Multivector Functions

Transcription

1 I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed i tems of multivecto algeba. The theoy is used hee to deive some poweful theoems which geealize well-kow theoems of potetial theoy ad the theoy of fuctios of a complex vaiable. Aalytic multivecto fuctios o E ae defied ad show to be appopiate geealizatios of aalytic fuctios of a complex vaiable. Some of thei basic popeties ae poited out. These esults have impotat applicatios to physics which will be discussed i detail elsewhee.. INTEGAL OF THE GADIENT OF A FUNCTION A multivecto fuctio f defied o a egio i E is said to be diffeetiable o if its gadiet f(x) exists i some sese at each poit x i. Iff ad g ae diffeetiable o, the gdv f +( ) + (g ) dvf = gdaf. (.) This fomula still holds if eithe f o g is a geealized fuctio [2] (distibutio), a fact which ofte simplifies itegatio. Hee it is used to itegate f. Let epeset the gadiet opeatig at the poit x ad the gadiet opeatig at the poit x.if = x x,the = = (.2) It ca be eadily veified that the equatio admits the paticula solutio = =. (.3) g() = δ() (.4) g() = Ω, (.5) whee Ω = 2π 2 Γ ( ) (.6) 2 is the aea of a uit sphee i E.Moeove,g is the gadiet of the scala fuctio G() = 2 (2 )Ω if 2 (.7a) = l 2π if =2. (.7b)

2 Thus g = G = G. (.8) The fuctio G is a paticula solutio of Laplace s equatio fo a poit souce i E. 2 G() =δ(). (.9) Now substitute the fuctio g just defied ito (.). By vitue of (.4), if x is i, the (g ) dv f = δ(x x ) if(x) =if(x ), (.0) whee the symbol i deotes the uitay taget (uit volume elemet) of. Sicei i =, (.) becomes { f(x )= ( ) dv f Ω i da f. (.) This shows that a multivecto fuctio diffeetiable i a egio of E is uiquely detemied by its gadiet i ad its value o the bouday of. By usig the fact that i =( ) i ad defiig the omal to the bouday of the egio by the equatio i da =, (.2) Eq. (.) ca be witte f(x )= { Ω f + f. (.3) Equatio (.3) implies that evey diffeetiable multivecto fuctio is the gadiet of aothe fuctio. Fo, by vitue of (.8), f(x )= ϕ(x ), (.4) whee, if 2, ϕ(x )= = { ( 2) Ω { ( 2) Ω f 2 ϕ f + C ϕ + C (.5) ad C(x )=0. (.6) The followig choice of C elimiates the tagetial deivative of ϕ i the secod tem of the ight of (.5): ( C(x )= ) ϕ. (.7) 2

3 So (.5) becomes ϕ(x )= { ( 2)Ω 2 ϕ [( + ) ϕ ] ϕ, (.8) a familia fomula fom potetial theoy, although, of couse, ϕ is a multivecto field hee. 2. ELATION TO COMPLEX AIABLE THEOY A vecto fuctio o E 2 is equivalet to a complex fuctio of a complex vaiable. Let f be such a fuctio. Takig g =i, Eq. (.) ca be witte dv f = dx f. (2.) So, if i, the f =0 (2.2) dx f(x) =0. (2.3) A vecto fuctio satisfyig (2.2) ad (2.3) is equivalet to a aalytic fuctio of a complex vaiable. Equatio (2.2) coespods to the Cauchy-iema equatios, ad (2.3) coespods to Cauchy s theoem. If f is aalytic except at poles x k i, the f(x) =2π k k δ(x x k ). (2.4) The esidue theoem is, obtaied by substitutig this ito (2.): dx f =2πi k, (2.5) k whee, of couse, k is the esidue of f at x k. Fo the case = 2, Equatio (.) ca be witte f(x )= 2π x x f(x) dx f(x). (2.6) 2πi x x Fo the special case f = 0, (2.4) educes to Cauchy s itegal fomula. It should be clea by ow that all of complex vaiable theoy ca be eadily fomulated i the laguage of multivecto calculus. 3

4 3. ANALYTIC MULTIECTO FUNCTIONS Whe fomulated i tems of multivecto calculus as i the last sectio, the otio of aalytic fuctio i complex vaiable theoy admits to a obvious geealizatio. Moeove, may theoems ad eve poofs ae essetially ualteed i the pocess. A multivecto fuctio f is hee called aalytic i a egio of E if fo x i. theoem: f(x) =0 (3.) By the fudametal theoem, this implies the geealizatio of Cauchy s da f =0 (3.2) o, sice i da = whee is the outwad omal, f =0. (3.3) If f is aalytic i except at poles x k,the f =Ω k δ(x x k ). (3.4) k This becomes the esidue theoem whe witte i itegal fom: f =Ω k. (3.5) Equatio (.3) yields the geealizatio of Cauchy s itegal fomula: f(x )= Ω k f. (3.6) This fomula eveals the fudametal popety of aalytic fuctios: If f is aalytic i, its value at evey poit i is uiquely detemied byitsvalueso. Fom (3.6) it follows that f(x ) ca be expaded i a powe seies. But this is ot the cucial popety of aalytic fuctios; may oaalytic fuctios ca also be expaded i a powe seies. No is the existece of the complex deivative d/dz a cucial popety of aalytic fuctios i E 2. The existece of d/dz is a expessio of the fact that the deivative with espect to a vecto is idepedet of its diectio; it follows diectly fom f = 0 ad the limitatio to two dimesios. This featue is peculia to two dimesios ad caot be geealized. But the essetial popeties of aalytic fuctios do ot deped o the existece of d/dz; they ae cotaied i Cauchy s itegal fomula, which geealizes quite icely to (3.6). Theefoe the use of d/dz should be eschewed. Fom (3.6) may theoems follow which ae staightfowad geealizatios of theoems i complex vaiable theoy: Liouville s theoem, the mea value theoem, the maximum 4

5 modulus piciple, etc. The poofs ae so simila to well-kow poofs based o Cauchy s itegal fomula that they eed ot be give hee. Othe popeties of aalytic fuctios ca be deived with the help of the fudametal theoem of calculus. Fo example, if f is aalytic i a egio of E,adif is a smooth oieted suface i with taget v, the dv f = dv f, (3.7) whee f = iv f is the omal deivative of f o. Usig the fact that = v + iv, the theoem follows immediately fom the idetity dv f = dv v f + dv f = dv f + dv f. A obvious coollay of (3.7): the itegal of the omal deivative of a aalytic fuctio ove ay closed suface vaishes. The otio of aalytic fuctio ca be pofitably geealized beyod what has bee discussed so fa. Let be a oieted suface with taget v. A fuctio f is said to be aalytic o if v f = O o. With this defiitio the theoy of aalytic fuctios ca be developed o cuved sufaces i much the same way as it is developed o flat sufaces i this pape. The theoy of aalytic fuctios of seveal complex vaiables ca be icopoated ito the theoy of multivecto fuctios i the followig way. Coside E 2 as a Catesia poduct of plaes, ad let i k be the taget 2-vecto to the kth plae. Note that i k is a uit imagiay fo the kth plae: i 2 k =. (3.8) The tagets commute with oe aothe: The pseudoscala i of E 2 ca be factoed ito the poduct The coodiate x of a poit i E 2 ca be expessed as the sum i j i k = i k i j. (3.9) i = i i 2 i. (3.0) x = x + x x, (3.) whee, fo k =, 2,...,, x k =(i k ) i k x = i k i k x. (3.2) 5

6 A fuctio f(x) oe 2 ca be egaded as a fuctio f(x,x 2,...,x )ofthe vaiables defied by (3.). If f is aalytic i a egio k of the kth plae, the ik f =0 (3.3) whe x k is i k. Let = 2 be the Catesia poduct of the egios k.ifx = x + x x is i, theby applicatios of (2.6), f(x )= i (2π) x x dx 2 x 2 x dx 2 2 x x dx f(x). (3.4) This is equivalet to Cauchy s itegal fomula fo a aalytic fuctio of complex vaiables. The ode of itegatios i (3.4) is immateial because of (3.9). Equatio (3.4) should be compaed with (3.6), which fo the peset case ca be witte f(x )= ( )! 2π This fomula applies because, by (3.), (x x ) x x f(x). (3.5) 2 i f(x )+ i2 f(x )+ + i f(x )= f(x )=0. (3.6) EFEENCES [] D. Hestees. J. Math. Aal. ad Appl. 24 (968), [2] I. M. Gel fad ad G. E. Shilov. Geealized Fuctios, ol.. Academic Pess, New Yok,