divergence operator and its inverse

8/9/2019 Divergence Operator and Its Inverse

1/76

Inverting the Divergence Operator

Mark HickmanDepartment of Mathematics & Statistics

University of [email protected]

53th Annual Meeting of the Australian Mathematical Society

Adelaide28 September - 1 October 2009

Divergence Operator

http://find/http://goback/


2/76

INTRODUCTION

Algebraic computing packages such as MAPLE andMATHEMATICA are adept at computing the integral of an

explicit expression in closed form (where possible). Neitherprogram has any trouble in, for example,

1

x log x dx = log (log x).

Divergence Operator

http://find/


3/76

INTRODUCTION

Algebraic computing packages such as MAPLE andMATHEMATICA are adept at computing the integral of an

explicit expression in closed form (where possible). Neitherprogram has any trouble in, for example,

1

x log x dx = log (log x).

MAPLE from release 9 onwards has a limited facility to handleexpressions such as

u vx − v u x

( u − v)2 dx =

v

u − v

(where u and v are understood to be functions of x).

Divergence Operator

http://find/


4/76

INTRODUCTION

However neither program can compute the “antiderivative” of exact expressions in more than one independent variable. For

example there are no inbuilt commands that would compute

u x vy − u xx vy− u y vx + u xy vx

= ∂

∂x

[ u vy − u x vy ] + ∂

∂y

[ u x vx − u vx ] .

Divergence Operator

http://find/


5/76

INTRODUCTION




= ∂

∂x

[ u vy − u x vy ] + ∂

∂y

[ u x vx − u vx ] .

In this last example, given a so-called differential function f,we wish to compute a vector field F such that

f = Div F.

Divergence Operator

http://find/


6/76

INTRODUCTION




= ∂

∂x

[ u vy − u x vy ] + ∂

∂y

[ u x vx − u vx ] .

In this last example, given a so-called differential function f,we wish to compute a vector field F such that

f = Div F.

Of course, such a vector field F will not exist for an arbitraryf. The existence (or non existence) and the computation of Foccurs in many situations.

Divergence Operator

http://find/


7/76

INTRODUCTION

The existence of F is resolved by the Euler operator.

Divergence Operator

http://find/


8/76

INTRODUCTION


The computation of F is resolved (in a theoretical sense, atleast) by the homotopy operator.

Divergence Operator



9/76

INTRODUCTION


The computation of F is resolved (in a theoretical sense, atleast) by the homotopy operator.

However, as will be demonstrated in this paper, the practicalimplementation of the homotopy operator to compute Finvolves a number of subtleties not readily apparent from its

definition.

Divergence Operator

N



10/76

NOTATION

The natural arena for our discussion is the jet bundle. Theindependent variables will be denoted generically by

x = (x1, x2, x3, . . . , xd).

Divergence Operator

N



11/76

NOTATION

The natural arena for our discussion is the jet bundle. Theindependent variables will be denoted generically by

x = (x1, x2, x3, . . . , xd).

For a unordered multi-indices (with non-negative components)I = (i1, i2, . . . , id) and J = ( j1, j2, . . . , jd) define

|I| = i1

+ i2

+ · · · + id

I! = i1! i2! · · · id!

xI = xi11 xi22 · · · x

idd

∂If

∂xI

= ∂|I|f

∂xi11 ∂x

i22 · · · ∂x

idd

I + J = (i1 + j1, i2 + j2, . . . , id + jd)I

J

=

i1

j1

i2

j2

· · ·

id

jd

=

I!

(I − J)! J!.

Divergence Operator

NO O

http://find/


12/76

NOTATION

We introduce a partial ordering on multi-indices. I J if I − Jhas no negative entries.

Divergence Operator

NOTATION

http://find/


13/76

NOTATION


In order to reduce the number of subscripts, we genericallydenote dependent variables by u and use the convention that

u

indicates summation over all dependent variables

Divergence Operator

NOTATION

http://find/


14/76

NOTATION


In order to reduce the number of subscripts, we genericallydenote dependent variables by u and use the convention that

u

indicates summation over all dependent variables

For each dependent variable u , let u I be the jet variableassociated with

∂I u

∂xI .

Divergence Operator

NOTATION

http://find/


15/76

NOTATION

The total derivative with respect to xi is given by

Di = ∂∂xi+

I, u

u I+ei ∂∂u I

where ei is the multi-index with 1 in the ith position, 0

elsewhere.

Divergence Operator

NOTATION

http://find/


16/76

NOTATION


Di = ∂∂xi+

I, u

u I+ei ∂∂u I


elsewhere.

The summation is over all non-negative multi-indices I and alldependent variables u . However there will only be a finitenumber of non-zero terms in this summation.

Divergence Operator

NOTATION

http://find/


17/76

NOTATION


Di = ∂∂xi+

I, u

u I+ei ∂∂u I


elsewhere.


In the one variable case, we will drop the subscript on D.

Divergence Operator

NOTATION

http://find/


18/76

NOTATION


Di = ∂∂xi+

I, u

u I+ei ∂∂u I


elsewhere.


In the one variable case, we will drop the subscript on D.

The divergence of a differential vector field F is given by

Div F =d

i=1

Di Fi.

Divergence Operator

NOTATION

http://find/


19/76

NOTATION

Finally, letDI = Di11 D

i22 · · · D

idd

where superscripts indicate composition.

Divergence Operator

EXISTENCE – THE EULER OPERATOR

http://find/


20/76

EXISTENCE THE EULER OPERATOR

DEFINITION

A scalar differential function f is exact or a divergence if and only if there exists a differential function F such that

f = Div F =d

i=1

Di Fi.

Divergence Operator


http://find/


21/76


DEFINITION

A scalar differential function f is exact or a divergence if and only if there exists a differential function F such that

f = Div F =d

i=1

Di Fi.

THEOREM (Olver, 1993, p. 248)

A necessary and sufficient condition for a function f to be exact is that

Eu f ≡I

(−1)I DI ∂f

∂u I = 0 (1 )

for each dependent variable u . Eu is called the Euler operator (or variational derivative) associated with the dependent variable u .

Divergence Operator


http://find/


22/76


EXAMPLE

Let f = u x vy − u xx vy − u y vx + u xy vx

then

Divergence Operator


http://find/


23/76


EXAMPLE

Let f = u x vy − u xx vy − u y vx + u xy vx

then

Eu f = − Dx∂f

∂u x+ D2x

∂f

∂u xx− Dy

∂f

∂u y+ Dx Dy

∂f

∂u xy

= − Dx vy − D2x vy + Dy vx + Dx Dy vx

= − vxy − vxxy + vxy + vxxy

= 0

Divergence Operator


http://find/


24/76

S

EXAMPLE

and

Ev f = − Dx∂f

∂vx− Dy

∂f

∂vy

= − Dx ( u xy − u y) − Dy ( u x − u xx)

= 0.

Thus f is exact.

Divergence Operator

COMPUTING THE INVERSE

http://find/


25/76

The question we wish to address is that given a differentialfunction f that is exact, compute F such that

f = Div F.

Of course F will not be unique.

Divergence Operator


http://find/


26/76

The question we wish to address is that given a differentialfunction f that is exact, compute F such that

f = Div F.

Of course F will not be unique.

DEFINITIONThe higher Euler operators are given by

EJu =IJ

(−1)I−J

I

J

DI−J

∂

∂u I(2 )

for each non-negative multi-index J and each dependent variable u .

Divergence Operator


http://find/


27/76

These operators can be easily implemented in both MAPLE andMATHEMATICA. Their importance lies in the following result.

Divergence Operator


http://find/


28/76


THEOREM

Let f be a differential function. Then

J DJ u EJu f = I u I

∂f

∂u I≡ Mu f (3 )

say.

Divergence Operator


http://find/


29/76


THEOREM

Let f be a differential function. Then

J DJ u EJu f = I u I

∂f

∂u I≡ Mu f (3 )

say.

If f is exact then the left hand side of (3) is a divergence (the

J = 0 term is zero).

Divergence Operator


http://find/


30/76

LEMMA

The operators Mu and Di

commute.

Divergence Operator

http://find/


31/76



32/76

LEMMA

The operators Mu and Di

commute.

For the sake of clarity, let us return briefly to the case of oneindependent variable. If f is exact then (3) reads

D ∞j=0

Dj

u Ej+1u f

= Mu f. (4)

Formally we wish to define

F = M−1u∞j=0

Dj

u Ej+1u f

to obtainf = D F.

Divergence Operator


http://find/


33/76

For this strategy to be successful, we must be able to solvethe equation Mu F = g. This equation is a first order linear

partial differential equation for F.

Divergence Operator


http://find/


34/76

For this strategy to be successful, we must be able to solvethe equation Mu F = g. This equation is a first order linear

partial differential equation for F.

PROPOSITION

Suppose

Mu F = g (5 )for some (given) differential function g. Then

F =

u g ◦ φuλ

dλ + χ (6 )

where

φu : u I → λ u I

u (7 )

and χ ∈ ker Mu.

Divergence Operator


http://find/


35/76

EXAMPLE

Let

g = v u x ( v − u )

( u + v)3 .

Then

F = u g ◦ φu

λ dλ = u v u x ( v − λ)

u (λ + v)3 dλ =

v u x

( u + v)2

withMu F = g

as expected.

Divergence Operator

THE “STANDARD” HOMOTOPY OPERATOR

http://find/


36/76

Note that this approach differs from the standard approach tohomotopy operators.

Divergence Operator


http://find/


37/76


There the homotopy

φ : u I → λ u I

is used and the integral becomes

10

g ◦ φ

λ dλ. (8)

Divergence Operator


http://find/


38/76


There the homotopy

φ : u I → λ u I

is used and the integral becomes

10

g ◦ φ

λ dλ. (8)

However, in many situations this integral is singular.

Divergence Operator


http://find/


39/76

Consider the almost trivial case

Mu F = 1.

Divergence Operator


http://find/


40/76


Mu F = 1.

In the standard approach we find

F = 1

0

1

λ

dλ

which is, of course, singular.

Divergence Operator


http://find/


41/76


Mu F = 1.


F = 1

0

1

λ

dλ


However, with the approach advocated here, we obtain

F = u 1

λ dλ = log u .

Divergence Operator


http://find/


42/76


Mu F = 1.


F =

10

1

λ

dλ


However, with the approach advocated here, we obtain

F = u 1

λ dλ = log u .

This frequent singular nature of the integral (8) is one of thesubtleties mentioned above.

Divergence Operator

THE ONE INDEPENDENT VARIABLE CASE

http://find/


43/76

Returning to the one independent variable case, let

Iu f = ∞j=0

Dj

u Ej+

1u f

(9)

Divergence Operator


http://find/


44/76


Iu f = ∞j=0

Dj

u Ej+

1u f

(9)

and

F = Hu f =

u (Iu f) ◦ φuλ

dλ. (10)

Divergence Operator


http://find/


45/76


Iu f = ∞j=0

Dj

u Ej+

1u f

(9)

and

F = Hu f =

u (Iu f) ◦ φuλ

dλ. (10)

Equation (4) isD Iu f = Mu f.

Divergence Operator


http://find/


46/76


Iu f = ∞j=0

Dj

u Ej+

1u f

(9)

and

F = Hu f =

u (Iu f) ◦ φuλ

dλ. (10)


By (6), we have

Mu F = Iu f

Divergence Operator


http://find/


47/76


Iu f = ∞j=0

Dj

u Ej+

1u f

(9)

and

F = Hu f =

u (Iu f) ◦ φuλ

dλ. (10)


By (6), we have

Mu F = Iu f

and thereforeD Mu F = Mu f.

Divergence Operator


http://find/


48/76

Since Mu and D commute, we obtain

Mu D F

= Mu f.

Divergence Operator


http://find/


49/76


Mu

D F = Mu

f.

However this only implies that

χ = f − D F ∈ kerMu.

Divergence Operator


http://find/


50/76


Mu

D F = Mu

f.



In the work of Hereman and his coworkers, this issue wascircumvented by requiring f to be polynomial with no explicitdependency on the independent variables. In this case thekernel of Mu is trivial.

Divergence Operator


http://find/


51/76


Mu

D F = Mu

f.



In the work of Hereman and his coworkers, this issue wascircumvented by requiring f to be polynomial with no explicitdependency on the independent variables. In this case thekernel of Mu is trivial.

When ker Mu is non-trivial, there are a number of issues to behandled.

Divergence Operator

THE CHOICE OF THE HOMOTOPY φ u

http://find/


52/76

EXAMPLE

Let

f = u xx u x

.

Now Iu f = 1 and so F = Hu f = log u . However

χ =

f −

D F =

uu xx − u 2x

uu x ∈ kerM

u.

We have, if anything, complicated matters.

Divergence Operator


http://find/


53/76

The issue here is that u does not occur explicitly in f. We canremedy this issue by choosing a different homotopy. In this

case, letφux : u I →

λ u I

u x

Divergence Operator


http://find/


54/76

The issue here is that u does not occur explicitly in f. We canremedy this issue by choosing a different homotopy. In this

case, letφux : u I →

λ u I

u x

Now we obtain

F = Hux f ≡ ux (I

uf) ◦ φ

uxλ dλ = log u

x.

Divergence Operator

THE KERNEL OF M u

N d d l h h “ d ”

http://find/


55/76

Next we need to deal with the “remainder” χ .

Divergence Operator

THE KERNEL OF M u

N d d l i h h “ i d ”

http://find/


56/76


Divergence Operator

THE KERNEL OF M u

N t d t d l ith th “ i d ”

http://find/


57/76


COROLLARY

χ ∈ ker Mu if and only if

χ ◦ φu = χ;

that is, χ = χ(ξI)

where ξI =

u I

u

(treating the jet variables that do not depend on u as constants).

Divergence Operator

THE KERNEL OF M u

N t d t d l ith th “ i d ”

http://find/


58/76


COROLLARY

χ ∈ ker Mu if and only if

χ ◦ φu = χ;

that is, χ = χ(ξI)

where ξI =

u I

u

(treating the jet variables that do not depend on u as constants).

Thus χ must be a function of the homogeneous coordinatesξI.

Divergence Operator

THE KERNEL OF M u

In this case we perform a change of coordinates

http://find/


59/76

In this case, we perform a change of coordinates

µ = log u .

Divergence Operator

THE KERNEL OF M u


http://find/


60/76


µ = log u .

Now

ξx = u x

u = µ x

ξxx = u

xx u = µ

xx + µ 2x

ξxxx = u xxx

u = µ xxx + 3µ xx µ x + µ

3x

...

and so χ is a function of derivatives µ I but not µ .

Divergence Operator

THE KERNEL OF M u


http://find/


61/76


µ = log u .

Now

ξx = u x

u = µ x

ξxx = u

xx u = µ

xx + µ 2x

ξxxx = u xxx

u = µ xxx + 3µ xx µ x + µ

3x

...

and so χ is a function of derivatives µ I but not µ .

We now compute the homotopy based on the dependentvariable µ , Hµx χ. since µ does not occur in χ.

Divergence Operator

THE KERNEL OF M u

If a remainder still exist it will be homogeneous in

http://find/


62/76

If a remainder still exist, it will be homogeneous in

µ I

µ x.

Divergence Operator

THE KERNEL OF M u


http://find/


63/76


µ I

µ x.

We repeat this process with the variable log µ x. At each stagewe reduce the number of variables that the remainder dependson. Thus the process will terminate with the remainder, if not

zero, depending only on the independent variable.

Divergence Operator

THE KERNEL OF M u


http://find/


64/76


µ I

µ x.


zero, depending only on the independent variable.Note that, despite the notation, the x subscript on thehomogeneous variable ξ does not indicate differentiation. Forexample

D ξx = ξxx − ξ2

x = ξxx.

Divergence Operator

THE KERNEL OF M u


http://find/


65/76


µ I

µ x.


zero, depending only on the independent variable.Note that, despite the notation, the x subscript on thehomogeneous variable ξ does not indicate differentiation. Forexample

D ξx = ξxx − ξ2

x

= ξxx.

Also note that if f ∈ ker Iu then, by (6), f ∈ kerMu and sowe perform the above change of variables.

Divergence Operator

THE KERNEL OF M u

EXAMPLE

http://find/


66/76

EXAMPLE

Let

f = u u

xx u 2x

.

Note that Iu f = 0. Rewriting f, we have

f = ξxx

ξ2x=

µ xx + µ 2x

µ 2x

and so

Iµ f = 1

µ x, F = Hµx f = −

1

µ x.

Now D F − f = − 1 = − D x and so

D

x −

1

µ x

= D

x −

u

u x

= f.

Divergence Operator

MORE THAN ONE DEPENDENT VARIABLE

Repeat process for each dependent variable until reminder

http://find/


67/76

p p preduces to 0.

EXAMPLELet

f = v( v vx u

2x + 2u v

2x u x − u v u xx vx − u v u x vxx)

u 2x v2x

.

(This example cannot be handled by MAPLE Release 13.) Notethat Iu f = 0. In this case we can either introduce homogeneousvariables or

Iv f = u v2

u x vxand Hv f =

u v2

u x vx.

Furthermore

D

u v2

u x vx

= f.

Divergence Operator

MORE THAN ONE INDEPENDENT VARIABLE

If f is exact then (3) becomes

http://find/


68/76

( )

I∈J

DI u EIu

f = Mu f

with 0 ∈ J.

Divergence Operator



http://find/


69/76

( )

I∈J

DI u EIu

f = Mu f

with 0 ∈ J.

Split the indexing set J

Jk = {I ∈ J : ik > 0 and ik = 0 for k < k }

for each k = 1, 2, . . . , d.

Divergence Operator



http://find/


70/76

I∈J

DI u EIu

f = Mu f

with 0 ∈ J.

Split the indexing set J

Jk = {I ∈ J : ik > 0 and ik = 0 for k < k }

for each k = 1, 2, . . . , d.

Clearly the Jk are disjoint whose union is J (there are many

possible choices for this split).

Divergence Operator


We now define

http://find/


71/76

Iku f = I∈JkDI−ek u EIu f .

Divergence Operator


We now define

http://find/


72/76


Let

Fk = Hku f ≡

u (Iku f) ◦ φuλ

dλ.

Divergence Operator


We now define

http://find/


73/76


Let

Fk = Hku f ≡

u (Iku f) ◦ φuλ

dλ.

As before, we have

dk=1

Dk Fk − f ∈ kerMu.

Divergence Operator


EXAMPLE

http://find/


74/76

Let

f = u 2x vx u y − v u

2x u xy + vx u y u xy − u x u

2y vxy

u 2x u 2y

.

We have

Fx = Hxu f = u u x u y vxy − 2u vx u y u xy − v u 3x + u x vx u 2y

u 3x u y

and

F

y = Hyu f

= u (2vx u xx − u x vxx)

u 3x

withDxF

x + DyFy = f.

Divergence Operator


EXAMPLE

http://find/


75/76

Note that if we use v we obtain

Gx = Hxv f = v u 2x + v u y u xy − u x u y vy

u 2x u y

and

G

y

= Hy

v f =

v u xx

u 2x

withDxG

x + DyGy = f.

Divergence Operator

ALGORITHM

procedure INVDIV(f) Input exact function f

http://find/


76/76

x := INDVAR(f) List of independent variables

seq (

F

k := 0, k ∈ x

) Initialize F

k

χ := f Initialize χfor u ∈ DEPVAR(f) do u a dependent variable

for k ∈ x dog = HOMOTOPY( u , χ, k ) Hku( χ)

Fk

:= Fk

+ g Update Fk

χ := χ − Dk g Update χif χ = 0 then return F F = [seq (Fk, k ∈ x)]end if

end for

end forreturn F, INVDIV(CHANGECOORD( χ)) Use homogeneous

coordinatesend procedure

Divergence Operator
http://find/

divergence operator and its inverse

Documents