inverse gradient operator

arXiv:0804.2239v3 [math-ph] 24 Aug 2010Inverse VectorOperators ShaonSahoo1 Department of Physics, Indian Institute of Science, Bangal ore 560012, India. Abstract In different branches of physics, we frequently deal with ve ctor del operator ( /vector∇). This del operator is generally used to findcurl or divergence of a vect or function or gradient of a scalar function. Inmanyimportantcases,weneedtoknowtheparentvectorwho securlordivergenceisknownorrequire tofindtheparentscalarfunctionwhosegradientisknown. Bu tthetaskisnotveryeasy,especiallyincase of finding vector potential whose curl is known. Here, ‘inver se curl’, ‘inverse divergence’ and ‘inverse gradient’ operators are defined tosolve those problems easi ly. Keywords del operator, curl, gradient, divergence, solenoidal vect or, vector potentials, curvilinear co- ordinate system. 1 Introduction In physical science, usefulness of the concepts of vector an d scalar potentials is indisputable. Theycanbeusedtoreplacemoreconventionalconceptsofmag neticandelectricfields. Infact, in modern era of science, these concepts of potentials becam e so useful and popular that their presence can be seen in virtually all disciplines related so mehow to ‘electromagnetism’! The operator which relates those potentials to the conventiona l fields is the so-called vector ‘del’ operator( /vector∇). Thisoperatorplaysaveryimportantroleinawiderangeof physicsbesidesbeing indispensable in electromagnetism. As for example, it is no w very common to see the opera- torinfluiddynamics,quantummechanics,statisticalmecha nics,classicalmechanics,theoryof relativity, thermodynamics, etc. There are three basic ope rations, namely, curl, divergence and gradient, which can be performed by the operator to relate di fferent quantities of importance. For the ease of discussion, let us divide the functions in two categories, (a) potential functions (thiscanbevectororscalar)and(b)fieldfunctions(againc anbevectororscalar). Abovemen- tioned operations are performed on potential functions to o btain corresponding field functions. While these operations are easy to perform, a back operation to get a potential function from a given field function is not always trivial. Since in some case s, it is convenient to replace one type of functions with others, one should be able to intercha nge these two types of functions comfortably. Unfortunately, a potential function is not un ique for a given field function and there is no general and uniqueprocedure to obtain a potentia l functionfrom a givenfield func- tion. In fact, due to this non-uniqueness, one may wonder whe ther it is at all possible to find any general procedure for this purpose. It is exactly the con cern addressed in this article. Here it is shownthat, it is possibleto find somegeneral procedure s in the form of inversevector op- erators, which can be applied easily to the field functions to obtain the corresponding potential functions. [email protected] 1 All the inverse operators are defined here in the orthogonal c urvilinear co-ordinate system [1]. Letu1,u2,u3bethethreemutuallyperpendicularfamiliesofco-ordinat esurfacesand ˆ e1, ˆe2, ˆe3 as unitvectorsnormalto respectivesurfaces. 2 InverseCurlOperator At firstwe mentionsomesymbols,used later. a.(/vector∇×)−1indicatesinversecurl operator. b.(/integraltext++/integraltext−)(∂u1)does integration w.r.t. u1, while other two variables of the integrand are to be treated fixed. If the integrand is the 3rd component (alo ng ˆe3) of a vector, then/integraltext+(∂u1)acts only on the part of the integrandhaving the variable u3. Similarly/integraltext−(∂u1) acts only on the part of the integrand does not contain the var iableu3. For example, let φ(u1,u2,u3) =Ψ1(u1,u2)+Ψ2(u1,u2,u3)be the component of a vector along ˆ e3, then, (p1/integraltext++p2/integraltext−)φ∂u1=p1/integraltext+Ψ2∂u1+p2/integraltext−Ψ1∂u1, withp1andp2being any two num- bers (u2,u3are tobetreated as constants). Nowtaketwovectors /vectorA=A1ˆe1+A2ˆe2+A3ˆe3and/vectorB=B1ˆe1+B2ˆe2+B3ˆe3, such that, /vectorB=/vector∇×/vectorA (1) Usinginversecurl operator, thisvectorpotential /vectorAcan beexpressed as, /vectorA=(/vector∇×)−1/vectorB (2) From eqn (1)itisobviousthat, /vector∇·/vectorB=/vector∇·(/vector∇×/vectorA)=0 ⇒1 h1h2h3[∂ ∂u1(h2h3B1)+∂ ∂u2(h3h1B2)+∂ ∂u3(h1h2B3)]=0 ⇒∂ ∂u1[c1(u+ 1)+c1(u− 1)]+∂ ∂u2[c2(u+ 2)+c2(u− 2)]+∂ ∂u3[c3(u+ 3)+c3(u− 3)]=0 ⇒∂ ∂u1c1(u+ 1)+∂ ∂u2c2(u+ 2)+∂ ∂u3c3(u+ 3)=0 (3) wherec1(u+ 1)andc1(u− 1)are two parts of the term h2h3B1, containing and not containing u1 respectively,such that, h2h3B1=c1(u+ 1)+c1(u− 1). Otherterms can be defined similarly. From eqn (1)weget, /vectorB=/vector∇×/vectorA=h1ˆe1h2ˆe2h3ˆe3 1 h1h2h3∂ ∂u1∂ ∂u2∂ ∂u3 h1A1h2A2h3A3(4) 2 ⇒B1ˆe1+B2ˆe2+B3ˆe3=ˆe1 h2h3[∂ ∂u2(h3A3)−∂ ∂u3(h2A2)] +ˆe2 h3h1[∂ ∂u3(h1A1)−∂ ∂u1(h3A3)] +ˆe3 h1h2[∂ ∂u1(h2A2)−∂ ∂u2(h1A1)] (5) We already have defined components of /vectorBin terms of c’s. Now, we have to choose h1A1,h2A2 andh3A3in termsof c’s insucha way that, r.h.s.oftheeqn (5)becomes sameas l.h.s. Let (byinspection), h1A1=k1/integraltextc2(u+ 2)∂u3+k2/integraltextc3(u+ 3)∂u2+k3/integraltextc2(u− 2)∂u3+k4/integraltextc3(u− 3)∂u2 Similarly, h2A2=k5/integraltextc3(u+ 3)∂u1+k6/integraltextc1(u+ 1)∂u3+k7/integraltextc3(u− 3)∂u1+k8/integraltextc1(u− 1)∂u3 and,h3A3=k9/integraltextc1(u+ 1)∂u2+k10/integraltextc2(u+ 2)∂u1+k11/integraltextc1(u− 1)∂u2+k12/integraltextc2(u− 2)∂u1 where,k1,k2,···,k12aresomeconstantsto bedetermined. (Note that, c1terms are not included for h1A1, since,h1A1does not appear in theexpression of 1st component B1in eqn (5). Also notethat, terms in the expressionof h1A1are in the integral formw.r.t.variables u2andu3butnotu1, asin eqn (5), h1A1is neverdifferentiated w.r.t. u1.) Now,seethecoefficient of ˆ e3(expressionof B3)onther.h.s.ofeqn (5), 1 h1h2[∂ ∂u1(h2A2)−∂ ∂u2(h1A1)] =1 h1h2[∂ ∂u1{k5/integraltextc3(u+ 3)∂u1+k6/integraltextc1(u+ 1)∂u3+k7/integraltextc3(u− 3)∂u1+k8/integraltextc1(u− 1)∂u3} −∂ ∂u2{k1/integraltextc2(u+ 2)∂u3+k2/integraltextc3(u+ 3)∂u2+k3/integraltextc2(u− 2)∂u3+k4/integraltextc3(u− 3)∂u2}] =1 h1h2[k5c3(u+ 3)+k6/integraltext∂ ∂u1c1(u+ 1)∂u3+k7c3(u− 3)+k8·0−k1/integraltext∂ ∂u2c2(u+ 2)∂u3 −k2c3(u+ 3)−k3·0−k4c3(u− 3)] =1 h1h2[2k5c3(u+ 3)−k1/integraltext{∂ ∂u1c1(u+ 1)+∂ ∂u2c2(u+ 2)}∂u3+2k7c3(u− 3)] [Takingk5=−k2,k7=−k4andk6=−k1] =1 h1h2[2k5c3(u+ 3)−k1/integraltext{−∂ ∂u3c3(u+ 3)}∂u3+2k7c3(u− 3)] [Usingeqn(3)] =1 h1h2[2k5c3(u+ 3)+k1c3(u+ 3)+2k7c3(u− 3)] =1 h1h2[3k5c3(u+ 3)+2k7c3(u− 3)] [Takingk1=k5] 3 =1 h1h2[3·1 3c3(u+ 3)+2·1 2c3(u− 3)] [Considering k5=1/3andk7=1/2] =1 h1h2[c3(u+ 3)+c3(u− 3)] =1 h1h2(h1h2B3) =B3 Similarlywecan showthat,if, k1=k5=k9=−k2=−k6=−k10=1/3 and,k3=k7=k11=−k4=−k8=−k12=1/2,then, 1 h2h3[∂ ∂u2(h3A3)−∂ ∂u3(h2A2)]=B1and1 h1h3[∂ ∂u3(h1A1)−∂ ∂u1(h3A3)]=B2 So, usingthesevaluesof k1,k2,···,k12, weget, A1=1 3h1[/integraltextc2(u+ 2)∂u3−/integraltextc3(u+ 3)∂u2]+1 2h1[/integraltextc2(u− 2)∂u3−/integraltextc3(u− 3)∂u2] A2=1 3h2[/integraltextc3(u+ 3)∂u1−/integraltextc1(u+ 1)∂u3]+1 2h2[/integraltextc3(u− 3)∂u1−/integraltextc1(u− 1)∂u3] A3=1 3h3[/integraltextc1(u+ 1)∂u2−/integraltextc2(u+ 2)∂u1]+1 2h3[/integraltextc1(u− 1)∂u2−/integraltextc2(u− 2)∂u1] Nowfrom eqn(2), weget, (/vector∇×)−1/vectorB=/vectorA=A1ˆe1+A2ˆe2+A3ˆe3 =ˆe1 3h1[/integraltextc2(u+ 2)∂u3−/integraltextc3(u+ 3)∂u2]+ˆe1 2h1[/integraltextc2(u− 2)∂u3−/integraltextc3(u− 3)∂u2] +ˆe2 3h2[/integraltextc3(u+ 3)∂u1−/integraltextc1(u+ 1)∂u3]+ˆe2 2h2[/integraltextc3(u− 3)∂u1−/integraltextc1(u− 1)∂u3] +ˆe3 3h3[/integraltextc1(u+ 1)∂u2−/integraltextc2(u+ 2)∂u1]+ˆe3 2h3[/integraltextc1(u− 1)∂u2−/integraltextc2(u− 2)∂u1] =−ˆe1 3h1/integraltext(∂u2)/integraltext(∂u3) c2(u+ 2)c3(u+ 3)+ˆe2 3h2/integraltext(∂u1)/integraltext(∂u3) c1(u+ 1)c3(u+ 3)−ˆe3 3h3/integraltext(∂u1)/integraltext(∂u2) c1(u+ 1)c2(u+ 2) −ˆe1 2h1/integraltext(∂u2)/integraltext(∂u3) c2(u− 2)c3(u− 3)+ˆe2 2h2/integraltext(∂u1)/integraltext(∂u3) c1(u− 1)c3(u− 3)−ˆe3 2h3/integraltext(∂u1)/integraltext(∂u2) c1(u− 1)c2(u− 2) =−1 3ˆe1 h1ˆe2 h2ˆe3 h3 /integraltext(∂u1)/integraltext(∂u2)/integraltext(∂u3) c1(u+ 1)c2(u+ 2)c3(u+ 3)−1 2ˆe1 h1ˆe2 h2ˆe3 h3 /integraltext(∂u1)/integraltext(∂u2)/integraltext(∂u3) c1(u− 1)c2(u− 2)c3(u− 3) 4 =−ˆe1 h1ˆe2 h2ˆe3 h3 /integraltext(∂u1)/integraltext(∂u2)/integraltext(∂u3) 1 3c1(u+ 1)+1 2c1(u− 1)1 3c2(u+ 2)+1 2c2(u− 2)1 3c3(u+ 3)+1 2c3(u− 3) =−ˆe1 h1ˆe2 h2ˆe3 h3 (1 3/integraltext++1 2/integraltext−)(∂u1) (1 3/integraltext++1 2/integraltext−)(∂u2) (1 3/integraltext++1 2/integraltext−)(∂u3) h2h3B1 h3h1B2 h1h2B3 Thisisthedesired expressionforourinversecurl operator . Since,/vector∇×(/vector∇φ)=0, so, it should be noted that, for a given solenoidal vector, corresponding vectorpotentialisnot unique,whichcan beingeneral expre ssedas, - /vectorA=(/vector∇×)−1/vectorB+/vector∇φ whereφisan arbitrary scalar function. 2.1 An Examplein CartesianSystem One example in Cartesian system would clarify the procedure of getting vector potential for a solenoidalvector. In this system, ˆ e1, ˆe2, ˆe3,u1,u2andu3are respectively replaced by ˆi,ˆj,ˆk,x, y, andz. Hereh1=h2=h3=1. Nowtakeasolenoidalvector /vectorB=(xyz+y2)ˆi+(xz+y)ˆj−z(1+yz/2)ˆk. (note/vector∇·/vectorB=0) So, vectorpotentialsof ˆBare,- /vectorA=(/vector∇×)−1/vectorB+/vector∇φ =−ˆi ˆj ˆk (1 3/integraltext++1 2/integraltext−)(∂x) (1 3/integraltext++1 2/integraltext−)(∂y) (1 3/integraltext++1 2/integraltext−)(∂z) xyz+y2xz+y −z−yz2/2+/vector∇φ =−ˆi[1 3/integraltext(−z−yz2/2)∂y+1 2/integraltext0∂y−1 3/integraltexty∂z−1 2/integraltextxz∂z] −ˆj[1 3/integraltextxyz∂z+1 2/integraltexty2∂z−1 3/integraltext(−z−yz2/2)∂x−1 2/integraltext0∂x] −ˆk[1 3/integraltexty∂x+1 2/integraltextxz∂x−1 3/integraltextxyz∂y−1 2/integraltexty2∂y]+/vector∇φ =ˆi[(zy+z2y2/4)/3+(yz/3+xz2/4)]−ˆj[(zx+yxz2/2)/3+(xyz2/6+y2z/2)] +ˆk[−(yx/3+x2z/4)+xy2z/6+y3/6]+/vector∇φ 5 2.1.1 Verificationofthe Result /vector∇×/vectorA=ˆi ˆj ˆk ∂ ∂x∂ ∂y∂ ∂z (zy+z2y2/4)/3−(zx+xyz2/2)/3−(yx/3+x2z/4) +(yz/3+xz2/4)−(xyz2/6+y2z/2) +(xy2z/6+y3/6)+/vector∇×(/vector∇φ) =ˆi(xyz+y2)+ˆj(xz+y)−ˆkz(1+yz/2)+0 =/vectorB 3 InverseDivergenceOperator Here we indicate this operator by (/vector∇·)−1. Now if /vectorAandφbe a vector and a scalar function respectively,suchthat φ=/vector∇·/vectorA=1 h1h2h3[∂ ∂u1(h2h3A1)+∂ ∂u2(h1h3A2)+∂ ∂u3(h1h2A3)],then/vectorAcan beexpressedas /vectorA=(/vector∇·)−1φ. Wecan nowdefine theoperatoras (by inspection), (/vector∇·)−1=k1ˆe1 h2h3/integraltexth1h2h3(∂u1)+k2ˆe2 h3h1/integraltexth1h2h3(∂u2)+k3ˆe3 h1h2/integraltexth1h2h3(∂u3) where,k1+k2+k3=1. 3.1 Verification Foragiven φ,/vectorA=(/vector∇·)−1φ =k1ˆe1 h2h3/integraltexth1h2h3φ(∂u1)+k2ˆe2 h3h1/integraltexth1h2h3φ(∂u2)+k3ˆe3 h1h2/integraltexth1h2h3φ(∂u3) Nownotethat,divergenceoftheobtainedvectorgivesthesc alarfunction φ, /vector∇·/vectorA=1 h1h2h3[k1h1h2h3φ+k2h1h2h3φ+k3h1h2h3φ] =(k1+k2+k3)φ=φ Since,/vector∇·(/vector∇×/vectorB) =0, so, it should be noted that for a given scalar function, cor responding potentialfunctionisnotunique,which can beingeneral exp ressedas, /vectorA=(/vector∇·)−1φ+/vector∇×/vectorB where/vectorBisany vector. 6 4 InverseGradientOperator Weindicatethisoperatorby (/vector∇)−1. Let/vectorA=A1ˆe1+A2ˆe2+A3ˆe3andφ(u1,u2,u3)beavectorandascalarfunction,suchthat /vectorA=/vector∇φ. Since, thevectorfield represented by /vectorAis conservative,thus,lineintegralof /vectorAis path indepen- dent. Choose(a,b,c)as initialpoint,such that φ(a,b,c)exists. Nowφ(u1,u2,u3)=(u1,u2,u3)/integraltextdφ (a,b,c)+c0[c0isintegrationconstant] =(u1,u2,u3)/integraltext(/vector∇φ)·/vectordr (a,b,c)+c0 =(u1,u2,u3)/integraltext/vectorA·/vectordr (a,b,c)+c0 =(a,b,u3)/integraltextA3h3du3 (a,b,c)+(a,u2,u3)/integraltextA2h2du2 (a,b,u3)+(u1,u2,u3)/integraltextA1h1du1 (a,u2,u3)+c0 So, φ=(/vector∇)−1/vectorA =u1/integraltext aA1h1∂u1+u2/integraltext bA2h2∂u2+u3/integraltext cA3h3∂u3+c0 u1=au1=a u2=b Here in the first integration u1is variable but other two ( u2,u3) are fixed, and so on for other twointegrations. 7 Note:The original work was done in 2002 and published in 2006 in Sci ence and Culture [2]. Thiswrite-upis almostsameas publishedone. References [1] Anytextonvectordeloperatorandcurvilinearco-ordin atesystem.Forexample: byGeorge B. Arfken and Hans J. Weber, Mathematical Methods For Physicists , Academic Press: 5th edition(2000);1stand 2ndchapters. [2] Shaon Sahoo, Sci.& Cult. ,72, (1-2), 89(2006). 8