ON LINE AND DOUBLE MULTIPLICATIVE INTEGRALS

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1 TWMS J. App. Eng. Mth. V.3 N pp ON LINE AN OUBLE MULTIPLIATIVE INTEGRALS A. BASHIROV 1 Abstrct. In the present pper the concepts of line nd double integrls re modified to the multiplictive cse. Two versions of the fundmentl theorem of clculus for line nd double integrls re proved in the multiplictive cse. Keywords: Multiplictive clculus multiplictive line integrl independence on pths Green s theorem. AMS Subject lssifiction: 26B12 97I40 97I Introduction In 1972 Grossmn nd Ktz [8] pointed out to different clculi clled non-newtonin clculi which modify the clculus creted by Isc Newton nd Gottfried Wilhelm Leibnitz in the 17th century. Since then number of works hs been done in this re. In Stnley [14] Bshirov et l. [2 3 4] nd Riz et l. [12] most populr non-newtonen clculus nmely multiplictive clculus is hndled. Some elements of stochstic multiplictive clculus re concerned in the works of Krndikr [10] nd letskii nd Teterin [7]. Bshirov nd Riz [5] studied complex multiplictive clculus. Another populr non- Newtonin clculus nmely bigeometric clculus is investigted in Volterr nd Hostinsky [15] Grossmn [9] Aniszewsk [1] Ksprzk et l. [11] Rybczuk et l. [13] órdov- Lepe [6]. In this pper we study functions of two vribles from multiplictive point of view. Our im is presenttion of two fundmentl theorems of multiplictive clculus for line nd double integrls. These theorems re useful deling with different pplictions of multiplictive clculus for exmple in studies on multiplictive complex clculus. One mjor nottion is tht the multiplictive versions of the concepts of Newtonin clculus will be clled concepts. For exmple derivtive is sme s multiplictive derivtive integrl s multiplictive integrl etc. As lwys for derivtive of the function f we use the symbol f distinguishing it from the ordinry derivtive f. But integrl of the function f is denoted by b f(x) dx which reflects the wy of its definition nd differs from the symbol of ordinry integrl. The sme kind of symbols will be used for multiplictive prtil derivtives line nd double integrls. 1 eprtment of Mthemtics Estern Mediterrnen University Gzimgus - North yprus e-mil: gmirz.bshirov@emu.edu.tr Mnuscript received Jnury TWMS Journl of Applied nd Engineering Mthemtics Vol.3 No.1 c Işık University eprtment of Mthemtics 2013; ll rights reserved. 103

2 104 TWMS J. APP. ENG. MATH. V.3 N Line Multiplictive Integrls Let f be positive function of two vribles defined on n open connected set in R 2 nd let be piecewise smooth curve in the domin of f. Tke prtition P = {P 0... P m } on nd let (ξ k η k ) be point on between P k 1 nd P k. enote by s k the rclength of from the point P k 1 to P k. According to the definition of *integrl from Bshirov et l. [3] define the integrl product P (f P) = f(ξ k η k ) s k. The limit of this product when mx{ s 1... s m } 0 independently on selection of the points (ξ k η k ) will be clled line *integrl of f in ds long for which we will use the symbol f(x y) ds. From f(ξ k η k ) s k m ln f(ξ kη k ) s k it is clerly seen tht the line *integrl of f long exist if f is positive function nd the line integrl of ln f long exists nd they re relted s f(x y) ds ln f(xy) ds. The following properties of line *integrls in ds cn be proved esily: ( () (f(x y) p ) ds = f(x y) ds) p p R (b) (f(x y)g(x y)) ds = f(x y) ds g(x y) ds (c) (f(x y)/g(x y)) ds = f(x y) ds/ g(x y) ds (d) f(x y) ds = f(x y) ds f(x y) ds = where = mens tht the curve is divided into two pieces t some interior point of [ b]. In similr wy we cn introduce the line *integrls in dx nd in dy nd estblish their reltion to the respective line integrls in the form f(x y) dx ln f(xy) dx nd f(x y) dy ln f(xy) dy. (1) lerly ll three kinds of line *integrls exist if f is positive continuous function. The bove mentioned properties of the line *integrls in ds re vlid for line *integrls in dx nd in dy s well. Additionlly ( f(x y) dx = f(x y) dx) 1 nd ( f(x y) dy = f(x y) dy) 1

3 while A. BASHIROV: ON LINE AN OUBLE MULTIPLIATIVE INTEGRALS 105 f(x y) ds = f(x y) ds where is the curve with the opposite orienttion. Moreover the following evlution formule for the line *integrls re lso esily seen: b ( () f(x y) ds = f(x(t) y(t)) x 2 +y 2) dt (b) (c) f(x y) dx = f(x y) dy = b b ( f(x(t) y(t)) x (t) ) dt ( f(x(t) y(t)) y (t) ) dt where {(x(t) y(t)) : t b} is suitble prmetriztion of nd b g(t)dt is the *integrl of g on the intervl [ b] (see Bshirov et l. [3]). It is lso suitble to denote f(x y) dx g(x y) dy = f(x y) dx g(x y) dy. In cses when is closed curve we write insted of. Exmple 2.1. Let c 0 nd let = {(x(t) y(t)) : t b} be piecewise smooth curve. Then c dx ln c dx (x(b) x()) ln c = c x(b) x(). In order to stte nd prove fundmentl theorem of clculus for line integrls we need in prtil derivtives. Recll tht derivtive of the function f of one vrible which is ssumed to be positive nd differentible in the ordinry sense is defined s ( ) 1 f(x + x) f x (x) = lim x 0 f(x) nd it hs the following reltion to the ofdinry derivtive: f (x) (ln f(x)) d dx ln f(x). Bsed on this it is nturl to define the prtil derivtives of the function f of two vribles s fx(x y) ln f(xy) x nd fy (x y) y ln f(xy). Theorem 2.1 (Fundmentl theorem of clculus for line *integrls). Let R 2 be n open connected set nd let = {(x(t) y(t)) : t b} be piecewise smooth curve in. Assume tht f is continuously differentible positive function on. Then fx(x y) dx fy (x y) dy = f(x(b) y(b))/f(x() y()). Proof. From the fundmentl theorem of clculus for line integrls fx(x y) dx fy (x y) dy This proves the theorem. (ln f x (xy) dx+ln f y (xy) dy) ([ln f] x(xy) dx+[ln f] y(xy) dy) ln f(x(b)y(b)) ln f(x()y()) = f(x(b) y(b))/f(x() y()).

4 106 TWMS J. APP. ENG. MATH. V.3 N Green s Theorem in Multiplictive Form As fr s line *integrls re concerned we cn present nother fundmentl theorem of *clculus relted to line *integrls tht is the Green s theorem in *form. Let f be bounded positive function f defined on the Jordn set R 2. Let Q = { k : k = 1... m} be prtition of. Tke ny (ξ k η k ) k nd let A k be the re of k. efine the integrl product P (f Q) = f(x(t) y(t)) A k. enote by d k the dimeter of k tht is d k = sup (x 1 x 2 ) 2 + (y 1 y 2 ) 2 where the supremum is tken over ll (x 1 y 1 ) (x 2 y 2 ) k. The limit of the integrl product when mx{d 1... d m } 0 independently on selection of the points (ξ k η k ) will be clled double *integrl of f on for which we will use the symbol f(x y) da. A reltion between double integrls nd double *integrls cn be esily derived s f(x y) da f(xy)da. The following properties of double *integrls cn lso be proved esily: ( () (f(x y) p ) da = f(x y) da) p p R (b) (f(x y)g(x y)) da = f(x y) da g(x y) da (c) (f(x y)/g(x y)) da = f(x y) da/ g(x y) da (d) f(x y) da = f(x y) da f(x y) da = where = mens tht 1 nd 2 re two non-overlpping Jordn sets with 1 2 =. Theorem 3.1 (Green s theorem in *form). Let f nd g be continuously differentible positive functions on simply connected Jordn set R 2 with the piecewise smooth nd positively oriented boundry. Then f(x y) dx g(x y) dy = (gx(x y)/fy (x y)) da. Proof. From the Green s theorem f(x y) dx g(x y) dy (ln f(xy) dx+ln g(xy) dy) This proves the theorem. = ([ln g] x(xy) [ln f] y(xy)) da (ln g x (xy) ln f y (xy)) da (g x(x y)/f y (x y)) da.

5 A. BASHIROV: ON LINE AN OUBLE MULTIPLIATIVE INTEGRALS onclusion Multiplictive clculus is n lterntive to Newtonin clculus. The growth relted problems in economics finnce cturil science etc. re better described nd solved in terms of multiplictive clculus rther thn Newtonin clculus. Since its estblishment s one of the non-newtonin clculi number of works re devoted to different pplictions of multiplictive clculus. In this pper the concepts of prtil derivtive line nd double integrls re considered from multiplictive point of view. Two fundmentl theorems of clculus for line nd double integrls re interpreted in terms of multiplictive clculus. References [1] Aniszewsk. (2007) Multiplictive Runge-Kutt method Nonliner ynmics [2] Bshirov A. E. nd Bshirov G. (2011) ynmics of literry texts Online Journl of ommuniction nd Medi Techniligies 1 (3) [3] Bshirov A. E. Kurpınr E. nd Özypıcı A. (2008) Multiplictive clculus nd its pplictions Journl of Mthemticl Anlysis nd Applictions 337 (1) [4] Bshirov A. E. Kurpınr E. Tndoǧdu Y. nd Özypıcı A. (2011) On modeling with multiplictive differentil equtions Applied Mthemtics - A Journl of hinees Universities 26(4) [5] Bshirov A. E. nd Riz M. (2011) On complex multiplictive differentition TWMS Journl of Applied Engineering Mthemtics 1(1) [6] órdov-lepe F. (2006) The multiplictive derivtive s mesure of elsticity in economics TEMAT-Theeteto Atheniensi Mthemtic 2(3) online. [7] letskii Yu. L. nd Teterin N. I. (1972) Multiplictive stochstic integrls Uspekhi Mtemticheskikh Nuk 27(2:164) [8] Grossmn M. nd Ktz R. (1972) Non-Newtonin lculus Lee Press Pigeon ove MA. [9] Grossmn M. (1983) Bigeometric lculus: A System with Scle-Free erivtive Archimedes Foundtion Rockport MA. [10] Krndikr R. L. (1982) Multiplictive decomposition of non-singulr mtrix vlued continuous semimrtingles The Annls of Probbility 10(4) [11] Ksprzk W. Lysik B. nd Rybczuk M. (2004) imensions Invrints Models nd Frctls Ukrinin Society on Frcture Mechnics SPOLOM Wroclw-Lviv Polnd. [12] Riz M. Özypıcı A. nd Kurpınr E. (2009) Multiplictive finite difference methods Qurterly of Applied Mthemtics 67 (4) [13] Rybczuk M. Kedzi A. nd Zielinski W. (2001) The concepts of physicl nd frctionl dimensions II. The differentil clculus in dimensionl spces hos Solutions Frctls [14] Stnley. (1999) A multiplictive clculus Primus IX(4) [15] Volterr V. nd Hostinsky B. (1938) Opertions Infinitesimles Lineres Hermn Pris. Agmirz Bsirov for photogrph nd biogrphy see TWMS Journl of Applied nd Engineering Mthemtics Volume 1 No