CHAPTER 1
Reduction o f Operators of Firs t Order to Canonical Form s
1. Firs t order equations. Le t u s begin wit h th e simples t example- a linea r firs t orde r
partial differentia l operato r wit h rea l coefficient s actin g o n rea l valued function s w(x) o f
X = (x
1,
•• , *") i n a n ope n se t O i n R
n
a s a directional derivative :
(1.1) Pw=fV(x)(aw/dxO .
i
At every poin t x i n H a smoothly varyin g (C°° ) vecto r a{x) = (A
1,
*• ,
an)
i s given,
and th e actio n o f P o n th e functio n w i s to differentiat e w i n the directio n o f a, th e
operator P i s also called a vector field . On e solve s the equatio n Pw - f b y finding "in -
tegral curves " of th e vecto r field, i.e . curve s wit h th e propert y tha t a t ever y poin t o n each
curve th e vecto r a a t tha t poin t i s tangent t o th e curve . I f suc h a curve i s given by
x =
x(f)
with t a s a real paramete r the n x{t) satisfie s th e syste m o f ordinar y differentia l equations ,
the "characteristi c equations'* ,
(1.2) dx!/dt=a1(x(t)), j= 1, , « .
From th e theor y o f ordinar y differentia l equation s i t follow s tha t throug h ever y poin t x
0
in n ther e passe s exactly on e integra l curve . I f w e consider th e restrictio n o f a functio n
vv(x) t o suc h a curve, on whic h i t become s a function o f f , w e find (using summatio n
convention)
dw _ dw dx f __
dt ~ " dx ^ It
Thus on th e curv e w satisfie s a simple ordinar y differentia l equatio n an d th e valu e of w
is then uniquel y determine d (a t leas t locally ) b y give n initia l value s on a hypersurface ( a sur -
face of dimensio n n ~- 1) which i s transversal t o th e vecto r field .
Locally on e ca n introduc e ne w independen t variable s y = (yx, •• ,'" ) whic h
"straighten out " th e integra l curve s an d reduc e th e differentia l operato r t o a particularl y
simple form ,
(1.3)
P=^VT^
**0-
The leadin g symbo l a J\. i s thus transforme d t o \r)
n
. An y solutio n o f Pw = 0 i s then
simply a n arbitrar y functio n o f (y l •• , vn _ 1 ) . W e see therefor e tha t th e loca l stud y o f
3
http://dx.doi.org/10.1090/cbms/017/02
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