Mathematics Primer for Vector Fields

Transcription

1 Mathematcs Prmer or Vector Felds B Dr Eugene Patrons Ths s the second n a seres o artcles dealng wth coaxal cables operatng n the requenc span rom drect current through the mcrowave regon In order to make the deas meanngul to the wdest audence ths artcle wll be devoted to the mathematcs necessar to properl descrbe phscal varables that have a dmenson or unt, a sze or magntude, and a drecton n space For the moment we wll be concerned wth phscal varables that all nto two dstnctl derent mathematcal categores The rst categor ncludes tems such as mass, charge, denst, pressure, and temperature among others These are scalar quanttes that ma have a partcular unt but otherwse have onl a numercal value or magntude and are subject to the rules o ordnar arthmetc and algebra The second categor ncludes tems such as dsplacement, veloct, acceleraton, orce, etc These are vector quanttes that possess a partcular unt, a scalar magntude, and a drecton n space as well Scalar and vector phscal varables both ma be called eld varables the change as unctons o poston n space and possbl o tme Fg 1 llustrates a graphcal technque or addng or subtractng two vectors that s amlar to nearl everone What we descrbe here s a more ormal technque or makng calculatons n a ull developed three-dmensonal space We wll emplo Cartesan coordnates at the outset, as these are the most amlar whle ntroducng other coordnate sstems as the stuaton dctates Unt vectors are dmensonless vectors havng a scalar magntude o one and whose sole role s to ndcate drecton along a coordnate axs In Cartesan coordnates the unt vectors are usuall wrtten ˆ, ˆj, and k ˆ ndcatng unt vectors along the x,, and z-axes, respectvel The hat smbol above each letter smbolzes that the vector s a unt vector havng onl drecton In the Cartesan coordnate sstem, as s true n an set o orthogonal coordnates, the unt vectors orm a mutuall perpendcular set An ordnar vector havng magntude and dmenson as well as drecton s oten dstngushed n prnted text b emplong a Fgure 1 Graphcal addton and subtracton o boldace smbol or b an arrow or bar over the chosen smbol In ths artcle vector quanttes wll be smbolzed b emplong a boldace letter smbol As an example, the general vector orce n Cartesan coordnates accordng to ths notatonal scheme would be wrtten as F! F ˆ ˆ ˆ x " F j" Fz k where F x s the scalar magntude o the orce along the x-coordnate axs and smlarl or F and F z These ndvdual scalar values ma be ether postve or negatve Vector quanttes havng lke dmensons ma be added or subtracted For example, consder two vector quanttes a! a ˆ ˆ ˆ x " a j" a zk and b! b ˆ ˆ ˆ x " b j" bzk that have the same dmensons or unts These two vectors ma be added to produce a new vector sa c! (a " b ) ˆ " (a " b )j ˆ " (a " b )kˆ wth the summaton proceedng on a component-b-component bass Smlarl, the vector b ma be subtracted rom a to produce a new vector sa d! (a # b ) ˆ " (a # b )j ˆ " (a # b )kˆ The magntude o a vector a s wrtten as a! a 2 x " a 2 " a 2 z!"#$%&'$()#*+,-/,00,1*2)/3*45*6&#,*7889 :;

2 Fgure 2 Scalar product o two Suppose the vector r represents dsplacement rom the orgn wth r! 3 ˆ " 4j ˆ " 12kˆ meters then r! 3 2 " 4 2 "12 2! 169!13 meters as the magntude o the total dsplacement rom the orgn Three derent methods are dened or products nvolvng The rst s the multplcaton o a vector b an ordnar scalar quantt Ths s smpl a scalng process as 10F s a vector n the drecton o F whose magntude has been ncreased b ten old The second s that o the scalar product o two The scalar product o two vectors elds a scalar quantt whose magntude s the product o the magntudes o the ndvdual vectors multpled b the cosne o the angle ncluded between the drectons o the ndvdual vectors as shown n Fg2 Let the two vectors be a and b as gven above The scalar product o these two vectors s wrtten as a$b wth ths product beng gven b a$ b! (a ˆ " a ˆj " a k) ˆ $ (b ˆ " b ˆj " b k) ˆ! a b ˆ $ ˆ " a b ˆ $ ˆj " a b ˆ $ kˆ " a b ˆj $ ˆ " a b ˆj $ ˆj " x x x x z x a b ˆj $ kˆ " a b kˆ $ ˆ " a b kˆ $ ˆj " a b kˆ $ kˆ! z z x z z z a b " a b " a b Ths s true because n the scalar product o a unt vector wth tsel the ncluded angle s zero or whch the cosne o the angle s unt and the scalar product o smlar unt vectors s thus unt On the other hand, the angle between dssmlar unt vectors s 90 or whch the cosne o the angle s zero and the scalar product s thus zero Thereore, ˆ $ ˆ! ˆ j$ ˆ j! k ˆ $ k ˆ! 1 and ˆ $ ˆ j! ˆ $ k ˆ! ˆ j$ ˆ! ˆ j$ k ˆ! k ˆ $ ˆ! k ˆ $ ˆ j! 0 It should be noted that a$b! b$a The process gven above has taken the magntude o a, multpled t b the magntude o b, and then multpled ths product b the cosne o the angle between the two The thrd method o multplng two vectors s termed the vector product The vector product s smbolzed b c! a % b where c s the product vector, a s the multplng vector, and b s the multplcand vector The vector product s tsel a vector havng a magntude equal to the product o the magntudes o the ndvdual vectors urther multpled b the sne o the angle ncluded between the drectons o the ndvdual vectors as shown n Fg 3 Two non-parallel vectors determne the orentaton o a plane and the product vector o these two vectors s perpendcular to ths plane and bears a rghthanded screw relatonshp wth the magned rotaton o the multplng vector toward the multplcand vector Ths s ndcated n Fg3 b the presence o the screwhead at the orgn As the tp o a s rotated towards b through the angle & a rght-handed screw would advance awa rom the reader As a consequence, Fgure 3 Vector product o two b % a!#(a% b)!#c I a! a ˆ ˆ ˆ x " a j" a zk and b! b ˆ ˆ ˆ x " b j" bzk, then the product vector c ma be calculated b drect expanson as was done n the scalar product case wth the derence that now one s takng the vector products o the unt vectors n each nstance rather than the scalar product as was the ormer case Now, o course, ˆ % ˆ! ˆj % ˆj! kˆ % kˆ! 0 whle ˆ % ˆj! k, ˆ ˆj % kˆ! ˆ, and kˆ % ˆ! ˆj Further, ˆ % kˆ! # ˆj, ˆj % ˆ! # k, ˆ and kˆ % ˆj! # ˆ Rather than calculatng b drect expanson a somewhat speeder calculaton s possble b means o the determnant!"#$%&'$()#*+,-/,00,1*2)/3*45*6&#,*7889 :5

3 ˆ ˆj kˆ a a a b b b Fgure 4 The calculaton o work nvolves a scalar product Evaluaton o the three b three determnant accordng to the usual rules elds c! (a b # b a ) ˆ " (a b # b a )j ˆ " (a b # b a )k ˆ Note that a and b correctl z z z x z x x x each are dsplacement vectors then the magntude o the vector c s the area o a parallelogram havng a and b as sdes The trul unque propert o vectors s the manner n whch the transorm rom one coordnate sstem to another Ths transormaton propert s such that the vector magntude and drecton reman unchanged under coordnate transormaton Ths wll be explored at the approprate tme Imagne that ou are exertng a constant orce o magntude F n pushng a loaded wheelbarrow across a horzontal surace through an extended dstance along the horzontal o an amount l The calculaton o the work done b our appled orce n the process s an example o the scalar product o two The orce that ou appl has both an upward component because ou must lt the handles o the wheelbarrow and a horzontal component n urgng the wheelbarrow n the orward horzontal drecton I the vector F represents the total orce and the vector l represents the subsequent dsplacement o the wheelbarrow along the horzontal then the work perormed s gven b the scalar product contaned n W! F $ l Ths s depcted n Fg 4 Notce n the gure that multplng the magntude o the orce b the cosne o the angle & selects the component o the orce that s along the horzontal so that the scalar amount o work s W=Fl cos(&) A more realstc example would be one where the orce that ou have to exert vares dependent on where ou are located along the horzontal path In such an nstance we must sum all o the ncrements o work perormed n brngng about the total amount o horzontal dsplacement Ths s done b ntegraton along the horzontal path gven b ' dw! ' F $ dl where represents the coordnates o the ntal pont and those o the nal pont denng the horzontal path Such an ntegral s called a lne ntegral An example o the vector product s depcted n Fg 5 where one desres to calculate the vector orce F experenced b a length l o straght wre conductng a stead current I n a regon over whch there s a unorm magnetc nducton descrbed b the vector B In ths nstance, a vector Il spatall descrbes the current carrng conductor Ths vector les along the length o the conductor and ponts n the postve sense o the current The orce experenced b the conductor s gven b Fgure 5 The magnetc orce on current F! Il % B and has a scalar magntude F= Il Bsn(&) Note that carrng conductor nvolves a vector the length o the conductor were parallel to the drecton o product wth the orce on the current the magnetc nducton the angle & would be zero and there element beng towards the reader would be no orce Note also rom the gure that the sense o the current were reversed, the drecton o the orce would reverse also Specal methods are also requred n the calculus o These methods nvolve the ntroducton o a vector derental operator termed del or sometmes nabla that s commonl smbolzed b an nverted Greek letter delta wrtten as ( In Cartesan coordnates the vector operator!"#$%&'$()#*+,-/,00,1*2)/3*45*6&#,*7889 :<

4 del appears as ˆ ) ˆ ) j kˆ ) (! " " where the ndcated dervatve operators are partal dervatves Ths operator ma be appled to both vector and scalar eld quanttes n a number o was Remember a eld quantt s a phscal varable that vares rom pont to pont n space and ma or ma not also var n tme Vector eld quanttes also have drecton whereas scalar eld quanttes have no drecton As an example o a scalar eld quantt suppose we know the temperature dstrbuton n a room where the z-coordnate descrbes elevaton and the x and coordnates pnpont locaton n a horzontal plane at some nstant o tme Ths knowledge s expressed n a mathematcal uncton T T s a uncton o the spatal coordnates so we sa T= T(x,,z) Wth ordnar heatng and coolng sstems we would expect that the temperature n a plane would not var dramatcall rom pont to pont but that t ma var rather markedl as we var the elevaton o the pont o observaton We seek the answer to the ollowng questons What s the drecton o the maxmum ncrease n the uncton T and what s the magntude o the rate o ncrease or slope along ths drecton? These questons are answered b the gradent o T wrtten as (T ˆ ) T ˆ ) T ˆ ) T ( T! " j " k We see that the gradent o the scalar temperature uncton s tsel a vector The drecton o ths vector s that o the maxmum rate o change o the uncton T and the magntude o ths vector s the maxmum slope o the uncton T Ths gradent operator wll be ver mportant when we stud the electrc eld and the assocated scalar electrc potental uncton The del operator ma be appled to a vector eld n two derent was The rst wa nvolves the calculaton o the dvergence o the vector eld quantt Suppose we have a vector eld that s descrbed b v = v (x,, z)! ˆx " ˆj " kz ˆ Ths s just a vector dsplacement eld whose magntude grows lnearl wth dstance rom the orgn The dvergence o ths vector eld s wrtten as ( $ v and s calculated as ˆ ) ˆ ) ˆ ) ˆ ˆ ˆ ( $ v! ( " j " k ) $ (x " j " kz)! " "! 3 Ths measures the combned growth rates o the ndvdual scalar components along the respectve coordnate axes o the vector ( $ v s tsel a scalar quantt The postve dvergence o a vector eld as s the case here ndcates the presence o a source o the eld A negatve dvergence would be ndcatve o a snk o the assocated vector eld The second wa o applng the del operator to a vector eld s through the calculaton o the curl o the vector eld quantt The curl o a vector eld s tsel a vector and measures the space rate o change that occurs at rght angles to the drecton o the orgnal vector at a gven pont Let v now be ˆv x " ˆjv ˆ " kvz The curl o v s wrtten as ( % v and Fgure 6 Graphcal s calculated as the determnant example o a vector ˆ ˆj kˆ eld havng curl ) ) ) (% v! v v v Ths wll be much more meanngul we examne a phscal example that s depcted n Fg 6 Ths gure depcts a veloct eld n a rver as a uncton o the elevaton above the bed o the center o the stream n a regon where the rver s straght and the rverbanks are parallel The elevaton above the bed!"#$%&'$()#*+,-/,00,1*2)/3*45*6&#,*7889 :9

5 s the z-coordnate and the stream low s along the postve x-coordnate The -axs s nto the paper and has a constant value o zero or the stream center As a result o vscous eects the stream veloct s zero on the rverbed and grows lnearl wth elevaton untl the surace s met The veloct eld n the regon o 0! z! 10 eet, the depth o the rver, s then v= ˆ a z " ˆj0 " k0 ˆ where a s a constant equal to 05 (mle hour -1 oot -1 ) Clearl, the stream veloct s n the drecton o the x-axs and has a value that onl depends on elevaton wth a maxmum value o 5 mles per hour Upon substtuton o v! îaz nto the determnant expressng the curl o v t s ound that (% v! ĵa Note that the curl o v at all ponts s Fgure 7 The lne ntegral o the gradent o a scalar eld s ndependent o the path everwhere perpendcular to v Fg 6 ndcates the presence o a paddle wheel adjacent to the veloct eld n the drawng The axs o the paddle wheel passes nto and out o the paper n ts present locaton Ths s a useul graphcal ad n determnng the exstence o a curl assocated wth a vector eld I the paddle wheel were exposed to the vector eld as shown the wheel would rotate n a clockwse drecton A rght-handed screw rotated n such a manner would advance nto the paper along the postve -axs that s n the drecton o the curl I the axs o the paddle wheel were along ether the x-axs or the z-axs, the veloct eld would not rotate the wheel We wll close ths artcle wth one o the undamental theorems o vector analss called the gradent theorem There are two other undamental theorems assocated wth the applcaton o the del operator called the dvergence theorem and curl theorem that we wll encounter next tme n connecton wth learnng about the electrc eld Fg 7 depcts a possble path between two ponts located n a three-dmensonal space such as the room mentoned earler where the scalar temperature eld s T = T(x,,z) The path depcted n Fg 7 ma be descrbed as accuratel as ou please b derental vector dsplacements dl each o whch s tangent to the actual path at each pont n queston Now (T s a vector whose magntude and drecton at an pont s that o the maxmum space rate o change o the temperature at that pont Thereore, the change n temperature when undergong the derental dsplacement dl wll be gven b the scalar product o (T wth the vector dl or dt = (T $dl The uncton dt s tsel a uncton o the space coordnates x,, and z and we smpl requre that t be an ntegrable uncton then we ntegrate dt along the path rom the ntal pont to the nal pont we obtan the result ' dt! ' (T $dl! T( ) # T() The concluson ndcates that the temperature derence between the endng pont and the startng pont depends onl on the derence between the temperature eld uncton evaluated at the ponts n queston and s ndependent o the path chosen between the two ponts n queston Fnall, the ntal pont and the nal pont are one and the same pont as mpled b the crcle on the ntegral smbol below, we ma conclude that the ntegral along a path that closes on tsel wll be (T $dl '! 0!"#$%&'$()#*+,-/,00,1*2)/3*45*6&#,*