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MathematicalPrependix
7
Sphericalcoordinates(
r;;
),use
r
tomeanthedistancetotheorigin,incontrasttocylindrical
whereitmeansthedistancetothe
z
-axis. Thetwoanglesaremeasuredfromthepositive
z
-axisand
aroundthe
z
-axisrespectively. Thecoordinatesurfacesarenow
r
=constant(sphere),
=constant
(a cone withapex x atthe origin), , and
=constant (a halfplane with one edge along the
z
-axis).
Ingeography, the latitude and longitude dene apointon thesurfaceoftheEarth. . Thelatitudeis
like
except thatlatitude is measured North and South from the equator r (zero to 90
each), and
is measured strictly South from the North Pole. . Longitude e is measured East and Westfrom the
Greenwichmeridian(zeroto180
ineachcase),and
ismeasuredinonedirectionstartingfromthe
x
-axis(0
2
).Justtokeepyouonyourtoes,somepeopleprefertouse 
+
.
!
Watchoutforvaryingconventionshere.Commonlyinmathbookstheroleof
and
inspherical
coordinatesarereversed,buttheconventionthatI’musingisthestandardinphysicsandengineering.
WhatisnotsoconventionalisthatIchooseistomaketheangle
thesameforpolar,cylindrical,and
sphericalcoordinates.Itistheanglearound the
z
-axis andinthe
x
-
y
plane. Youdohavetowatch
outforconventionsused elsewherebecause you willoftenndthat
isusedforthis angleinpolar
and cylindricalcoordinates. . I’m m tryingtobeconsistenthere,using
forthesameangleinallthree
coordinatesystems.
Aswithrectangularandpolarcoordinates,youcanndtherelationshipsbetweeneachpairof
thesecoordinates. Forrectangularandcylindrical,itisthesameaswithrectangularandpolar,because
z
isthesameforeach. Justusetheequations(0.12)again.
Fortheconversionsbetweenrectangularandsphericalcoordinates,
r
=
p
x
2+
y
2+
x
2
; 
=cos
1
z
r
; 
=tan
1
(
y=x
)
z
=
r
cos
; x
=
r
sin
cos
; y
=
r
sin
sin
(0
:
13)
Thesecondlineofthispairisthesetofequationsthatyouencountermostoften.
Fortheconversionsbetweencylindricalandspherical I’llleaveitasanexercise,problem 0.17,
butyouseeimmediatelythatyouhaveaprobleminnotation:whatis
r
? I’veusedthesamenotation
r
inbothsystems,andthatcanbeconfusing. Thereisnostandardwaytodothis. SomewilldoasI
havedone;otherschoosetheGreeksymbol
forthecylindricalradius;somewillchoose
s
;somewill
choose
R
,andthereareprobablyothernotationsthatIdon’tknowabout.Indthatusing
r
forboth
rarelycausesconfusionthough,becauseitisusuallyclearfromcontextwhichoneyoumean. Onthe
rareoccasionsthatIdoneedtomakethedistinction,Icommonlyuse
r
?
toindicatetheperpendicular
distancetotheaxis(thecylindrical
r
).
x
y
z
r
Fig.0.4
Intwodimensionsyoucandrawthecoordinategridasasetoflinesparallel
tothe
x
and
y
axes,andinthreedimensionsyouwoulddothesamething,with
linesparalleltothethreeaxes.Theonlyplaceyouwillwanttodothishowever
isinyourmind,becausethedrawingonpaperwillbecomesoclutteredthatyou
can’tseewhatyouhavedone.
Insphericalcoordinatesthegridisformedoflinesandarcsofcircles:Hold
thetwocoordinates
=constant,
=constant, and youhavearadial line,
theraystartingfromtheorigin. Forthenextpairofcoordinates,
r
=constant,
=constantistheintersectionofasphereandahalf-planethathasoneedge
alongthe
z
-axis. Itdeneshalfofagreatcircle,andthatisthe
coordinate
curve. Thethirdpairofcoordinatesare
r
=constant,
=constant,andthey
formtheintersectionofasphereandaconewhoseaxisisalong
z
.Thisdenes
the
coordinatecurveasasmallcircleparalleltothe
x
-
y
plane.
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MathematicalPrependix
8
ThismapoftheEarth*showstheprimemeridianatzerolongitude,correspondingtothezero
pointfor
.Similarly,thelinesofconstantlatitudeshowtherelation:latitude=j90
j(NorS).
Fig.0.5
The equations forsimplecurves canlookvery dierentinthe various coordinate systems. . In
rectangularandinplanepolarcoordinatestheequationsforstraightlinesarerespectively
y
=
mx
+
b
and
r
=
a
sec(
0
)
(0
:
14)
Equationsforacirclecancomeinseveralforms
x
2
+
y
2
=
R
2
;
or
r
=
R;
or
r
=
a
cos
;
or
r
2
=
a
+
br
cos(
0
)
(0
:
15)
Anellipseis
x
2
a
2
+
y
2
b
2
=1
;
or
r
=
1
A
+
B
cos
(
B<A
)
;
or
x
=
a
cos
y
=
b
sin
;
or
r
1
+
r
2
=2
a
(0
:
16)
Someofthesearefamiliar. Probablysomearenot.Thelastoneisintwo-centerbipolarcoordinates|
notoneofyourstandards.
You can see e many y examples of f pictures of functions in n rectangularand d polarcoordinates in
The Famous Curves Index. . Insomecases,they y haveJavaversions thatallowyoutoplay with the
parameters.
www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html
AreasandVolumes
Oneofthetraditionalusesforintegrationistocomputesareasandvolumes.Evenwhenthat’snotthe
aim,youstillneedtoknowhowtosetupsuchproblems. Whenyoucomputemomentsofinertiaora
centerofmassit’satoolyouwillneed.
Fortwodimensionstherectangularandpolarcoordinates areallyouneed,alongwiththetwo
pictures.
www.worldatlas.com
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MathematicalPrependix
9
x
y
r
r
Fig.0.6
Thetwoshadedregionsarerectanglesdenedrespectivelybythelinesat
x
,
x
+
x
,
y
,and
y
+
y
orbetween
r
and
r
+
r
,
,
+
. Well,inthepolarcaseonlyapproximatelyrectangular,butin
thelimitthatthetwosidesgotozeroit’strue. Theareaofarectangleistheproductofitssides,so
theareaelementsare
A
=
x
y
!
dA
=
dxdy
and
A
=
rr
!
dA
=
rdrd
(0
:
17)
Theareaofacircleis
Z
dA
=
Z
R
0
rdr
Z
2
0
d
=2
Z
R
0
rdr
=
R
2
A comment t on n notation. . You u may y be accustomed to seeing multiple integrals written dierently,
somethinglike
Z
R
0
Z
2
0
rddr
withthe convention that youworkfrom the insideout. . There’s s nothingwrong with thatnotation,
butInditlessconfusingifIwritethedierentialnexttotheintegralsignanditsassociatedlimits.
Multiplication is commutative and associative,so
rddr
=
drdr
=
drrd
. Youthen n integrate
righttoleft.
Inthreedimensions,thevolumeelementinrectangularcoordinatesisjustasitisintwodimen-
sions,butwithanextrafactorof
z
.Incylindricalcoordinates,itisthesameastwo-dimensionalpolar
exceptforthesameextrafactorforthedistanceinthe
z
-direction
dV
=
dxdydz
dV
=
rdrddz
(0
:
18)
Thesphericalcaserequiresalittlemoredrawing.
x
y
z
r
sin
rd
r
sin
d
Fig.0.7
Fix
r
foramomentandthepieceofareaonthesurfaceofconstant
r
isboundedby(
;
+
d
)
and (
;
+
d
). Again,thisformsarectanglein n the sameway that
dr
and
rd
formarectangle
inplanepolarcoordinateseventhoughallfoursidesarereallyarcsofcircles. Thesidesare
rd
and
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MathematicalPrependix
10
r
sin
d
. Intherstcasetheradiusofthecircleis
r
,withcenterattheorigin. Inthesecondcase
theradiusis
r
sin
withcenteronthe
z
-axis.
dA
=
r
2
sin
dd
d
=sin
dd
(0
:
19)
Whataboutthevolume? Justmultiplythisrectangleby
dr
togetthevolumeoftherectangularbox
withthisareaasabase.Thecombination
d
showsupsomuchithasitsownsymbol,and iscalled
\solidangle".
dV
=
r
2
sin
drdd
x
y
z
r
sin
rd
r
sin
d
dr
Fig.0.8
Thevolumeofasphereisthen
Z
dV
=
Z
R
0
r
2
dr
Z
0
sin
d
Z
2
0
d
=2
Z
R
0
r
2
dr
Z
0
sin
d
=2
Z
R
0
r
2
dr
2=
4
3
R
3
LimitsofIntegration
Averycommonproblemyoufacewithmultipleintegralsistogureoutthelimitsofintegration. Then
whathappenstothoselimitswhenyoudotheintegralsinadierentorder.Thekeyis
drawapictureofthedomainofintegration
and read thelimits from the picture. . Even n in plane rectangular r coordinates s you will get frustrated
withoutapicturetoguideyou.
Thesimplestexample:whatistheareaofarighttriangle,doneasadoubleintegral?Dividethe
areaintolittlepiecesofarea,
A
’s,andaddthemtogether,nallytakingalimitaseachpiecegoes
tozero. Nowyoumustdecide: : inwhatsystematic c orderareyougoingto add the pieces together?
Certainlynotrandomly.
A
x
i
y
j
a
b
y
=
bx=a
Fig.0.9
Ifthetypicalpieceofareais
A
=
x
i
y
j
thenwhenyoudothesum
P
i;j
x
i
y
j
youmust
choosetheorderofsummation. Intherstpictureyouarerstsumming
x
i
X
j
y
j
the
y
’sgofrom0to
bx
i
=a
,sothisis 
x
i
.
bx
i
a
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MathematicalPrependix
11
Thesumon
x
i
becomesanintegralfrom0to
a
,
R
a
0
dxbx=a
.Thesumintheotherorderandusing
thesecondpicturestartswithasumonthe
x
i
andyouhavetoreadthoselimitsfromthepicture,
startingat
ay
j
=b
andendingat
a
.
A
=
Z
a
0
dx
Z
bx=a
0
dy
=
Z
b
0
dy
Z
a
ay=b
dx
=
ab
2
(0
:
20)
0.4Vectors
Thissectionappearsinmostchaptersfromfouron.
Velocity,acceleration,momentum,angularmomentum,electriceld,magneticeld,gravitationaleld,
force,torque,angularvelocity.Andthesearejustvectorsthatappearinthis book.
Thegeneraldenitionofavectoristhatitisanelementofavectorspace,butforhereandnow
thatismoregeneralitythanneeded.Thealgebraofvectorsintwoorthreedimensionsstartsfromthe
pictures
~
A
~
B
~
C
~
A
~
B
~
D
~
C
=
~
A
+
~
B
~
D
=
~
A
~
B
and these e pictures apply whether r you u are talking about gravity y or momentum m or r any y of f the other
vectorslistedabove. Thereis s amathematicaltheoremguaranteeingthis,sayingthatonceyouknow
youareworkingwiththreedimensionalvectors,thentheyareallessentiallythesame. Inmathematical
jargon,isomorphic. Thesamefortwodimensions(orseven). Thismeansthatyoudon’thavetolearn
everythingdierentlyfordierentsortsofvectors|theyallbehavethesameway.
Does avelocityvectorlooklike alinewithanarrowhead attached? ? Veryfewcarstoday y have
arrowsstickingoutofthefrontend,designedtoskewerpedestrians,*buttheimportofthisstatement
aboutvectorsisthatitdoesn’tmatter.Youcanusethesepicturestomodelvelocityormagneticelds
oranyothervectorandyouareguaranteedthattheyallgivethesameresults. Thisiswhyyoustudy
thegeometryofvectorsasasubjectofitsown. Youdon’thavetorelearnitwhenyouencounterthe
nextsortofvector.
~
A
~
B
~
A
~
B
~
A
~
B
=
AB
sin
~
A
.
~
B
=
AB
cos
(0
:
21)
Thevectorproduct(crossproduct)oftwovectors
~
A
and
~
B
isanothervectorperpendiculartothese
twoandhavingmagnitude
AB
sin
,where
istheindicatedanglebetweenthem. Thescalarproduct
(dotproduct)isthescalar
AB
cos
.HereIamusingtheconventionthatthemagnitudeofthevector
~
A
istheletter
A
. Seetheproblems0.24{0.32forsomereviewandpractice.
Thevectorproductas juststated is ambiguousbecause therearetwodirectionsthatareper-
pendiculartothetwoinputvectors,andtheyareinoppositedirections. Whichonetopickischosen
byconvention: theright-handrule.
* butndapictureofa1950-eraOldsmobile!
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MathematicalPrependix
12
Whyarethesethecorrectdenitionsforproductsofvectors? Whyacosineinonecaseanda
sineintheother?Probablywhenthesubjectwasbeinginventedtheseweren’ttherstattemptsmade
indeningproducts,butthesearetheonesthathavegoodproperties|especiallythedistributivelaw.
5
.
(3+4)=5
.
3+5
.
4=35,andforvectors
~
A
.
~
B
+
~
C
=
~
A
.
~
B
+
~
A
.
~
C;
~
A
~
B
+
~
C
=
~
A
~
B
+
~
A
~
C
(0
:
22)
Whatabouttheothersortsof multiplicationlaws: : commutative e andassociative? ? 5
.
4=4
.
5
and
~
A
.
~
B
=
~
B
.
~
A;
but
~
A
~
B
~
B
~
A
so the commutativerule works forthedotproduct butnotforthecross product. . Itanti-commutes
instead.Fortheassociativelaw,5
.
(3
.
4)=(5
.
3)
.
4=60,but
~
A
.
(
~
B
.
~
C
) doesn’tmakeanysense. . Thedotproductisbetweenvectors.
Forthecrossproduct,
~
A
(
~
B
~
C
)makessense,butitisnotassociative,andyoucanseethateasily
bywritingasimpleexample.
^
x
(^
x
^
y
)=^
x
^
z
^
y;
but
(^
x
^
x
)^
y
=0^
y
=0
TheclosestthatthecrossproductcomestoassociativityistheJacobiidentity,
~
A
(
~
B
~
C
)+
~
B
(
~
C
~
A
)+
~
C
(
~
A
~
B
)=0
(0
:
23)
Itdoesn’tshowupveryofteninordinaryvectoralgebra,butyoucannditusedneartheendofchapter
four.
Ausefulidentityinvolvingthistriplecrossproductis
~
A
~
B
~
C
=
~
B
~
A
.
~
C
~
C
~
A
.
~
B
(0
:
24)
(Dotheselasttwoequationsworkwiththesimpleexample^
x
(^
x
^
y
)ofafewlinesback?)
Andthetriplescalarproductobeystheidentities
~
A
.
~
B
~
C
=
~
A
~
B
.
~
C
interchangedotandcross
~
A
.
~
B
~
C
=
~
B
.
~
C
~
A
=
~
C
.
~
A
~
B
cyclicpermutationoffactors
(0
:
25)
Bases,UnitVectors
Thegeometryofvectorscanbecomecumbersometomanipulate,especiallywhenyouhavemanyvectors
involved. Thetechniqueofcomponentsisawaytoturnmuchofthegeometryintoalgebraandeven
arithmetic. Itisto o vectors whatanalytic geometry is toplanegeometry. . In n rectangularcoordinates
thesystemshouldbefamiliar. Intwodimensionspicktwovectors,oneparalleltothe
x
-axisandone
parallel tothe
y
-axis. Callthem ^
x
and ^
y
,thepairformingabasis,andmake themhave magnitude
one. Notonemeteroronesecond,justonedimensionless.
x
y
^x
^
y
^
x
^
y
~
A
A
x
^
x
A
y
^
y
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MathematicalPrependix
13
Youcanwriteanyvectorintheplaneasthesumoftwoothervectors,oneparallelto^
x
andoneparallel
to ^
y
. Theformerisamultipleof ^
x
(positiveornegative)andthelatterisamultipleof ^
y
. Thetwo
componentsofthevector
~
A
arethetwonumbers
A
x
and
A
y
,withunitsasappropriate.
Thepointofwritingasinglevectorintermsoftwoothervectors(threeothersinthreedimensions)
istochangegeometryintoalgebraandarithmetic.Itistochangetheproblemofdierentiatingvectors
intothemorefamiliarproblemofdierentiatingordinaryfunctions.Thebasisvectorsyouwillencounter
inthisbookaremostlyorthogonalandnormalizedtoone. Thatisaconveniencenotanecessity,and
inothercontextsyoucanmakeotherchoicestodeneabasis.
Thedotproductoftwovectorsiseasytocomputeintermsofthesebasisvectors.
~
A
.
~
B
=
A
x
^
x
+
A
y
^
y
+
A
z
^
z
.
B
x
^
x
+
B
y
^
y
+
B
z
^
z
=
A
x
B
x
^
x
.
^
y
+
A
x
B
y
^
x
.
^
y
+
A
x
B
z
^
x
.
^
z
+
A
y
B
x
^
y
.
^
y
+
A
y
B
y
^
y
.
^
y
+
A
y
B
z
^
y
.
^
z
+
A
z
B
x
^
z
.
^
y
+
A
z
B
y
^
z
.
^
y
+
A
z
B
z
^
z
.
^
z
=
A
x
B
x
+
A
y
B
y
+
A
z
B
z
(0
:
26)
The simplicityappearsbecausethebasisvectorsareorthonormal. . Thatis,eachhas s unitmagnitude
andtheyaremutuallyperpendicular.
Doestheproductruleforderivativesapplytotheproductofvectors?Yes,andyoucanseewhy
bydierentiatingthecomponentform:
d
dt
~
A
.
~
B
=
dA
x
dt
B
x
+
A
x
dB
x
dt
+=
d
~
A
dt
.
~
B
+
~
A
.
d
~
B
dt
Thesameanalysisappliestothecrossproduct.
IndexNotation
Aconvenientandpowerfulnotationtomanipulatethecomponentsofvectors:Denotethebasisvectors
byauniednotation,
~e
1
,
~e
2
,and
~e
3
insteadofrespectively ^
x
,^
y
,and^
z
. Youthenwritethevectorsas
~
A
=
A
1
~e
1
+
A
2
~e
2
+
A
3
~e
3
and
~
B
=
B
1
~e
1
+
B
2
~e
2
+
B
3
~e
3
Thatthesevectors,
~e
i
,areorthogonalandunitvectorstranslatesto
~e
1
.
~e
2
=0
;
~e
3
.
~e
3
=1
;
andallothersuchcombinations
Introduceastandardnotation:
ij
=
1 if
i
=
j
0 if
i
6=
j
then
~e
i
.
~e
j
=
ij
(0
:
27)
wheretheindices
i
and
j
takeonanyandallofthevalues1,2,3. ThisiscalledtheKroneckerdelta
symbol.
The dot productisexactlythesameas inEq.(0.26),onlythe nalresult justhas subscripts
1
;
2
;
3insteadof
x;y;x
. Itdoesn’tseemtobeworthallthetroubledoesit? Itis. Firstaversionof
thecalculationinthisnotation.
~
A
.
~
B
=
X
i
A
i
~e
i
.
X
j
B
j
~e
j
=
X
i
X
j
A
i
B
j
~e
i
.
~e
j
=
X
i
X
j
A
i
B
j
ij
MathematicalPrependix
14
Nowdothesumon
j
X
j
B
j
ij
=
B
i
Remember:
ij
=0forthetwotermswhere
j
6=
i
Therestis
P
i
A
i
B
i
,exactlyasinEq.(0.26).
Thesummationconventionwas introducedby Einsteinwhen itbecamecleartohim thatin
manipulationswiththisnotationforthecomponents,asummationsymbol
e.g.
P
i
)alwaysappeared
wheneveranindexwasrepeatedinasingleterm. Sowhybothertowriteit? Theconventionisthat
wheneveranindexisrepeatedinasingletermitistobesummed.Theimmediatelyprecedingcalculation
isthenshortenedto
~
A
.
~
B
=
A
i
~e
i
.
B
j
~e
j
=
A
i
B
j
~e
i
.
~e
j
=
A
i
B
j
ij
=
A
i
B
i
(0
:
28)
Thisconvention alsomeansthatif you encounteranexpression with three identicalindices,suchas
A
jk
=
B
i
C
ik
C
ji
thengobackandndyourmistake. Thiscan’thappen.
Withthisconventionyouhaveforexample,
ij
v
j
=
v
i
.
Howdoyou handle cross productsin this notation? ? You u need to introduceanotherpiece of
notation.
~e
i
.
~e
j
~e
k
=
e
ijk
thealternatingsymbol
(0
:
29)
Thisis1iftheindices
i;j;k
aredierentfromeach other,i.e.apermutation of123. . It’s+1for
123 and any cyclic permutation of 123, i.e.231 and 312. . Interchange e any y two o (e.g.132) and d you
get  1. . If f any two indices are the same then n this s is zero. . You u can n easily y verify y that
e
ijk
equals
1
2
(
i
j
)(
j
k
)(
k
i
),thoughthisidentityismoreofacuriositythananythingparticularlyuseful.
~
A
~
B
=
~e
i
e
ijk
A
j
B
k
;
e
ijk
e
imn
=
jm
kn
jn
km
(0
:
30)
Theseareeasytocheck. Atworstyoucanjustlookatallthepossiblecases. Elegantderivationscan
comelater.ThesecondisequivalenttoEq.(0.24).
0.5Dierentiation
There’snochapterinthisbookinwhichthisisnotused.
Somuchofthissubjectdependsonknowingwhatderivativesareandhowtomanipulatethemthatit’s
worthspendingsomespacetoreviewthesubject(andmaybeintheprocesstointroducesomeideas
youhaven’tseen).
Thestandarddenitionofthederivativeofafunctionofonevariableis
df
dx
(
x
)= lim
x
!0
f
(
x
+
x
f
(
x
)
x
(0
:
31)
The\prime"notation,
f
0
(
x
),forthederivativeisusefultoo,butifthat’stheonlynotationyouuseitcan
hinderyoutothepointofincapacity. TheLeibnitznotationofEq.(0.31)lendsitselftomanipulation
whilethe
f
0
doesnot.Foranimmediateexample,themostcommonmethodyouwillusetodierentiate
anythingisthechainrule,andthatisremarkablyobscureinonenotationwhilebeingquiteintuitivein
theother.
h
(
x
)=
f
g
(
x
)
!
h
0
(
x
)=
f
0
(
g
)
g
0
or
dh
dx
=
df
dg
dg
dx
(0
:
32)
MathematicalPrependix
15
Howdoyouderivethis?ThedenitionofthederivativeandtheLeibnitzformdictatethemanipulations
youmustuse. Startfromthedenition:
h
(
x
+
x
h
(
x
)
x
=
h
(
x
+
x
h
(
x
)
g
(
x
+
x
g
(
x
)
.
g
(
x
+
x
g
(
x
)
x
(0
:
33)
orinashorterform,
h
x
=
h
g
.
g
x
As
x
!0,theincrementin
g
mustapproachzeroalso,asotherwiseitsderivativewouldnotexistand
thechainrulewouldnotapply.* ThesecondfactorinEq.(0.33)becomes,inthelimitthat
x
!0,
thederivative
dg=dx
. Also,becauseinthesamelimit
g
=
g
(
x
+
x
g
(
x
)!0,therstfactor
becomes
df=dg
,andthatendsthederivation.Acommonwaytowritethecompositionoffunctionsis
touse,insteadof
h
(
x
)=
f
g
(
x
)
,thenotation
h
=
f
g
.
Apossiblyunfamiliarruleoccurswhenyouencounterthederivativeofafunctioninwhichthe
independentvariableappearsinseveralunrelatedplaces. Forexample,
f
1
(
x
)=
x
x
;
or
f
2
(
x
)=
Z
x
0
t
p
x
tdt
(0
:
34)
Thinkofeitheroftheseasafunction
f
(
x;x
),wheretherst
x
isoneofthetwo
x
’sin
x
x
andthe
second
x
istheother. Nowstartfromthedenitionandmanipulate,addingandsubtractingthesame
term.
f
(
x
+
x;x
+
x
f
(
x;x
)
x
=
f
(
x
+
x;x
+
x
f
(
x;x
+
x
)+
f
(
x;x
+
x
f
(
x;x
)
x
=
f
(
x
+
x;x
+
x
f
(
x;x
+
x
)
x
+
f
(
x;x
+
x
f
(
x;x
)
x
As
x
!0,thesecondtermbecomesthedenitionofthederivativeof
f
withrespecttothesecond
x
. Inthesamelimitthersttermbecomesthederivativewithrespecttotherst
x
. (Thisassumes
thatall the derivatives are continuous,so thatas s 
x
! 0 the factthat the rst dierentiation is
approachedfromnear
x
insteadofat
x
willnotmatter.)
Anotationyouwillseeinsomecalculustextscanbeusefulhere.
d
dx
f
(
x;x
)=
D
1
f
(
x;x
)+
D
2
f
(
x;x
)
(0
:
35)
Here
D
1
meansdierentiatewithrespecttotherstargumentand
D
2
withrespecttothesecond.
Thisderivationshowsthatif
x
shows upintwodierentplaces,youdierentiatewithrespect
tooneofthemandthenwithrespecttothenextoneandaddthetworesults. Thefamiliarproduct
ruleisaspecialcaseofthis.
d
dx
f
(
x
)
g
(
x
)=
df
dx
g
+
f
dg
dx
(0
:
36)
Inthelessfamiliarcases,Eq.(0.34),
df
1
dx
=
d
dx
x
x
=
xx
x
1
+ln
xx
x
=
x
x
(1+ln
x
)
* Whatisthederivativeofthefunction\
h
(
x
)=
x
"atthepoint
x
=0? Oneofcourse. Nowlet
f
(
x
)=1
=x
and
g
(
x
)=1
=x
,thenitlookslike
h
=
f
g
,butneither
f
0
(0)nor
g
0
(0)exist.
MathematicalPrependix
16
and
df
2
dx
=0+
1
2
Z
x
0
t
p
x
t
dt
ParametrizedDierentiation
Onceyouseetheexplanationofthisitseemsremarkablesimple,butifyouencounterthephenomenon
inanunfamiliarcontextyoumaynotthinkofit.
If
u
isafunctionoftimeand
v
isafunctionoftime,whatis
du=dv
?
Attimes
t
and
t
+
t
u
=
u
(
t
+
t
u
(
t
) and d 
v
=
v
(
t
+
t
v
(
t
)
then
u
v
=
u=
t
v=
t
andthelimitas
t
!0is
du
dv
=
du=dt
dv=dt
also
=
dt=dv
dt=du
(0
:
37)
Example
Inpolarcoordinates,
x
=
r
cos
and
y
=
r
sin
. Whatis
dy=dx
asyougoaroundacircle? Here
r
isconstant,so
dy
dx
=
dy=d
dx=d
=
r
cos
r
sin
= cot
Itis easytocheckthisatangles such as
=0or
=
4etc. Thistechniquewillshowupin n amore
complicatedexampleinchapternine,relativity,whencomputingrelativevelocityandacceleration.
0.6Velocity,Acceleration
Thissectionappearsinmostchaptersfromfouron.
Thedenitionsofvelocityandaccelerationare
~v
=
d~r
dt
=
_
~r
and
~a
=
d~v
dt
=
_
~v
Thisdoesn’tdependonyourcoordinatesystem,beitrectangular,polar,spherical,oroblatespheroidal.
Computingthemindierentcoordinatesystemcangettechnicalthough. Fortunatelythreecommon
cases (rectangular, , polar, cylindrical) ) areeasy. . Sphericalis s more involved but still manageable,and
therestarerareenoughthatyoucanlearnthemwhenorifyouneedthem. The\dot"notationover
theletterislike the
0
socommonincalculustexts,butithasaparticularmeaning. Italways means
dierentiationwithrespecttotime.
x
y
^
x
^
y
r
^
r
^
Fig.0.10
r
=
p
x
2
+
y
2
tan
=
y=x
Rectangularisfamiliar
~r
=
x
^
x
+
y
^
y
+
z
^
z;
so
~v
_
x
^
x
+_
y
^
y
_
z
^
z
and
~a
=
x
^
x
+
y
^
y
+
z
^
z
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