Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4.
. Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (, ) hus the mass of 4-gams can be epessed as M (,3) 4 gam he mass s unchanged f the aes s tansfomed,.e., O n geneal we wte M (,3) M (4,3.5) 4gam M (, ) M (, ) (.) he quanttes that ae nvaant unde coodnate tansfomaton ae called scalas, othewse the ae called vectos.
.3 Coodnate ansfomaton (.) vecto can be epessed elatve to the tad as. Relatve to a new tad ``` havng a dffeent oentaton fom that of the tad, the same vecto can be epessed as (.3) It s clea now that we can wte (.4) In mat notaton, Eq.(.4) can be epessed as
(.5) he 3-b- 3 mat n Eq.(.5) s called the tansfomaton mat, o the otaton mat. In summaton notaton we wte (.6),,3 3 s called the decton cosnes of the `-as elatve to the -as. Eample. pont P s epesented n the (,, 3 ) sstem b P(,,3). he same pont s epesented as P(`, `, `3) whee has been otated towad aound -as b 30 o. Fnd the otaton mat and detemne P(`, `, `3) Souton Fom the fgue, the otaton mat s gven b
0.866 0.5 0 0.5 0.866 0 0 0 cos30 cos0 cos90 cos60 cos30 cos90 cos90 cos90 cos0 Usng Eq.(.5) we fnd. 0.866 0.5.37 0.5 0.866 Note that the length of the poston vecto s nvaant unde tansfomaton,.e., 3.74 3 3
Elementa Scala & Vecto Opeatons he Scala Poduct: he scala poduct of two vectos s a scala defned as cos, (.7) he scala poduct obe the commutatve and the dstbutve laws as C he Unt Vectos: (.8) C (.9) he ae vectos havng a magntude of unt and dected along a specfc coodnate as. he unt vecto along the adal decton s gven b e R R R (.9)
If an two unt vectos ae othogonal e e (.0) he Vecto Poduct: he vecto poduct of two vectos s a thd vecto defned as C (.) he component of the vecto C s gven b C, (.) he smbol s called the pemutaton smbol (Lev-Cvta denst) wth 0 - f f f anndesequal toan othe nde thendecesfom an even pemutaton thendecesfom an odd pemutaton (.3)
Usng Eq.(.), the components of C can be evaluated as C 3 3 3 3 3 3 33 he onl nonvanshng tems ae 3 & 3,.e., C Smlal, we fnd 3 3 3 3 3 3 3 3 3 3 C 3 3 C 3 he magntude of the vecto C s defne also as C sn, (.4 Geometcall, C s the aea of the paallelogam defned b the two vectos & )
commutatve not C C C C cos volume of paallelepped ule bac cab C C C
.6 Gadent Opeato (Dectonal Devatve) Let be a scala functon of 3-vaables,.e., (,, ) d d d d hs tells us how vaes as we go a small dstance (d, d, d) awa fom the pont (,, ). Let us ewte the above equaton as d wth o d ds s called the gadent of the functon. d d d
he smbol as s called the del opeato. It can be wtten e (.5) he gadent opeato can opeate on a scala functon, can be used n a scala poduct wth a vecto functon (dvegence), o can be used n a vecto poduct wth a vecto functon (cul),.e., gade dv cul e e (.6) (.7) (.8)
he gadent ponts n the decton of mamum ncease of the functon. he magntude gves the slope along ths mamal decton. s dected pependcula to the suface =constant. he successve opeaton of the gadent poduces the Laplacan opeato (.8) he Laplacan of a scala functon s wtten as (.9)
Integal Calculus b Lne Integal he lne ntegal s epessed as dl a dl whee s a vecto functon and s an nfntesmal dsplacement vecto along a path fom pont a to pont b. If the path foms a closed loop, a ccle s put on the ntegal sgn,.e., dl If the lne ntegal s ndependent on the path followed, the vecto consevatve Suface Integal S ds he suface ntegal s epessed as ds whee s a vecto functon and s an nfntesmal element of aea. gan f the suface s closed we put a ccle on the ntegal sgn, that s ds ds s called he decton of s pependcula to the suface an dected outwad fo closed sufaces and abta fo open sufaces.
Volume Integal he volume ntegal s epessed as V d whee s a vecto functon and d s an nfntesmal element of volume. he Dvegence heoem (he Guass s heoem) v d S ds It states that the ntegal of a dvegence ove a volume s equal to the value of the functon at the bounda. Stoes' heoem S ds dl Snce the bounda lne fo an closed suface shn down to a pont, then S ds 0
Sphecal coodnates (,,) Cuvlnea Coodnates : s the dstance fom the ogn (fom 0 to ) : the pola angle, s the angle between and the -as (fom 0 to ) : the amuthal angle s the angle between the poecton of to the - plane and the -as (fom 0 to ) he elaton between the Catesan coodnates and the sphecal coodnates can be wtten as sn cos sn sn sn cos he unt vectos assocated wth the sphecal coodnates ae elated to the coespondng unt vectos n the Catesan coodnates as e e e sn cos sn sn cos cos cos cos sn sn sn cos
he nfntesmal dsplacement vecto n sphecal coodnates s epessed as dl d d sn d he volume element s epessed as d dl dl dl sn ddd Fo the suface elements we have ds dl dl dd sconstant ds sn dl dl dd sconstant ds3 dl dl sn dd sconstant o fnd the volume of a sphee of adus R we have V d R 0 0 0 sn d d d 4 3 R 3
o fnd the gadent n sphecal coodnates let = (,, ) so ) ( sn d d d dl d () d d d d Equatng the above two equatons we get, sn sn o sn Smlal, one can fnd the dvegence and the cul n sphecal coodnates
sn sn sn sn sn sn sn sn sn he Laplacan s defned as
Clndcal coodnates (,, ) : s the dstance fom the -as (fom 0 to ) : the amuthal angle s the angle between and the -as (fom 0 to ) : the dstance fom the - plane (fom - to ) he elaton between the Catesan coodnates and the clndcal coodnates can be wtten as cos sn he unt vectos assocated wth the clndcal coodnates ae elated to the coespondng unt vectos n the Catesan coodnates as cos sn sn cos
he nfntesmal dsplacement vecto n clndcal coodnates s epessed as dl d d d he volume element s epessed as d dl dl dl ddd Fo the suface elements we have ds dl dl d d ds dl dl dd ds dl dl d d 3 sconstant sconstant sconstant
he Del he Dvegence he Cul sn he Laplacan
Poston, Veloct, and cceleaton vectos: In Catesan coodnates, the poston vecto s wtten as e e e 3 Whle, the veloct, and the acceleaton vectos ae d d d d d v e e e 3 e a e Note that the unt vectos n Catesan coodnates ae constant n tme (both n magntude and decton) In sphecal coodnates, the poston vecto s wtten as e Whle, the veloct vecto s v d e e de
d e ut e sn cos sn sn cos e d d e cos cos sn sn cos sn sn cos sn cos cos cos sn sn sn sn cos e sn e v e e sn e and the acceleaton vecto s dv de de a e e e de sn e cos e sn e sn de but e de cos e and sn e cos e a sn e sn cos e sn sn cos e
In clndcal coodnates, the poston vecto s wtten as e e Whle, the veloct vecto s v but de sn cos e and the acceleaton vecto s dv de a e e usng a de e de e e cos sn v de e e e e e e e e