.4. This works for 1-forms on Rn:

! , P1dx1 + P1dx2 + · · · + Pndxn

g_dxi , dxi

where

g(t) =

6664

x1(t)

x2(t)

...

xn(t)

7775

and g_Pi = Pi _ g

Remark 5.5.5. We can say that we use composition with g on the function

part, each Pi goes to Pi _g, and we use composition with g0 on the differential

part, to get g_!.

Remark 5.5.6. In particular we recover the case where n = 2 and

g(t) =

x(t)

y(t)

5.5. GREEN’S THEOREM AGAIN 91

g_dx =

dt; g_dy =

dt.

So if ! = Pdx + Qdy

g_! = (P _ g)

dt + (Q _ g)

P _ g

+ Q _ g

dt.

Remark 5.5.7. We can do the same thing with maps of I2, the unit square,

into Rn, and differential 2-forms on Rn getting pulled back to I2:

Definition 5.14. If g : I2 −! U _ R2 is a smooth embedding, and if ! is a

2-form on U,

! , Pdx ^ dy

g_! , (P _ g) dx ^ dy

and _

# _

where

I2 g

_ −! U

x(s, t)

y(s, t)

allows us to calculate dx ^ dy in terms of ds ^ dt.

This gives:

dx ^ dy =

ds +

−

ds ^ dt

g_(!) = P _ g

−

ds ^ dt

Remark 5.5.8. We again use composition with g on the function part, and

with its derivative on the differential part.

92 CHAPTER 5. GREEN’S THEOREM

Proposition 5.5.1. If g : I2 −! U _ R2 is a 1 − 1 smooth embedding a.e,

and if ! is a differential 2-form on U

g_! =

g(I2)

“Proof” the result is to automatically give the usual change of variable

formula. _

Remark 5.5.9. This generalises to the case where

g : Im −! U _ Rn

is a smooth embedding a.e. Then g_ takes k-forms on U to k-forms on Im

by (1) composition with g to get the function part and (2) composition with

Dg to get the dxi turned into dtj and for ! any k-form

g_! =

g(Ik)

I am not going to prove the claim in general, it is basically the change of

variable formula. Note that there is no need to take special account of the

sign or to take absolute values of numbers, since this is taken care of by

the dx ^ dy terms. It is a good idea to get the thing into standard shape

before actually integrating however, or you can get the sign wrong. Fubini’s

theorem needs some changes before it works for integrating 2-forms.

Proposition 5.5.2. If ! is a smooth 0-form on R2 and

c : I −! R2

is an embedding then:

d(c_(!) = c_(d!)

Proof: A 0-form is just a function and I shall call it f to make it more

friendly sounding for those of you made nervous by greek letters. If we write

c : I −! R2

x(t)

y(t)

then since f, being a 0-form, has no differentials to bother about, c_(f) = f _c

and

d(f _ c) =

d(f _ c)

5.5. GREEN’S THEOREM AGAIN 93

is the derived 1-form.

This is, using the chain rule:

It might be useful to remember where we are and rewrite this as:

c(t)

Now over U,

df =

dx +

Now applying c_ to this we evaluate the function part at c(t) and fix up the

differentials using the derivative of c, which gives us the line preceding. _

Remark 5.5.10. We can go up a dimension and do this for maps which

embed squares in R2. The argument is almost the same

Proposition 5.5.3. If ! is a differential 0-form on U _ R2 and c : I2 −! U

is a smooth embedding,

d(c_(!)) = c_(d!)

Proof A 0-form is just a function, call it f

d(c_f) = D(f _ c) ( definition of c_)

= Df _ Dc (chain rule)

= c_df ( definition of c_)

Remark 5.5.11. The notation used here is very condensed and it is probably

a good idea to write it out in old fashioned terms so I give the proof again:

Proposition 5.5.4. Repeat: If f : U −! R is a function defined on some

set U _ R2 and if c : I2 −! U is a smooth embedding of the unit square in

U, then

d(c_(f)) = c_(df)

Proof Since f is a 0-form there are no differentials to bother about, and

c_(f) = f _ c which is another 0-form, this time on I2.

94 CHAPTER 5. GREEN’S THEOREM

The exterior derivative applied to 0-forms is just the ordinary derivative and

for f _ c is, if we write:

c : I2 −! U _

just

@(f _ c)

ds +

@(f _ c)

which we shall write out explicitly as

ds +

Using the chain rule. Now

df =

dx +

and applying c_ to this gives us the preceding line. _

Remark 5.5.12. This works also for differential 1-forms:

Proposition 5.5.5. If ! is a differential 1-form on U _ R2 and c : I2 −! U

is a smooth embedding then

d(c_!) = c_d!

Proof If

! , Pdx + Qdy

d! =

−

dx ^ dy.

define

_c : I2 −! U

then

Dc =

5.5. GREEN’S THEOREM AGAIN 95

and _

# _

c_! = P

__ _

ds +

| {z }

__ _

ds +

| {z }

P _ c

+ Q _ c

ds +

P _ c

__ _

+ Q _ c

d(c_!) =

(P _ c)

+ (Q _ c)

−

(P _ c)

+ (Q _ c)

ds ^ dt

(P _ c)

@2x

@s@t

@ (P _ c)

+ (Q _ c)

@2y

@s@t

@ (Q _ c)

− (P _ c)

@2x

@s@t

−

@ (P _ c)

− (Q _ c)

@2y

@t@s

−

@ (Q _ c)

ds ^ dt

Notice that of these eight terms, the first and fifth cancel and the third and

seventh cancel.

Using _

@ (P _ c)

_ "

(chain rule) and likewise for Q _ c,

d(c_!) =

@s + @P

− @x

@t + @P

+@y

@s + @Q

− @y

@t + @Q

5ds ^ dt

_ @Q

_@x

@t − @y

− @P

_@x

@t − @x

+@Q

_@y

@s − @y

+ @P

_@x

@s − @x

ds ^ dt

The last two terms are zero so this reduces to:

dc_(!) =

−

_ _

−

ds ^ dt

= c_d!

96 CHAPTER 5. GREEN’S THEOREM

Remark 5.5.13. It works just as well on Rn but there are more terms. It

also works for differential k-forms for any k < n on U _ Rn. As it stands

it is a rather tedious but straightforward calculation: the sort of thing that

makes you feel like a real mathematician at relatively low cost. You probably

get the general idea by now.

Remark 5.5.14. after that moderately painful part the rest is easy:

Definition 5.15. boundary operator If U _ Rn is any set, a boundary

point of U is a point such that every open ball on it intersects both U and

the set complement of U, Rn \ U. The set of all boundary points of U is

written @U and @ is called the boundary operator.

Remark 5.5.15. Now I pull the rabbit out of the hat:

Proposition 5.5.6. Green’s Theorem Let ! be a differential 1−form on

U _ R2 (U open) and let D _ U be any region which is parametrised by a

smooth embedding a.e. c : I2 −! U.

Then Z

! =

d!.

Proof

! =

@(c(I2))

! (definition of D)

_I2

c_! (by the change of variables

formula and adding four curves.)

dc_! (by Green’s Theorem for a square)

c_d! (by Proposition 5.5.5)

c(I2)

d! (by the change of variables formula)

d! (definition of D)

Remark 5.5.16. Now I bow deeply and you clap and throw money (notes

only).

5.5. GREEN’S THEOREM AGAIN 97

Remark 5.5.17. This gives Greens Theorem for quite a lot of shapes in R2.

We can actually note that c does not have to be smooth everywhere: if c

is continuous and invertible and is smooth except at a finite set of points,

with inverse smooth except at a finite set of points, this will not change any

integrals.

So Green’s Theorem also works on D2, by an exercise I gave a while back.

Remark 5.5.18. The results given can be strengthened considerably. But

the present form serves our purposes.

Remark 5.5.19. We can almost prove the result that was stated to be too

hard at the end of the last chapter. I state it again but in modern language:

Proposition 5.5.7. If ! is a smooth 1-form on U _ R2 which is closed, and

if U is connected and simply connected, then ! is exact.

(or in translation into old-fashioned language, if F = Pi + Qj is a vector

field on R2 and @Q/@x−@P/@y = 0 and U is connected and has no holes in

it, then F is conservative.)

Almost Proof

Take any continuous simple (1-1) loop in U; then this can be expressed as

a map from @I2 to U. Since U is simply connected we can extend this to a

continuous 1-1 map ˜ f from I2 to U.

If this were smooth almost everywhere we could apply Green’s Theorem to

the interior and since d! = 0 we can conclude that the integral around the

loop must be zero. This would be enough to conclude that every path integral

depends only on its endpoints, which would give us the required result.

Unfortunately we have no guarantee that ˜ f is smooth. To get around this

we could rather laboriously prove that every continuous map can be approximated

by a smooth map, and argue that the line integrals along the

non-smooth arcs are approximated by the line integrals around the smooth

approximation, and likewise for the surface integrals. This can be done, but

it is a lot of work and we don’t have time for it. Too bad. We conclude

therefore that the result looks plausible, but is a hard one to prove. _

98 CHAPTER 5. GREEN’S THEOREM

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