A =(A x,a y,a z ) (1.1) A 2 x + A2 y + A2 z (1.2)

Transcription

1 Chpter 1 Vector Anlysis Problem Set #1: 1.2, 1.3, 1.9, 1.1, 1.11, 1.15, 1.18, 1.2 (Due Thursdy Jn. 23rd) Problem Set #2: 1.14, 1.17, 1.28, 1.29, 1.3,1.33, 1.39, 1.43, 1.46 (Due Tuesdy Feb. 18th) 1.1 Vector Algebr Vectors Vector quntities (or three-vectors) re denoted by boldfce letters A, B,... in contrst to sclr quntities denoted by ordinry letters A, B,... For exmple, in Crtesin coordintes vector hs length (or mgnitude) A =(A x,a y,a z ) (1.1) A A = A 2 x + A2 y + A2 z (1.2) which is sclr. Sclrs re rel numbers or elements in spce R nd vectors re elements in spce R 3.ForvectorsA, B,... R 3 nd ngle θ between A nd B one cn define: 1. Addition: which is commuttive A + B (A x + B x,a y + B y,a z + B z ) (1.3) A + B = B + A (1.4) 4

2 CHAPTER 1. VECTOR ANALYSIS 5 ssocitive nd defines inverse (or minus) vector where the zero vector is (A + B)+C = A +(B + C) (1.5) A +( A) (1.6) (,, ). (1.7) Geometriclly the ddition is understood by prllel trnsporting vector B so tht it strts where the vector A ends. Then the vector A+B points from the beginning of vector A to the end of vector B. 2. Multipliction by sclr: which is distributive A (A x,a y,a z ) (1.8) (A + B) =A + B (1.9) where R is the sclr. Geometriclly the resulting vector A is vector pointing in the sme direction (or in the opposite direction if <) s vector A but whose mgnitude is times lrger (or smller if < 1). 3. Dot product (or sclr product): A B AB cos θ (1.1) which is commuttive nd distributive A B =B A (1.11) A (B + C) =A B + A C. (1.12) Geometriclly the dot product mesures the length of the vector A when projected to the direction of B times B or equivlently the length of the vector B when projected to the direction of A times A. 4. Cross product (or vector product): A B AB sin θˆn (1.13)

3 CHAPTER 1. VECTOR ANALYSIS 6 where the vector ˆn hs unit length (unit vector) which is non-commuttive (or nti-commuttive) nd distributive ˆn =1 (1.14) A B = B A (1.15) A (B + C) =A B + A C. (1.16) Geometriclly the mgnitude of vector A B is the re of prllelogrm generted by A nd B nd points in the direction ˆn perpendiculr both A nd B using the right-hnd-rule (just convention). Note tht the cross product exist only in three nd seven dimensionl spces Components It is convenient to write vectors in the components form A =(A x,a y,a z )=A xˆx + A y ŷ + A z ẑ (1.17) where ˆx, ŷ nd ẑ re unit vectors in the direction of positive x, y nd z xes. Then, 1. Addition: A+B =(A xˆx + A y ŷ + A z ẑ )+(B xˆx + B y ŷ + B z ẑ )=(A x + B x ) ˆx +(A y + B y ) ŷ +(A z + B z ) ẑ (1.18) 2. Multipliction by sclr: A = (A xˆx + A y ŷ + A z ẑ )=(A x ) ˆx +(A y ) ŷ +(A z ) ẑ (1.19) 3. Dot product: A B = (A xˆx + A y ŷ + A z ẑ ) (B xˆx + B y ŷ + B z ẑ )= = A xˆx (B xˆx + B y ŷ + B z ẑ )+A y ŷ (B xˆx + B y ŷ + B z ẑ )+A z ẑ (B xˆx + B y ŷ + B z ẑ = (A xˆx B xˆx )+(A y ŷ B y ŷ )+(A z ẑ B z ẑ )=A x B x + A y B y + A z B z (1.2) since nd Note tht A = ˆx ˆx = ŷ ŷ = ẑ ẑ =1 (1.21) ˆx ŷ = ŷ ẑ = ẑ ˆx =. (1.22) A 2 x + A2 y + A2 z = A A. (1.23)

4 CHAPTER 1. VECTOR ANALYSIS 7 4. Cross product: A B = (A xˆx + A y ŷ + A z ẑ ) (B xˆx + B y ŷ + B z ẑ )= since = A xˆx (B xˆx + B y ŷ + B z ẑ )+A y ŷ (B xˆx + B y ŷ + B z ẑ )+A z ẑ (B xˆx + B y ŷ + = (A y B z A z B y ) ˆx +(A z B x A x B z ) ŷ +(A x B y A y B x ) ẑ ˆx ŷ ẑ = det A x A y A z B x B y B z ˆx ˆx = ŷ ŷ = ẑ ẑ = (1.25) nd (for the right-hnded coordinte system) ˆx ŷ = ẑ ŷ ẑ = ˆx ẑ ˆx = ŷ (1.26) ŷ ˆx = ẑ ẑ ŷ = ˆx ˆx ẑ = ŷ. (1.27) It follows tht the so-clled sclr triple product C (A B) = (A y B z A z B y ) C x +(A z B x A x B z ) C y +(A x B y A y B x ) C z = C x C y C z = det A x A y A z (1.28) B x B y B z is nothing but the volume of prllelepiped generted by A, B nd C nd A (B C) =B (C A) =C (A B). (1.29) There is lso vector triple product A (B C) =B (A C) C (A B), (1.3) but you re not required to memorize these formuls since they cn lwys be re-derived from the components representtion of vectors Nottions Let us now introduce some nottions: 1. Position vector is vector r = xˆx + yŷ + zẑ (1.31)

5 CHAPTER 1. VECTOR ANALYSIS 8 describes position of point (x, y, z) reltive to the origin (whose coordintes re (,, )). Its mgnitude is nd unit vector in the direction of r is r = r = x 2 + y 2 + z 2 (1.32) ˆr = r r = xˆx + yŷ + zẑ x2 + y 2 + z 2. (1.33) 2. Seprtion vector is vector s r r =(x x ) ˆx +(y y ) ŷ +(z z ) ẑ (1.34) describes position of point (x, y, z) reltive to the origin (whose coordintes re (x,y,z )). Its mgnitude is s = r r = (x x ) 2 +(y y ) 2 +(z z ) 2 (1.35) nd unit vector in the direction of s is ŝ = s s = (x x ) ˆx +(y y ) ŷ +(z z ) ẑ (x x ) 2 +(y y ) 2 +(z z ) 2. (1.36) 3. Displcement vector is n infinitesiml vector dr ˆx + dyŷ + dzẑ (1.37) describes displcement from point (x, y, z) to point (x +, y + dy, z + dz). Whtisthemgnitudeofdr? Isthereunitvectorinthedirection of dr? 1.2 Differentil Clculus Grdient Consider function of single vrible f(x) then one cn expnd it round some point x s ( ) df (x) f(x) =f(x )+ (x x )+ 1 ( ) d 2 f(x) (x x x=x 2 2 ) (1.38) x=x

6 CHAPTER 1. VECTOR ANALYSIS 9 If we only keep the liner term thn ( ) df (x) f(x) f(x ) (x x ) (1.39) x=x which in differentil form is simply df = ( ) df. (1.4) Similrly the function of three vribles f(x, y, z) to the liner order in expnsion is ( ) ( ) ( ) f(x, y, z) f(x, y, z) f(x, y, z) f(x, y, z) f(x,y,z ) (x x )+ (y y )+ x=x y=y (1.41) or in differentil form ( ) ( ) ( ) f f f df = + dy + dz. (1.42) This cn lso be rewritten s dot product of two vectors (( ) ( ) ( ) ) f f f df = ˆx + ŷ + ẑ (ˆx + dyŷ + dzẑ ) ( f =, f, f ) (, dy, dz) ((... =,...,... ) ) f (, dy, dz). (1.43) where in the lst line vector-like opertor cts on the function f to produce vector.thevectorisclledgrdientoff defined s ( f f =, f, f ) ( ) ( ) ( ) f f f = ˆx + ŷ + ẑ. (1.44) Geometriclly grdient f is vector pointing in the direction of ( locl) mximum increse of function f nd its mgnitude gives the rte of the increse. For instnce t locl mxim, minim or sddle point, the grdient of function is zero vector. It is lso useful to define vector-like opertor known s del (or nbl) opertor (...,...,... ). (1.45) x=x

7 CHAPTER 1. VECTOR ANALYSIS 1 Then the grdients cn be produced by cting with nbl on functions (... f =,...,... ) ( f f =, f, f ) (1.46) where is treted s vector quntity nd f is treted s sclr quntity Divergence nd Curl One cn lso imgine vector function which hs three vlues A x (x,y,z ), A y (x,y,z ) nd A z (x,y,z ) in ech point in spce or equivlently there is three-component vector v(x,y,z ) ttched to ech point (x,y,z ). Mthemticlly speking vectors re elements of tngent spce t given point (x,y,z ) T (x,y,z )M on mnifold (in our cse 3 dimension Eucliden spce M = R 3 )ndthevectorfieldsreelementsoftngentbundle v T M. Onecnlsothinkofthenblopertorsvectorfieldopertor lthough it is not usully clled this wy ( [... ], (x,y,z ) [ ]..., (x,y,z ) [ ] ).... (1.47) (x,y,z ) (Tke your time nd think wht it mens). We shll omit writing (x,y,z ), but it is ssumed tht the sclr nd vector quntities re fields. Given sclr fields f, g,... vector fields v, u,... nd vector opertor one cn do the usul vector mnipultions t ech point seprtely to produce new fields: Addition: A + B =(A x + B x,a y + B y,a z + B z ) (1.48) 1. Multipliction by sclr: or = A =(A x,a y,a z ) (1.49) (...,...,... ) ( =,, ) (1.5) which is grdient of sclr field which is vector field s we hve lredy mentioned. 2. Dot product (or sclr product): A B = A x B x + A y B y + A z B z (1.51)

8 CHAPTER 1. VECTOR ANALYSIS 11 or A = (...,...,... ) (A x,a y,a z )=. (1.52) which is sclr field clled divergence of vector A. Geometriclly the divergence mesures the mount by which the lines of vector field diverge from ech other. 3. Cross product (or vector product): ˆx ŷ ẑ A B =det A x A y A z (1.53) B x B y B z or ˆx ŷ ẑ A =det (1.54) A x A y A z which is vector field clled curl of vector A. Geometricllythecurl mesures the mount by which the lines of vector field curl round given point. According to Helmholtz theorem the knowledge of divergence A nd of curl A of some vector field A is sufficient to determine the vector field itself (given tht both A nd A fll off fster thn 1/r 2 s r ). Using definitions of grdient 1.5, divergence 1.52 nd curl 1.54 it is stright-forwrd to derive different product rules (b) = b + b (A B) = (A ) B +(B ) A (A) = A + A (A B) = B ( A) A ( B) (A) = ( A) A ( ) (A B) = (B ) A (A ) B + A ( B) B ( A)(1.55) nd quotient rules ( ) ( b ) A ( ) A = = = b b b 2 ( A) A ( ) 2 ( A)+A ( ) 2. (1.56)

9 CHAPTER 1. VECTOR ANALYSIS 12 From definitions one cn lso derive expressions for second derivtives the most useful of which is Lplcin opertor 2 ( ). (1.57) It is lso extremely importnt to remember tht the curl of grdient or divergence of curl is lwys zero ( ) = (,, ) ( A) =. (1.58) Mxwell Equtions Consider sclr field which is single function (in three spce nd one time dimensions) ρ(t,x,y,z ). (1.59) nd three-vector fields which is collection of three function (in three spce nd one time dimensions) B (B x,b y,b z ) E (E x,e y,e z ) J (J x,j y,j z ). (1.6) It looks bit odd nd in fct (s you might suspect) there is more nturl object (the so-clled four-vector potentil) which is n element of tngent bundle of the four dimensionl mnifold describing the spce-time. Then to promote these vectors to vector fields we should imgine they re functions of sptil x,y,z nd temporl t coordintes B(t,x,y,z ) (B x (t,x,y,z ),B y (t,x,y,z ),B z (t,x,y,z )) E(t,x,y,z ) (E x (t,x,y,z ),E y (t,x,y,z ),E z (t,x,y,z )) J(t,x,y,z ) (J x (t,x,y,z ),J y (t,x,y,z ),J z (t,x,y,z )). In writing the fields we usully omit the ugly looking (t,x,y,z ) but the dependence on spce nd time coordintes is lwys implied. Now if we think of ρ nd J s the electric chrge nd electric current density (fields) nd of electric E nd mgnetic B fields then there re the so-clled Mxwell equtions which relte these fields to ech other. In SI units the fmous equtions tke the following form

10 CHAPTER 1. VECTOR ANALYSIS 13 E = ρ ϵ (Guss s lw) (1.61) B E c 2 t = µ J (Ampere s lw) (1.62) E + B t = (Frdy s lw) (1.63) B = (Guss s lw) (1.64) where c = 1 ϵµ is the speed of light. Why light? Becuse light is nothing but the wves of electric E nd mgnetic B field (or for short electromgnetic wves) propgting in spce. These equtions cn be derived from vritionl principle nd interested students will be encourged to do so t the end of the course. It turnsout tht the fundmentl fields re not the electric E nd mgnetic B fields, but the sclr V nd vector A potentil (fields). In terms of these potentil fields E = V A (1.65) t B = A. (1.66) The two potentils combined form four-vector (V,A) which is the element of the tngent bundle of our four-dimensionl spce-time. And to derive (1.61,1.62,1.63,1.64) fromfirstprinciples(i.e. vritionl principle) one should strt with prticulr Lgrngin written in terms of V nd A nd vry it with respect to V nd A. Werenotgoingtodothis, but we will ssume tht the Mxwell equtions give correct description of electricity nd mgnetism for mcroscopic chrges, currents, distnces, energies, etc. The Mxwell equtions describe how chrges (sttionry ρ or moving J) generte the electric E nd mgnetic B fields, but do not describe how the chrges move due to electric nd mgnetic forces. For tht you needn dditionl eqution known s Lorentz force lw: F = q (E + v B). (1.67) Eqution (1.67) together with equtions(1.61,1.62,1.63,1.64) describe everything there is to know in this course, but before we strt let us reviewthe mthemtics of integrl clculus.

11 CHAPTER 1. VECTOR ANALYSIS Integrl Clculus Integrls For function of single vrible f(x) there is only one type of integrl b f(x) (1.68) but for function of three vribles f(x, y, z) one cn integrte over line (or pth), over re (or surfce) nd over volume: Pth integrl P v(l) dl (1.69) where the integrl is tke over some pth P from point l = to point l = b. (ThesubscriptP is often dropped, but it is lwys implied tht the integrl is over some pth.) For exmple, work required to move prticle long some pth is given by W = F(l) dl (1.7) where F is force cting on the prticle. Surfce integrl S v(l) d (1.71) where the integrl is tke over some surfce S (lso often omitted subscript) nd d is n infinitesiml ptch of re with direction perpendiculr to the surfce (lousy, but common nottion) with lso sign mbiguity in the definition. For closed surfces v(l) d (1.72) the convention is tht the infinitesiml re vector points outwrds. For exmple, consider surfce integrl of v(x, y, z) =2xzˆx +(x +2)ŷ + y(z 2 3)ẑ (1.73) over cubicl box with side 2. Then there will be six contributions to the integrl

12 CHAPTER 1. VECTOR ANALYSIS x =2nd d = dydzˆx implies v d =2xzdydz =4zdydz, nd 2 2 v d =4 dy zdz =16. (1.74) 2. x =nd d = dydzˆx implies v d = 2xzdydz =, nd v d =. (1.75) 3. y =2nd d = dzŷ implies v d =(x +2)dz, nd 2 2 v d = (x +2) dz =12. (1.76) 4. y =nd d = dzŷ implies v d = (x +2)dz, nd 2 2 v d = (x +2) dz = 12. (1.77) 5. z =2nd d = dyẑ implies v d = y(z 2 3)dy = ydy, nd 2 2 v d = ydy =4. (1.78) 6. z =nd d = dyẑ implies v d = y(z 2 3)dy =3ydy, nd 2 2 v d =3 ydy =12. (1.79) And the totl flux is v d = = 32. (1.8) Volume integrl f 3 = f(x, y, z)dydz (1.81) V V which is nothing but triple integrl which cn be tken in ny order ( ( ) ) ( ( ) ) f(x, y, z) dy dz = f(x, y, z)dz dy =... (1.82)

13 CHAPTER 1. VECTOR ANALYSIS Exct differentils The fundmentl theorem describes how to integrte functions which re exct differentils. For exmple, if then x=b x= F (x) = df (x) (1.83) F (x) = f(b) f(). (1.84) In higher dimensions this results generlizes to integrting function over n rbitrry pth F (x, y, z) = f(x, y, z). (1.85) b ( f) dl = f(b) f() (1.86) connecting points nd b. For exmple, if the force in eqution(1.7) is conservtive (e.g. grvittionl force, but not friction force) F = V (1.87) then the work only depend on the vlue of the potentil in the initil nd finl points W = b F(l) dl = V () V (b). (1.88) nd over closed pths the work is lwys zero W = F(l) dl =. (1.89) Note tht it is lwys possible to rewrite curl-less vector fields s grdient F = F = V (1.9) nd thus the pth independence of work (1.88) ndvnishingofworkfor closed pths (1.89) would follow utomticlly.and if the vector field is not curl-less we cn still rewrite it s F F = V + A.

14 CHAPTER 1. VECTOR ANALYSIS 17 Astrightforwrdgenerliztionofthesmeideledstothe Guss s (or divergence) theorem ( v) 3 = v d (1.91) nd to Stokes s (or curl) theorem ( v) d = S V S P v dl. (1.92) Roughly speking the Guss s theorem (1.91) describesthetwowysofclculte the number of (vector v) fieldlinesenteringgivenvolumeminus the number of field lines leving the volume. On cn clculte it either by integrting divergence of the field lines over the volume (s on the left hnd side), or by integrting the flow of the field lines over the surfce (s on the right hnd side). Similrly the Stokes s theorem (1.92) describes the different wys how swirling of the field lines cn be clculte. One cn clculte it by integrting the curl of field lines over the re (s on the left hnd side), or by the integrting the rottion of field lines s we go round boundry (s on the right hnd side). For vector field (1.73) v(x, y, z) =2xzˆx +(x +2)ŷ + y(z 2 3)ẑ (1.93) we cn find v =2z +2yz (1.94) v =det ˆx ŷ ẑ xz x +2 y(z 2 3) =(z 2 3)ˆx +2xŷ + ẑ. (1.95) It is now esy to check tht (2z +2yz) dydz =16+16=32 (1.96) is the sme s totl flux through the boundry (1.8) scorrectlypredicted by Guss s theorem (1.91). Moreover for the fce of the cube (y =) 2 2 ( (z 2 3)ˆx + ŷ 2x + ẑ ) ŷdz = 2 2 2xdz =8 (1.97)

15 CHAPTER 1. VECTOR ANALYSIS 18 or + z=2 z= (2xzˆx ˆx ) + (2xzˆx ( ˆx )) + ++ x=2 x= ( y(z 2 3)ẑ ẑ ) dz + ( y(z 2 3)ẑ ( ẑ ) ) dz = 2 (4xˆx ( ˆx )) + = 2 4x = 8 (1.98) in greement with Stokes s theorem (1.92). In conclusion, let us consider n exct differentil of product of two functions, d dg (fg)=f + g df, (1.99) then we cn integrte both sides to obtin or b d (fg) = b [fg] b = b b ( f dg ) = ( f dg ( f dg b ) + ) + b b ( g df ( g df ), ), (1.1) ( g df ) +[fg] b. (1.11) Thus we cn replce integrtion of function f dg with integrting function g df plus boundry term which is often set to zero. Expression (1.11) is known s integrtion by prts. Although trivil the integrtion by prts is n extremely useful tool which cn lso be generlized to more complicted integrls described bove with use of either Guss s or Stokes s theorems. For exmple, implies nd V (fa) =f A + A f (1.12) f ( A) 3 = A ( f) 3 + V S fa d (1.13) (fa) =f ( A) A ( f) (1.14)

16 CHAPTER 1. VECTOR ANALYSIS 19 implies S f ( A) d = (A ( f)) d + V P Generlized functions fa dl. (1.15) There is specil clss of functions known s generlized functions (or distribution functions). The most useful exmple of such function is the (Dirc) δ-function. Strictly speking it is not function s it only mkes sense to tlk bout δ-function when it is inside of n integrl. In one dimension it cn be defined by the following expression δ(x )f(x) f() (1.16) where f(x) is n rbitrry function.sometimes it is convenient to express δ-function s derivtive of the Heviside step function, i.e. δ(x) = d H(x). (1.17) In three dimensions it is defined s product of three delt functions so tht or δ (3) (r) =δ(x)δ(y)δ(z) (1.18) δ(x )δ(y b)δ(z c)f(x, y, z)dydz = f(, b, c) (1.19) f(r)δ(r ) 3 = f(). (1.11) One cn think of δ-function s probbility distribution for point prticle locted t since the integrl of the entire spce is exctly one 1.4 Trnsformtions Simple trnsformtions δ (3) (r ) 3 =1. (1.111) Clerly the choice of the reference frme or coordintes system (i.e. origin, xes, hndedness, etc ) is rbitrry, nd we wnt the lws of physics not

17 CHAPTER 1. VECTOR ANALYSIS 2 to depend on this choice. In other words if we mke predictions ofhow given system should behve in one coordinte system then we should hve rulehowtomkepredictionsinnothercoordintesystem. For tht we need rule how to trnsform different quntities from one system to nother. In fct ll of the quntities (such s sclrs, vectors, tensors, spinors, etc) re distinguished from other by the wy they trnsform under chnges of coordintes. Wht re the possible trnsformtions in Eucliden three dimensionl spce (denoted by 3D)? There re: 3trnsltions(or shifts) long ˆx, ŷ nd ẑ directions 3rottionsfrom ˆx to ŷ,fromŷ to ẑ nd from ẑ to ˆx. These re linerly independent trnsformtions (i.e. there re no non-zero liner combintions of these six trnsformtions which leves the system untrnsformed), but one cn produce other linerly dependent trnsformtions by forming liner combintions of these six trnsformtions (e.g.shift by-5 meters long ŷ,rottebyπ/5 from ẑ to ˆx nd then rotte by π/7 from ˆx to ŷ ). How mny linerly independent trnsformtions in 1D? 2D? 4D? nd? In n dimensions there re n trnsltions nd s rottion mny rottions s there re distinct pirs of xis (rottions from x to y, fromx toz, etc.) Thus there re (n 1) + (n 2) = n + n(n 1) 2 = n(n +1) 2 n(n 1). 2 independent trnsformtions. Liner combintion of trnsltions cn be described by trnsltion vector T =(T x,t y,t z ) (1.112) of the old coordinte system (x, y, z) to new coordinte system (x,y,z ). Note tht the trnsltion vector is lso expressed in the old (unprimed) coordintes. Then sclrs (e.g. A) ndvectors(e.g. A = (A x,a y,a z )) trnsforms into A nd A =(A x,a y,a z ) such tht nd A = A (1.113) A = A. (1.114)

18 CHAPTER 1. VECTOR ANALYSIS 21 This is just sttement of the fct tht vectors prllel trnsported in the Eucliden spce do not chnge. For brevity of nottions (nd to confuse reders) the primes re often dropped either for the newly trnsformed vector (s in books on generl reltivity) or for the new coordintes (s in book on electrodynmics) so tht A x = A x A y = A y A z = A z (1.115) The nottions re confusing, but it should lwys be cler from the context whether we re in the old (unprimed) or in the new (primed) coordintes system. Liner combintion of rottions cn be described by rottion mtrix R xx R xy R xz R yx R yy R yz R zx R zy R zz cos φ 1 sin φ 1 sin φ 1 cos φ 1 1 (1.116) + 1 cosφ 2 sin φ 2 sin φ 2 cos φ 2 + for some ngles φ 1, φ 2 nd φ 3.Thensclrsndvectortrnsforms cos φ 3 sin φ 3 1 sin φ 3 cosφ 3 A = A (1.117) 3 A i = R ij A j (1.118) j=1 where it is ssumed tht i =1, 2, 3 nd it stnds correspondently for either x, y, z. or using the Einstein summtion convention (lwys sum over repeted indices) A i = R ija j. (1.119) For more complicted objects such s tensors the trnsformtion lw would be written s M ij = 3 R ik R jl M kl (1.12) k,l=1 or (with Einstein summtion convention) simply M ij = R ikr jl M kl. (1.121).

19 CHAPTER 1. VECTOR ANALYSIS 22 Clerly, the simplest trnsformtion rule is for sclr quntities - they do not chnge under coordinte trnsformtions Generl trnsformtions So fr we hd been using Crtesin coordintes, but nothing cn stop us from describing the points on different mnifolds using other coordintes system. The two most useful exmples re the so-clled sphericl nd cylindricl coordintes both of which re generliztions of two dimensionl polr coordintes to our three dimensionl spce: Sphericl coordintes x = r sin θ cos φ y = r sin θ sin φ z = r cos θ (1.122) where r (, ) is the rdil distnce,θ (,π) is the inclintion ngle nd φ [, 2π) is zimuthl ngle. Cylindricl coordintes x = s cos φ y = s sin φ z = z (1.123) where s (, ) is the rdil direction projected to x y plne nd φ [, 2π) is the sme zimuthl ngle s in sphericl coordintes. Then ny vector in Crtesin coordintes (x, y, z) A = A xˆx + A y ŷ + A z ẑ (1.124) cn be expressed in terms of the new coordintes (r, θ, φ) or (s, φ, z) A = A rˆr + A θˆθ + Aφ ˆφ (1.125) A = A rˆr + A φ ˆφ + Az ẑ (1.126) nd vise vers. The trnsformtion mtrix is clled Jcobin ncnbe clculted for ny trnsformtion. For exmple, using the inverse Jcobin mtrix

20 CHAPTER 1. VECTOR ANALYSIS 23 Ĵ 1 = r r r θ θ θ φ φ φ = sin θ cos φ rcos θ cos φ r sin θ sin φ sin θ sin φ rcos θ sin φ rsin θ cos φ cos θ r sin θ one cn expressed vectors in new coordintes 1 A, B 1, C (1.127) (1.128) 1 in terms of old coordintes sin θ cos φ Ĵ 1 A sin θ sin φ = sin θ cos φˆx + sin θ sin φŷ +cosθẑ. Ĵ 1 B cos θ r cos θ cos φ r cos θ sin φ = r cos θ cos φˆx + r cos θ sin φŷ r sin θẑ. r sin θ r sin θ sin φ Ĵ 1 C r sin θ cos φ = r sin θ sin φˆx + r sin θ cos φŷ. (1.129) which cn be normlized to define ˆr cos φˆx + sin θ sin φŷ +cosθẑ ˆθ cos θ cos φˆx +cosθ sin φŷ sin θẑ. ˆφ sin φˆx + cosφŷ. (1.13) In fct these normliztion constnts (1, r nd r sin θ) reimportntsthey ppers in generl infinitesiml displcement dl = drˆr + rdθˆθ + r sin θdφˆφ (1.131) often written in terms of the so-clled metric tensor 1 dl 2 = dr 2 + rdθ 2 + dφ 2 = r 2 (1.132) r 2 sin 2 θ Similrly for cylindricl coordintes x = s cos φ y = s sin φ z = z (1.133)

21 CHAPTER 1. VECTOR ANALYSIS 24 the inverse Jcobin mtrix is Ĵ 1 = s s s φ φ φ = cos φ s sin φ sin φ s cos φ (1.134) 1 nd Ĵ 1 A Ĵ 1 B Ĵ 1 C cos φ sin φ s sin φ s cos φ 1 =cosφˆx + sin φŷ = s sin φˆx + s cos φŷ = ẑ. (1.135) which cn be normlized to define with infinitesiml displcement nd metric tensor ŝ cos φˆx + sin φŷ ˆφ sin φˆx +cosφŷ ẑ ẑ. (1.136) dl = dsŝ + sdφ ˆφ + dzẑ (1.137) dl 2 = ds 2 + s 2 dφ 2 + dz 2 = 1 s 2. (1.138) 1 Note tht to trnsform from crtesin coordintes to sphericl or cylindricl coordintes one should strts with r = x 2 + y 2 + z ( 2 ) z θ = rccos x2 + y 2 + z 2 ( y φ = rctn x) (1.139)

22 CHAPTER 1. VECTOR ANALYSIS 25 or s = x 2 + y ( 2 y φ = rctn x) z = z (1.14) nd clcultes the Jcobin mtrix Ĵ = r θ φ r θ φ r θ φ (1.141) but the logic is exctly the sme. One cn lso rewrite grdients, divergencies nd curls in terms of new coordintes, nd the simplest of ll is the grdient: or f = f f ˆr + r r θ ˆθ f + r sin θ φ ˆφ (1.142) f = f s ŝ + f s φ ˆφ + f ẑ. (1.143) When trnsforming divergence nd curl we must trnsform both the nbl opertor nd the vector which is tedious but strightforwrd exercise which leds to the formuls listed in ny Electrodynmics book.