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Title: mathes formulas
Description: esy way to learn mathes formulas. a group of mathes formulas.
Description: esy way to learn mathes formulas. a group of mathes formulas.
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1300 Math Formulas
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Preface
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qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíìÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä=
ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI=
ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ=
Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã=
kìãÄÉê= pÉíëI= ^äÖÉÄê~I= dÉçãÉíêóI= qêáÖçåçãÉíêóI= j~íêáÅÉë=
~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI=
aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK==
qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ=
ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí=
Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK===
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Contents
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1 krj_bo=pbqp=
NKN= pÉí=fÇÉåíáíáÉë==1=
NKO= pÉíë=çÑ=kìãÄÉêë==5=
NKP= _~ëáÅ=fÇÉåíáíáÉë==7=
NKQ= `çãéäÉñ=kìãÄÉêë==8=
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2 ^idb_o^=
OKN= c~ÅíçêáåÖ=cçêãìä~ë==12=
OKO= mêçÇìÅí=cçêãìä~ë==13=
OKP= mçïÉêë==14=
OKQ= oççíë==15=
OKR= içÖ~êáíÜãë==16=
OKS= bèì~íáçåë==18=
OKT= fåÉèì~äáíáÉë==19=
OKU= `çãéçìåÇ=fåíÉêÉëí=cçêãìä~ë==22=
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3 dbljbqov=
PKN= oáÖÜí=qêá~åÖäÉ==24=
PKO= fëçëÅÉäÉë=qêá~åÖäÉ==27=
PKP= bèìáä~íÉê~ä=qêá~åÖäÉ==28=
PKQ= pÅ~äÉåÉ=qêá~åÖäÉ==29=
PKR= pèì~êÉ==33=
PKS= oÉÅí~åÖäÉ==34=
PKT= m~ê~ääÉäçÖê~ã==35=
PKU= oÜçãÄìë==36=
PKV= qê~éÉòçáÇ==37=
PKNM= fëçëÅÉäÉë=qê~éÉòçáÇ==38=
PKNN= fëçëÅÉäÉë=qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==40=
PKNO= qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==41=
iii
PKNP= háíÉ==42=
PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43=
PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45=
PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46=
PKNT= oÉÖìä~ê=eÉñ~Öçå==47=
PKNU= oÉÖìä~ê=mçäóÖçå==48=
PKNV= `áêÅäÉ==50=
PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53=
PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54=
PKOO= `ìÄÉ==55=
PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56=
PKOQ= mêáëã==57=
PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58=
PKOS= oÉÖìä~ê=móê~ãáÇ==59=
PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61=
PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62=
PKOV= mä~íçåáÅ=pçäáÇë==63=
PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66=
PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68=
PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69=
PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70=
PKPQ= péÜÉêÉ==72=
PKPR= péÜÉêáÅ~ä=`~é==72=
PKPS= péÜÉêáÅ~ä=pÉÅíçê==73=
PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74=
PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75=
PKPV= bääáéëçáÇ==76=
PKQM= `áêÅìä~ê=qçêìë==78=
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QKN= o~Çá~å=~åÇ=aÉÖêÉÉ=jÉ~ëìêÉë=çÑ=^åÖäÉë==80=
QKO= aÉÑáåáíáçåë=~åÇ=dê~éÜë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==81=
QKP= páÖåë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==86=
QKQ= qêáÖçåçãÉíêáÅ=cìåÅíáçåë=çÑ=`çããçå=^åÖäÉë==87=
QKR= jçëí=fãéçêí~åí=cçêãìä~ë==88=
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QKS= oÉÇìÅíáçå=cçêãìä~ë==89=
QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90=
QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90=
QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91=
QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92=
QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93=
QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94=
QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94=
QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95=
QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97===
QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98=
QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99=
QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102=
QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103=
QKOM= qêáÖçåçãÉíêáÅ=bèì~íáçåë==106=
QKON= oÉä~íáçåë=íç=eóéÉêÄçäáÅ=cìåÅíáçåë==106=
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RKN= aÉíÉêãáå~åíë==107=
RKO= mêçéÉêíáÉë=çÑ=aÉíÉêãáå~åíë==109=
RKP= j~íêáÅÉë==110=
RKQ= léÉê~íáçåë=ïáíÜ=j~íêáÅÉë==111=
RKR= póëíÉãë=çÑ=iáåÉ~ê=bèì~íáçåë==114=
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6 sb`qlop=
SKN= sÉÅíçê=`ççêÇáå~íÉë==118=
SKO= sÉÅíçê=^ÇÇáíáçå==120=
SKP= sÉÅíçê=pìÄíê~Åíáçå==122=
SKQ= pÅ~äáåÖ=sÉÅíçêë==122=
SKR= pÅ~ä~ê=mêçÇìÅí==123=
SKS= sÉÅíçê=mêçÇìÅí==125=
SKT= qêáéäÉ=mêçÇìÅí=127=
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TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130=
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TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131=
TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139=
TKQ= `áêÅäÉ==149=
TKR= bääáéëÉ==152=
TKS= eóéÉêÄçä~==154=
TKT= m~ê~Äçä~==158=
TKU= qÜêÉÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==161=
TKV= mä~åÉ==165=
TKNM= píê~áÖÜí=iáåÉ=áå=pé~ÅÉ==175=
TKNN= nì~ÇêáÅ=pìêÑ~ÅÉë==180=
TKNO= péÜÉêÉ==189=
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UKN= cìåÅíáçåë=~åÇ=qÜÉáê=dê~éÜë==191=
UKO= iáãáíë=çÑ=cìåÅíáçåë==208=
UKP= aÉÑáåáíáçå=~åÇ=mêçéÉêíáÉë=çÑ=íÜÉ=aÉêáî~íáîÉ==209=
UKQ= q~ÄäÉ=çÑ=aÉêáî~íáîÉë==211=
UKR= eáÖÜÉê=lêÇÉê=aÉêáî~íáîÉë==215=
UKS= ^ééäáÅ~íáçåë=çÑ=aÉêáî~íáîÉ==217=
UKT= aáÑÑÉêÉåíá~ä==221=
UKU= jìäíáî~êá~ÄäÉ=cìåÅíáçåë==222=
UKV= aáÑÑÉêÉåíá~ä=léÉê~íçêë==225=
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VKN= fåÇÉÑáåáíÉ=fåíÉÖê~ä==227=
VKO= fåíÉÖê~äë=çÑ=o~íáçå~ä=cìåÅíáçåë==228=
VKP= fåíÉÖê~äë=çÑ=fêê~íáçå~ä=cìåÅíáçåë==231=
VKQ= fåíÉÖê~äë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==237=
VKR= fåíÉÖê~äë=çÑ=eóéÉêÄçäáÅ=cìåÅíáçåë==241=
VKS= fåíÉÖê~äë=çÑ=bñéçåÉåíá~ä=~åÇ=içÖ~êáíÜãáÅ=cìåÅíáçåë==242=
VKT= oÉÇìÅíáçå=cçêãìä~ë==243=
VKU= aÉÑáåáíÉ=fåíÉÖê~ä==247=
VKV= fãéêçéÉê=fåíÉÖê~ä==253=
VKNM= açìÄäÉ=fåíÉÖê~ä==257=
VKNN= qêáéäÉ=fåíÉÖê~ä==269=
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VKNO= iáåÉ=fåíÉÖê~ä==275=
VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285=
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NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==295=
NMKO= pÉÅçåÇ=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==298=
NMKP= pçãÉ=m~êíá~ä=aáÑÑÉêÉåíá~ä=bèì~íáçåë==302=
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NNKN= ^êáíÜãÉíáÅ=pÉêáÉë==304=
NNKO= dÉçãÉíêáÅ=pÉêáÉë==305=
NNKP= pçãÉ=cáåáíÉ=pÉêáÉë==305=
NNKQ= fåÑáåáíÉ=pÉêáÉë==307=
NNKR= mêçéÉêíáÉë=çÑ=`çåîÉêÖÉåí=pÉêáÉë==307=
NNKS= `çåîÉêÖÉåÅÉ=qÉëíë==308=
NNKT= ^äíÉêå~íáåÖ=pÉêáÉë==310=
NNKU= mçïÉê=pÉêáÉë==311=
NNKV= aáÑÑÉêÉåíá~íáçå=~åÇ=fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë==312=
NNKNM= q~óäçê=~åÇ=j~Åä~ìêáå=pÉêáÉë==313=
NNKNN= mçïÉê=pÉêáÉë=bñé~åëáçåë=Ñçê=pçãÉ=cìåÅíáçåë==314=
NNKNO= _áåçãá~ä=pÉêáÉë==316=
NNKNP= cçìêáÉê=pÉêáÉë==316=
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NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318=
NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319=
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qÜáë=é~ÖÉ=áë=áåíÉåíáçå~ääó=äÉÑí=Ää~åâK=
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Chapter 1
Number Sets
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1
...
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2
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4
...
^ ⊂ f=
^ ⊂ ^=
^ = _ =áÑ= ^ ⊂ _ =~åÇ= _ ⊂ ^
...
NUMBER SETS
=
=====
=
Figure 1
...
=
7
...
=
`çããìí~íáîáíó=
^∪_ = _∪^=
^ëëçÅá~íáîáíó=
^ ∪ (_ ∪ ` ) = (^ ∪ _ ) ∪ ` =
fåíÉêëÉÅíáçå=çÑ=pÉíë=
` = ^ ∪ _ = {ñ ö ñ ∈ ^ ~åÇ ñ ∈ _} =
=
=
=====
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Figure 2
...
=
10
...
NUMBER SETS
11
...
=
13
...
aáëíêáÄìíáîáíó=
^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I=
^ ∩ (_ ∪ ` ) = (^ ∩ _ ) ∪ (^ ∩ ` ) K=
fÇÉãéçíÉåÅó=
^ ∩ ^ = ^ I==
^∪^ = ^=
açãáå~íáçå=
^ ∩ ∅ = ∅ I=
^∪f= f=
fÇÉåíáíó=
^ ∪ ∅ = ^ I==
^∩f= ^
=
15
...
17
...
`çãéäÉãÉåí=
^′ = {ñ ∈ f ö ñ ∉ ^}
=
`çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå
^ ∪ ^′ = f I==
^ ∩ ^′ = ∅ =
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aÉ=jçêÖ~å∞ë=i~ïë
(^ ∪ _ )′ = ^′ ∩ _′ I==
(^ ∩ _ )′ = ^′ ∪ _′ =
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aáÑÑÉêÉåÅÉ=çÑ=pÉíë
` = _ y ^ = {ñ ö ñ ∈ _ ~åÇ ñ ∉ ^} =
=
3
CHAPTER 1
...
=
19
...
_ y ^ = _ ∩ ^′
21
...
^ y _ = ^ =áÑ= ^ ∩ _ = ∅
...
=
23
...
^′ = f y ^
25
...
NUMBER SETS
1
...
27
...
=
29
...
k~íìê~ä=åìãÄÉêëW=k=
tÜçäÉ=åìãÄÉêëW= kM =
fåíÉÖÉêëW=w=
mçëáíáîÉ=áåíÉÖÉêëW= w + =
kÉÖ~íáîÉ=áåíÉÖÉêëW= w − =
o~íáçå~ä=åìãÄÉêëW=n=
oÉ~ä=åìãÄÉêëW=o==
`çãéäÉñ=åìãÄÉêëW=`==
=
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k~íìê~ä=kìãÄÉêë
`çìåíáåÖ=åìãÄÉêëW k = {NI OI PI K} K=
tÜçäÉ=kìãÄÉêë
`çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= k M = {MI NI OI PI K} K=
fåíÉÖÉêë
tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW=
w + = k = {NI OI PI K}I=
w − = {KI − PI − OI − N} I=
w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K=
o~íáçå~ä=kìãÄÉêë
oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==
~
n = ñ ö ñ = ~åÇ ~ ∈ w ~åÇ Ä ∈ w ~åÇ Ä ≠ M K=
Ä
fêê~íáçå~ä=kìãÄÉêë
kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK
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5
CHAPTER 1
...
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32
...
k⊂ w⊂n⊂ o ⊂ `=
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===
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Figure 5
...
NUMBER SETS
1
...
^ÇÇáíáîÉ=fÇÉåíáíó=
~+M=~ =
=
35
...
`çããìí~íáîÉ=çÑ=^ÇÇáíáçå=
~ +Ä= Ä+~ =
37
...
aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå=
~ − Ä = ~ + (− Ä) =
=
39
...
41
...
jìäíáéäáÅ~íáîÉ=fÇÉåíáíó=
~ ⋅N = ~ =
jìäíáéäáÅ~íáîÉ=fåîÉêëÉ=
N
~ ⋅ = N I= ~ ≠ M
~
=
jìäíáéäáÅ~íáçå=qáãÉë=M
~ ⋅M = M
=
`çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=
~ ⋅Ä = Ä⋅~
=
=
7
CHAPTER 1
...
44
...
aÉÑáåáíáçå=çÑ=aáîáëáçå=
~
N
= ~⋅ =
Ä
Ä
=
=
=
1
...
=
47
...
áN = á =
á O = −N =
á P = −á =
áQ = N=
áR = á =
á S = −N =
á T = −á =
áU = N =
á Q å +N = á =
á Q å+ O = −N =
á Q å + P = −á =
á Qå = N =
ò = ~ + Äá =
`çãéäÉñ=mä~åÉ=
=
8
CHAPTER 1
...
=
49
...
=
51
...
=
53
...
~ = ê Åçë ϕ I= Ä = ê ëáå ϕ ==
=
9
CHAPTER 1
...
55
...
=
mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=
~ + Äá = ê(Åçë ϕ + á ëáå ϕ) =
jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=
fÑ= ~ + Äá =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå=
ê = ~ O + ÄO =EãçÇìäìëFI==
Ä
ϕ = ~êÅí~å =E~êÖìãÉåíFK=
~
=
57
...
mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
ò N ⋅ ò O = êN (Åçë ϕN + á ëáå ϕN ) ⋅ êO (Åçë ϕO + á ëáå ϕO ) =
= êNêO [Åçë(ϕN + ϕO ) + á ëáå(ϕN + ϕO )] =
`çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
|||||||||||||||||||||
ê(Åçë ϕ + á ëáå ϕ) = ê[Åçë(− ϕ) + á ëáå(− ϕ)] =
=
59
...
NUMBER SETS
60
...
=
62
...
=
64
...
1 Factoring Formulas
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oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==
k~íìê~ä=åìãÄÉêW=å=
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65
...
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67
...
=
69
...
=
71
...
~ O − ÄO = (~ + Ä)(~ − Ä) =
~ P − ÄP = (~ − Ä)(~ O + ~Ä + ÄO ) =
~ P + ÄP = (~ + Ä)(~ O − ~Ä + ÄO ) =
~ Q − ÄQ = (~ O − ÄO )(~ O + ÄO ) = (~ − Ä)(~ + Ä)(~ O + ÄO ) =
~ R − ÄR = (~ − Ä)(~ Q + ~ P Ä + ~ O ÄO + ~ÄP + ÄQ ) =
~ R + ÄR = (~ + Ä)(~ Q − ~ P Ä + ~ O ÄO − ~ÄP + ÄQ ) =
fÑ=å=áë=çÇÇI=íÜÉå=
~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K − ~Äå −O + Äå −N ) K==
fÑ=å=áë=ÉîÉåI=íÜÉå==
~ å − Äå = (~ − Ä)(~ å −N + ~ å −O Ä + ~ å −P ÄO + K + ~Äå−O + Äå −N ) I==
12
CHAPTER 2
...
2 Product Formulas
73
...
=
75
...
=
77
...
=
79
...
=
81
...
ALGEBRA
2
...
=
83
...
=
(~Ä)ã = ~ ã Äã =
85
...
=
87
...
=
~ M = N I= ~ ≠ M =
89
...
=
=
=
=
=
14
CHAPTER 2
...
4 Roots
=
91
...
=
å
~ ã Ä = åã ~ ã Äå =
93
...
=
95
...
=
(~ )
å
ã
( ~)
å
å
é
= å ~ ãé =
=~=
åé
97
...
=
å
~ =~ =
99
...
=
ã
å
ã
~ = ãå ~ =
( ~)
å
~ ãé =
ã
= å ~ã =
15
CHAPTER 2
...
=
~± Ä =
102
...
=
=
=
2
...
105
...
107
...
109
...
ALGEBRA
110
...
äçÖ ~ å ñ = äçÖ ~ ñ =
å
=
äçÖ Å ñ
112
...
äçÖ ~ Å =
=
äçÖ Å ~
=
114
...
içÖ~êáíÜã=íç=_~ëÉ=NM=
äçÖ NM ñ = äçÖ ñ =
=
116
...
äçÖ ñ =
äå ñ = MKQPQOVQ äå ñ =
äå NM
=
N
118
...
ALGEBRA
2
...
iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=
Ä
~ñ + Ä = M I= ñ = − K==
~
=
120
...
aáëÅêáãáå~åí=
a = ÄO − Q~Å =
=
122
...
~ñ O + Äñ = M I= ñ N = M I= ñ O = − K=
~
=
Å
124
...
`ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==
ó P + éó + è = M I==
O
18
CHAPTER 2
...
7 Inequalities
s~êá~ÄäÉëW=ñI=óI=ò=
~ I ÄI ÅI Ç
I=ãI=å=
oÉ~ä=åìãÄÉêëW=
~N I ~ O I ~ P I KI ~ å
aÉíÉêãáå~åíëW=aI= añ I= aó I= aò ==
=
=
126
...
ALGEBRA
127
...
=
129
...
=
131
...
=
133
...
fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå=
~ Ä
> K=
ã ã
=
135
...
fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= < K=
ã ã
=
137
...
fÑ= M < ~ < Ä =~åÇ= å < M I=íÜÉå= ~ å > Äå K=
=
139
...
I==
~Ä ≤
O
ïÜÉêÉ= ~ > M =I= Ä > M X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= ~ = Ä K==
=
N
141
...
ALGEBRA
142
...
fÑ= ~ñ + Ä > M =~åÇ= ~ > M I=íÜÉå= ñ > − K=
~
=
Ä
144
...
~ñ O + Äñ + Å > M =
=
=
~ > M=
=
=
=
=
a>M=
=
=
=
a=M=
=
=
=
a
=
ñ < ñ N I= ñ > ñ O =
=
ñ N < ñ I= ñ > ñ N =
=
=
−∞< ñ <∞=
=
21
~
=
ñN < ñ < ñ O =
=
ñ ∈∅ =
=
=
ñ ∈∅ =
=
=
=
CHAPTER 2
...
=
147
...
=
149
...
=
fÑ= ñ < ~ I=íÜÉå= − ~ < ñ < ~ I=ïÜÉêÉ= ~ > M K=
fÑ= ñ > ~ I=íÜÉå= ñ < −~ =~åÇ= ñ > ~ I=ïÜÉêÉ= ~ > M K=
fÑ= ñ O < ~ I=íÜÉå= ñ < ~ I=ïÜÉêÉ= ~ > M K=
fÑ= ñ O > ~ I=íÜÉå= ñ > ~ I=ïÜÉêÉ= ~ > M K=
151
...
=
=
=
2
...
dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=
åí
ê
^ = ` N + =
å
=
22
CHAPTER 2
...
páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=
fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë=
Ñçêãìä~=ëáãéäáÑáÉë=íçW=
í
^ = `(N + ê ) K=
=
155
...
1 Right Triangle
=
iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä=
eóéçíÉåìëÉW=Å=
^äíáíìÇÉW=Ü=
jÉÇá~åëW= ã ~ I= ã Ä I= ã Å =
^åÖäÉëW= α I β =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
^êÉ~W=p=
=
=
=
=
Figure 8
...
α + β = VM° =
=
24
CHAPTER 3
...
ëáå α =
~
= Åçë β =
Å
=
158
...
í~å α =
~
= Åçí β =
Ä
=
Ä
160
...
ëÉÅ α = = Åçë ÉÅ β =
Ä
=
162
...
móíÜ~ÖçêÉ~å=qÜÉçêÉã=
~ O + ÄO = Å O =
=
164
...
=
25
CHAPTER 3
...
Ü O = ÑÖ I===
ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==
=
O
O
~
Ä
166
...
=
Å
167
...
o = = ã Å =
O
=
~ +Ä−Å
~Ä
169
...
~Ä = ÅÜ =
=
=
26
CHAPTER 3
...
p =
~Ä ÅÜ
=
=
O
O
=
=
=
3
...
=
172
...
Ü O = ÄO −
O
~
=
Q
27
CHAPTER 3
...
i = ~ + OÄ =
=
175
...
3 Equilateral Triangle
=
páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=
^äíáíìÇÉW=Ü=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 12
...
Ü =
~ P
=
O
=
28
CHAPTER 3
...
o = Ü =
=
P
P
=
~ P o
N
178
...
i = P~ =
=
180
...
4 Scalene Triangle
E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=
=
=
páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=
~ +Ä+Å
pÉãáéÉêáãÉíÉêW= é =
==
O
^åÖäÉë=çÑ=~=íêá~åÖäÉW= αI βI γ =
^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Ü ~ I Ü Ä I Ü Å =
jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ã ~ I ã Ä I ã Å =
_áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= αI βI γ W= í ~ I í Ä I í Å =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
^êÉ~W=p=
=
=
29
CHAPTER 3
...
=
181
...
~ + Ä > Å I==
Ä + Å > ~ I==
~ + Å > Ä K=
=
183
...
jáÇäáåÉ=
~
è = I= è öö ~ K=
O
=
=
=
=====
Figure 14
...
GEOMETRY
185
...
187
...
189
...
Ü ~ =
31
CHAPTER 3
...
Ü ~ = Ä ëáå γ = Å ëáå β I=
Ü Ä = ~ ëáå γ = Å ëáå α I=
Ü Å = ~ ëáå β = Ä ëáå α K=
=
Ä +Å ~
− I==
O
Q
O
O
~ + Å ÄO
− I==
ãO =
Ä
O
Q
O
O
~ + Ä ÅO
O
− K=
ãÅ =
O
Q
192
...
=
O
O
O
193
...
í O =
I==
~
(Ä + Å )O
Q~Åé(é − Ä)
íO =
I==
Ä
(~ + Å )O
Q~Äé(é − Å )
íO =
K=
Å
(~ + Ä)O
=
32
CHAPTER 3
...
p =
=
=
=
3
...
33
CHAPTER 3
...
Ç = ~ O ==
=
197
...
ê = =
O
199
...
p = ~ =
=
=
=
O
3
...
=
201
...
GEOMETRY
202
...
i = O(~ + Ä) =
=
204
...
7 Parallelogram
=
páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä=
aá~Öçå~äëW= ÇN I Ç O =
`çåëÉÅìíáîÉ=~åÖäÉëW= αI β =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
^äíáíìÇÉW=Ü==
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=====
=
=
Figure 18
...
α + β = NUM° =
206
...
GEOMETRY
207
...
i = O(~ + Ä) =
209
...
8 Rhombus
=
páÇÉ=çÑ=~=êÜçãÄìëW=~=
aá~Öçå~äëW= ÇN I Ç O =
`çåëÉÅìíáîÉ=~åÖäÉëW= αI β =
^äíáíìÇÉW=e=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=====
=
Figure 19
...
GEOMETRY
210
...
Ç + Ç = Q~ =
O
N
O
O
O
=
212
...
ê = = N O =
=
O
Q~
O
=
214
...
p = ~Ü = ~ ëáå α I==
N
p = ÇNÇ O K=
O
=
=
=
O
3
...
GEOMETRY
=
=
Figure 20
...
è =
217
...
10 Isosceles Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
^êÉ~W=p=
=
=
38
CHAPTER 3
...
=
218
...
Ç = ~Ä + Å =
=
N
O
220
...
p =
⋅ Ü = èÜ =
O
=
=
=
=
=
=
221
...
GEOMETRY
3
...
=
223
...
è =
=Å=
O
=
225
...
GEOMETRY
226
...
i = O(~ + Ä) = QÅ =
=
(~ + Ä) ~Ä = èÜ = ÅÜ = iê ==
~+Ä
229
...
o =
O
3
...
GEOMETRY
=
=
Figure 23
...
~ + Ä = Å + Ç =
~+Ä Å+Ç
231
...
i = O(~ + Ä) = O(Å + Ç ) =
=
=
=
Å+Ç
~+Ä
⋅Ü =
⋅ Ü = èÜ I==
O
O
N
p = ÇNÇ O ëáå ϕ K=
O
233
...
13 Kite
=
páÇÉë=çÑ=~=âáíÉW=~I=Ä=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉëW= αI βI γ =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
42
CHAPTER 3
...
=
234
...
i = O(~ + Ä) =
=
=
236
...
14 Cyclic Quadrilateral
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
43
CHAPTER 3
...
=
237
...
míçäÉãó∞ë=qÜÉçêÉã=
~Å + ÄÇ = ÇNÇ O =
239
...
o =
I==
Q (é − ~ )(é − Ä)(é − Å )(é − Ç )
i
ïÜÉêÉ= é = K=
O
=
N
241
...
GEOMETRY
3
...
=
242
...
i = ~ + Ä + Å + Ç = O(~ + Å ) = O(Ä + Ç ) =
=
O
ÇN Ç O − (~ − Ä) (~ + Ä − é )
O
I==
Oé
i
ïÜÉêÉ= é = K==
O
=
O
O
244
...
GEOMETRY
N
245
...
16 General Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=======
=
Figure 27
...
α + β + γ + δ = PSM° =
247
...
GEOMETRY
N
248
...
17 Regular Hexagon
=
páÇÉW=~=
fåíÉêå~ä=~åÖäÉW= α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 28
...
α = NOM° =
=
250
...
GEOMETRY
251
...
i = S~ =
=
O
~ P P
I==
O
i
ïÜÉêÉ= é = K=
O
=
=
=
253
...
18 Regular Polygon
=
páÇÉW=~=
kìãÄÉê=çÑ=ëáÇÉëW=å=
fåíÉêå~ä=~åÖäÉW= α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
48
CHAPTER 3
...
=
254
...
α =
å−O
⋅ NUM° =
O
=
å−O
⋅ NUM° =
O
=
256
...
ê = ã =
~
O í~å
π
å
= oO −
~O
=
Q
=
258
...
p =
Oπ
åo
ëáå I==
å
O
O
p = éê = é o O −
~O
I==
Q
49
CHAPTER 3
...
19 Circle
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
`ÜçêÇW=~=
pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=
q~åÖÉåí=ëÉÖãÉåíW=Ö=
`Éåíê~ä=~åÖäÉW= α =
fåëÅêáÄÉÇ=~åÖäÉW= β =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
α
260
...
=
50
CHAPTER 3
...
~N~ O = ÄNÄO =
=
=
=
Figure 31
...
ÉÉN = ÑÑN =
=
=
=====
=
Figure 32
...
Ö O = ÑÑN =
=
51
CHAPTER 3
...
=
264
...
=
265
...
p = πo O =
πÇ
io
=
==
Q
O
O
=
52
CHAPTER 3
...
20 Sector of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW= α =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 35
...
ë = oñ =
268
...
i = ë + Oo =
=
270
...
GEOMETRY
3
...
=
271
...
Ü = o −
Qo O − ~ O I= Ü < o =
O
=
273
...
GEOMETRY
O
O
N
[ëo − ~(o − Ü )] = o απ − ëáå α = o (ñ − ëáå ñ ) I==
O
O NUM°
O
O
p ≈ Ü~ K=
P
274
...
22 Cube
=
bÇÖÉW=~==
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
===
Figure 37
...
Ç = ~ P =
=
~
276
...
GEOMETRY
277
...
p = S~ =
O
=
279
...
23 Rectangular Parallelepiped
=
bÇÖÉëW=~I=ÄI=Å==
aá~Öçå~äW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
=====
Figure 38
...
Ç = ~ O + ÄO + Å O =
281
...
s = ~ÄÅ ==
=
=
56
CHAPTER 3
...
24 Prism
=
i~íÉê~ä=ÉÇÖÉW=ä=
eÉáÖÜíW=Ü=
i~íÉê~ä=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
=====
Figure 39
...
p = p i + Op_ K==
=
284
...
i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã=
p i = éä I==
ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK=
=
57
CHAPTER 3
...
s = p_ Ü =
=
287
...
25 Regular Tetrahedron
=
qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=Ä~ëÉW= p_ =
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 40
...
Ü =
O
~=
P
=
58
CHAPTER 3
...
p_ =
P~ O
=
Q
=
290
...
s = p_ Ü =
K==
P
S O
=
=
=
O
3
...
GEOMETRY
=
=
Figure 41
...
ã = ÄO −
~O
=
Q
=
293
...
p i = å~ã = å~ QÄO − ~ O = éã =
Q
O
=
295
...
p = p_ + p i =
=
N
N
297
...
GEOMETRY
3
...
=
298
...
GEOMETRY
299
...
p i =
=
O
=
301
...
s = pN + pNpO + pO =
P
=
O
Üp Ä Ä Üp
303
...
28 Rectangular Right Wedge
=
páÇÉë=çÑ=Ä~ëÉW=~I=Ä=
qçé=ÉÇÖÉW=Å=
eÉáÖÜíW=Ü=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
62
CHAPTER 3
...
=
N
(~ + Å ) QÜO + ÄO + Ä ÜO + (~ − Å )O =
O
=
305
...
p = p_ + p i =
=
ÄÜ
(O~ + Å ) =
307
...
p i =
3
...
GEOMETRY
308
...
=
309
...
o =
~ O
=
O
=
64
CHAPTER 3
...
p = O~ O P =
=
~P O
312
...
=
313
...
o =
)
~ P P+ R
=
NO
(
)
~
O R+ R =
Q
=
315
...
s =
=
NO
=
=
(
)
65
CHAPTER 3
...
317
...
o =
)
=
(
)
~ P N+ R
=
Q
=
(
)
319
...
s =
=
Q
=
=
=
(
)
3
...
GEOMETRY
eÉáÖÜíW=e=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=====
=
Figure 47
...
p i = Oπoe =
=
Ç
322
...
s = p_ e = πo O e =
=
=
=
67
CHAPTER 3
...
31 Right Circular Cylinder with
an Oblique Plane Face
=
o~Çáìë=çÑ=Ä~ëÉW=o=
qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= ÜN =
qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= Ü O =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 48
...
p i = πo(ÜN + Ü O ) =
=
O
Ü − ÜO
325
...
GEOMETRY
O
ÜN − Ü O
O
326
...
s =
O
=
=
=
3
...
69
CHAPTER 3
...
e = ã O − o O =
=
πãÇ
329
...
p_ = πo O =
=
Ç
N
331
...
s = p_ e = πo O e =
P
P
=
=
=
3
...
GEOMETRY
=
=
Figure 50
...
e = ã O − (o − ê ) =
=
o
334
...
O = O = â O =
pN ê
=
336
...
p = pN + pO + p i = π o O + ê O + ã(o + ê ) =
=
Ü
338
...
s =
N+ â + âO =
N + + =
P ê ê P
=
=
=
O
[
(
]
)
[
71
]
CHAPTER 3
...
34 Sphere
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
Figure 51
...
p = Qπo O =
=
Q
N
N
341
...
35 Spherical Cap
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉW=ê=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= p_ =
^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= p` =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
72
CHAPTER 3
...
=
ê O + ÜO
342
...
p_ = πê O =
=
344
...
p = p_ + p` = π(Ü O + Oê O ) = π(OoÜ + ê O ) =
=
π
π
346
...
36 Spherical Sector
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê=
eÉáÖÜíW=Ü=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
73
CHAPTER 3
...
=
347
...
s = πo O Ü =
P
=
kçíÉW= qÜÉ= ÖáîÉå= Ñçêãìä~ë= ~êÉ= ÅçêêÉÅí= ÄçíÜ= Ñçê= ±çéÉå≤= ~åÇ=
±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK=
=
=
=
3
...
GEOMETRY
=
=====
=
Figure 54
...
pp = OπoÜ =
=
350
...
s = πÜ(PêNO + PêOO + Ü O )=
S
=
=
=
3
...
GEOMETRY
=
=
Figure 55
...
p i =
πo O
α = Oo O ñ =
VM
=
353
...
s =
πoP
O
α = oP ñ =
OTM
P
=
=
=
3
...
GEOMETRY
=
=======
=
Figure 56
...
s = π~ÄÅ =
P
=
=
=
Prolate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ > Ä F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
~ ~êÅëáå É
356
...
s = πÄO~ =
P
=
77
CHAPTER 3
...
p = OπÄ Ä +
ÄÉ L ~
ïÜÉêÉ= É =
ÄO − ~ O
K=
Ä
=
Q
359
...
40 Circular Torus
=
j~àçê=ê~ÇáìëW=o=
jáåçê=ê~ÇáìëW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
78
CHAPTER 3
...
=
360
...
s = OπOoê O =
=
=
79
=
Chapter 4
Trigonometry
=
=
=
=
^åÖäÉëW= α I= β =
oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó==
tÜçäÉ=åìãÄÉêW=â=
=
=
4
...
N ê~Ç =
=
363
...
N D =
=
365
...
=
=
=
=
=
NUM°
≈ RT°NT DQR? =
π
π
ê~Ç ≈ MKMNTQRP ê~Ç =
NUM
π
ê~Ç ≈ MKMMMOVN ê~Ç =
NUM ⋅ SM
π
ê~Ç ≈ MKMMMMMR ê~Ç =
NUM ⋅ PSMM
^åÖäÉ=
EÇÉÖêÉÉëF=
^åÖäÉ=
Eê~Çá~åëF=
M=
PM= QR= SM= VM= NUM= OTM= PSM=
M=
π
=
S
π
=
Q
80
π
=
P
π
=
O
π=
Pπ
=
O
Oπ =
CHAPTER 4
...
2 Definitions and Graphs of Trigonometric
Functions
=
=
=
=
Figure 58
...
ëáå α =
ó
=
ê
=
368
...
í~å α =
ó
=
ñ
=
370
...
TRIGONOMETRY
371
...
ÅçëÉÅ α =
ê
=
ó
=
373
...
=
374
...
TRIGONOMETRY
=
=
Figure 60
...
q~åÖÉåí=cìåÅíáçå=
π
ó = í~å ñ I= ñ ≠ (Oâ + N) I= − ∞ ≤ í~å ñ ≤ ∞K =
O
=
=
=
Figure 61
...
TRIGONOMETRY
376
...
=
377
...
TRIGONOMETRY
=
=
Figure 63
...
`çëÉÅ~åí=cìåÅíáçå==
ó = Åçë ÉÅ ñ I= ñ ≠ âπ K=
=
Figure 64
...
TRIGONOMETRY
4
...
Signs of Trigonometric Functions
379
...
=
nì~Çê~åí=
=
f=
ff=
fff=
fs=
páå
α=
H=
H=
=
=
`çë
α=
H=
=
=
H=
q~å
α=
H=
=
H=
=
`çí
α=
H=
=
H=
=
pÉÅ
α=
H=
=
=
H=
`çëÉÅ=
α=
H=
H=
=
=
=
=
Figure 65
...
TRIGONOMETRY
4
...
=
α° = α ê~Ç =
M=
M=
π
=
PM=
S
π
=
QR=
Q
π
=
SM=
P
π
=
VM=
O
Oπ
=
NOM=
P
NUM=
π=
Pπ
=
OTM=
O
PSM= Oπ =
=
=
=
=
=
=
=
=
=
=
=
=
=
O
=
O
P
=
O
Åçë α =
N=
P
=
O
O
=
O
N
=
O
N=
M=
P
=
O
M=
N
− =
O
− N=
− N=
M=
ëáå α =
M=
N
=
O
í~å α = Åçí α
M=
∞=
N
=
P=
P
ëÉÅ α =
N=
O
=
P
ÅçëÉÅ α =
∞=
O=
N=
N=
P=
N
=
P
O=
O
=
P
M=
∞=
N=
∞=
O=
O=
M=
N
P
∞=
− N=
O
=
P
∞=
M=
∞=
M=
∞=
− N=
N=
M=
∞=
N=
∞=
− P=
87
−
−O=
CHAPTER 4
...
=
α° = α ê~Ç =
π
=
NR=
NO
ëáå α =
Åçë α =
í~å α =
Åçí α =
S− O
=
Q
S+ O
=
Q
O− P =
O+ P =
R−O R
=
R
R+O R =
NU=
π
=
NM
R −N
=
Q
NM + O R
Q
PS=
π
=
R
NM − O R
Q
R +N
=
Q
RQ=
Pπ
=
NM
R +N
=
Q
NM − O R
Q
TO=
Oπ
=
R
NM + O R
Q
R −N
=
Q
TR=
Rπ
=
NO
S+ O
=
Q
S− O
=
Q
=
=
=
4
...
ëáå O α + Åçë O α = N =
=
384
...
ÅëÅ O α − Åçí O α = N =
=
ëáå α
386
...
TRIGONOMETRY
387
...
í~å α ⋅ Åçí α = N =
=
N
389
...
ÅçëÉÅ α =
=
ëáå α
=
=
=
4
...
=
=
=
=
=
=
=
β=
−α=
VM° − α =
VM° + α =
NUM° − α
NUM° + α
OTM° − α
OTM° + α
PSM° − α
= PSM° + α
ëáå β =
− ëáå α =
+ Åçë α =
+ Åçë α =
+ ëáå α =
− ëáå α =
− Åçë α =
− Åçë α =
− ëáå α =
+ ëáå α =
89
Åçë β =
+ Åçë α =
+ ëáå α =
− ëáå α =
− Åçë α =
− Åçë α =
− ëáå α =
+ ëáå α =
+ Åçë α =
+ Åçë α =
í~å β =
− í~å α =
+ Åçí α =
− Åçí α =
− í~å α =
+ í~å α =
+ Åçí α =
− Åçí α =
− í~å α =
+ í~å α =
Åçí β =
− Åçí α =
+ í~å α =
− í~å α =
− Åçí α =
+ Åçí α =
+ í~å α =
− í~å α =
− Åçí α =
+ Åçí α =
CHAPTER 4
...
7 Periodicity of Trigonometric Functions
=
392
...
Åçë(α ± Oπå ) = Åçë α I=éÉêáçÇ= Oπ =çê= PSM° K=
=
394
...
Åçí(α ± πå ) = Åçí α I=éÉêáçÇ= π =çê= NUM° K=
=
=
=
4
...
ëáå α = ± N − Åçë O α = ±
α
O =
=
α
N + í~å O
O
N
(N − Åçë Oα ) = O Åçë O α − π − N =
O
O Q
O í~å
=
=
397
...
í~å α =
ëáå α
ëáå Oα
N − Åçë Oα
= ± ëÉÅ O α − N =
=
=
Åçë α
N + Åçë Oα
ëáå Oα
90
CHAPTER 4
...
Åçí α =
=
α
N
O=
400
...
ÅëÅ α =
= ± N + Åçí O α =
α
ëáå α
O í~å
O
=
=
=
N + í~å O
4
...
ëáå(α + β) = ëáå α Åçë β + ëáå β Åçë α =
=
403
...
Åçë(α + β ) = Åçë α Åçë β − ëáå α ëáå β =
=
405
...
TRIGONOMETRY
406
...
í~å(α − β ) =
=
408
...
Åçí(α − β) =
í~å α + í~å β
=
N − í~å α í~å β
í~å α − í~å β
=
N + í~å α í~å β
N − í~å α í~å β
=
í~å α + í~å β
N + í~å α í~å β
=
í~å α − í~å β
=
=
=
4
...
ëáå Oα = O ëáå α ⋅ Åçë α =
=
411
...
í~å Oα =
=
=
O
N − í~å α Åçí α − í~å α
=
Åçí O α − N Åçí α − í~å α
413
...
TRIGONOMETRY
4
...
ëáå Pα = P ëáå α − Q ëáå P α = P Åçë O α ⋅ ëáå α − ëáåP α =
=
415
...
ëáå Rα = R ëáå α − OM ëáå P α + NS ëáå R α =
=
417
...
Åçë Qα = U Åçë Q α − U Åçë O α + N =
=
419
...
í~å Pα =
=
N − P í~å O α
=
Q í~å α − Q í~å P α
421
...
í~å Rα =
=
N − NM í~å O α + R í~å Q α
=
Åçí P α − P Åçí α
423
...
Åçí Qα =
==
Q í~å α − Q í~å P α
=
93
CHAPTER 4
...
Åçí Rα =
N − NM í~å O α + R í~å Q α
=
í~å R α − NM í~å P α + R í~å α
=
=
=
4
...
ëáå
α
N − Åçë α
=
=±
O
O
=
427
...
í~å
α
N − Åçë α
ëáå α
N − Åçë α
=±
=
=
= ÅëÅ α − Åçí α =
O
N + Åçë α N + Åçë α
ëáå α
=
429
...
13 Half Angle Tangent Identities
=
α
O =
430
...
TRIGONOMETRY
α
O=
431
...
í~å α =
α
N − í~å O
O
=
α
N − í~å O
O=
433
...
14 Transforming of Trigonometric
Expressions to Product
=
434
...
ëáå α − ëáå β = O Åçë
α+β
α −β
=
Åçë
O
O
α +β
α −β
=
ëáå
O
O
=
436
...
Åçë α − Åçë β = −O ëáå
α +β
α −β
=
ëáå
O
O
=
95
CHAPTER 4
...
í~å α + í~å β =
=
439
...
Åçí α + Åçí β =
=
441
...
Åçë α + ëáå α = O Åçë − α = O ëáå + α =
Q
Q
=
π
π
443
...
í~å α + Åçí β =
=
Åçë α ⋅ ëáå β
=
Åçë(α + β )
445
...
N + Åçë α = O Åçë O =
O
=
α
447
...
TRIGONOMETRY
π α
448
...
N − ëáå α = O ëáå O − =
Q O
=
=
=
4
...
ëáå α ⋅ ëáå β =
Åçë(α − β) − Åçë(α + β )
=
O
=
451
...
ëáå α ⋅ Åçë β =
=
453
...
Åçí α ⋅ Åçí β =
=
455
...
TRIGONOMETRY
4
...
ëáå O α =
=
457
...
ëáå Q α =
=
459
...
ëáå S α =
=
461
...
Åçë P α =
=
463
...
Åçë R α =
=
465
...
TRIGONOMETRY
4
...
fåîÉêëÉ=páåÉ=cìåÅíáçå==
ó = ~êÅëáå ñ I= − N ≤ ñ ≤ N I= −
π
π
≤ ~êÅëáå ñ ≤ K=
O
O
=
=
=
Figure 66
...
fåîÉêëÉ=`çëáåÉ=cìåÅíáçå==
ó = ~êÅÅçë ñ I= − N ≤ ñ ≤ N I= M ≤ ~êÅÅçë ñ ≤ π K=
=
99
CHAPTER 4
...
=
468
...
100
CHAPTER 4
...
fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå==
ó = ~êÅ Åçí ñ I= − ∞ ≤ ñ ≤ ∞ I= M < ~êÅ Åçí ñ < π K=
=
=====
Figure 69
...
fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==
π π
ó = ~êÅëÉÅ=ñ I ñ ∈ (− ∞I − N] ∪ [NI ∞ )I ~êÅ ëÉÅ ñ ∈ MI ∪ I πK
O O
=
Figure 70
...
TRIGONOMETRY
471
...
=
=
4
...
ñ=
M=
N
=
O
PM° =
SM° =
O
−
O
~êÅëáå ñ = M° =
~êÅÅçë ñ = VM°
N
−
ñ=
O
− PM°
~êÅëáå ñ =
− QR°
=
NOM°
~êÅÅçë ñ =
NPR° =
=
O
=
O
QR° =
QR° =
P
−
O
P
O
SM°
PM°
VM°
M° =
− N=
=
− VM°
=
NUM°
NRM° =
=
− SM°
102
N=
=
=
CHAPTER 4
...
ñ=
M=
P
P
N=
~êÅí~å ñ =
M° =
PM°
QR°
SM°
~êÅ Åçí ñ = VM°
SM°
QR°
PM°
P= −
P
P
4
...
~êÅëáå(− ñ ) = − ~êÅëáå ñ =
=
π
475
...
~êÅëáå ñ = ~êÅÅçë N − ñ O I= M ≤ ñ ≤ N K=
=
477
...
~êÅëáå ñ = ~êÅí~å
O
N− ñ
=
N− ñO
I= M < ñ ≤ N K=
ñ
=
480
...
~êÅÅçë(− ñ ) = π − ~êÅÅçë ñ =
103
− P=
− QR°
− SM° =
=
NPR°
NOM° =
NRM° =
=
− PM°
=
=
=
479
...
TRIGONOMETRY
482
...
~êÅÅçë ñ = ~êÅëáå N − ñ O I= M ≤ ñ ≤ N K=
=
484
...
~êÅÅçë ñ = ~êÅí~å
N− ñO
I= M < ñ ≤ N K=
ñ
=
N− ñO
I= − N ≤ ñ < M K=
ñ
486
...
~êÅÅçë ñ = ~êÅ Åçí
ñ
N− ñO
I= − N ≤ ñ ≤ N K=
=
488
...
~êÅí~å ñ = − ~êÅ Åçí ñ =
O
=
ñ
=
490
...
~êÅí~å ñ = ~êÅÅçë
N+ ñO
=
N
I= ñ ≤ M K=
492
...
TRIGONOMETRY
493
...
~êÅí~å ñ = − − ~êÅí~å I= ñ < M K=
ñ
O
=
N
495
...
~êÅí~å ñ = ~êÅ Åçí − π I= ñ < M K=
ñ
=
497
...
~êÅ Åçí ñ = − ~êÅí~å ñ =
O
=
N
I= ñ > M K=
499
...
~êÅ Åçí ñ = π − ~êÅëáå
N+ ñO
=
ñ
=
501
...
~êÅ Åçí ñ = ~êÅí~å I= ñ > M K=
ñ
=
N
503
...
TRIGONOMETRY
4
...
505
...
507
...
21 Relations to Hyperbolic Functions
508
...
510
...
512
...
1 Determinants
=
513
...
MATRICES AND DETERMINANTS
514
...
p~êêìë=oìäÉ=E^êêçï=oìäÉF=
=
=
Figure 72
...
k-íÜ=lêÇÉê=aÉíÉêãáå~åí=
~NN ~NO K ~Nà
~ ON ~ OO K ~ O à
K K K K
ÇÉí ^ =
~ áN ~ á O K ~ áà
K K K K
~ åN ~ å O K ~ åà
K ~Nå
K ~ Oå
K K
=
K ~ áå
K K
K ~ åå
=
517
...
MATRICES AND DETERMINANTS
518
...
i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï=
å
ÇÉí ^ = ∑ ~ áà` áà I= á = NI OI KI å K=
à=N
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå=
å
ÇÉí ^ = ∑ ~ áà` áà I= à = NI OI KI å K==
á =N
=
=
=
5
...
qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ=
ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK=
~ ~ O ~N ÄN
=
==
= N
ÄN ÄO ~ O ÄO
=
521
...
fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ=
ÇÉíÉêãáå~åí=áë=òÉêçK=
~N ~N
= M=
~O ~O
=
109
CHAPTER 5
...
fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó=====
~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í=
Ñ~ÅíçêK=
â~ N âÄN
~ ÄN
=â N
=
~ O ÄO
~ O ÄO
=
524
...
3 Matrices
=
525
...
pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å × å K==
=
527
...
^=ëèì~êÉ=ã~íêáñ= ~ áà =áë=ëâÉï-ëóããÉíêáÅ=áÑ= ~ áà = −~ àá K==
=
[ ]
110
CHAPTER 5
...
aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç=
ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK==
=
530
...
^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK=
=
=
=
5
...
qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ=
çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== ã × å ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ=
Éèì~äK=
=
533
...
MATRICES AND DETERMINANTS
íÜÉå==
~NO + ÄNO K ~ Nå + ÄNå
~NN + ÄNN
~ +Ä
~ OO + ÄOO K ~ Oå + ÄOå
ON ON
K=
^+_=
M
M
M
~ ãN + ÄãN ~ ã O + Äã O K ~ ãå + Äãå
=
534
...
jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë=
qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ=
åìãÄÉê= çÑ= Åçäìãåë= áå= íÜÉ= Ñáêëí= áë= Éèì~ä= íç= íÜÉ= åìãÄÉê= çÑ=
êçïë=áå=íÜÉ=ëÉÅçåÇK==
=
fÑ=
~NN ~NO K ~Nå
~
~ OO K ~ Oå
I==
^ = ~ áà = ON
M
M
M
~ ãN ~ ã O K ~ ãå
ÄNN ÄNO K ÄNâ
Ä
ÄOO K ÄO â
I=
_ = Äáà = ON
M
M
M
ÄåN Äå O K Äåâ
=
=
=
=
=
[ ]
[ ]
[ ]
112
CHAPTER 5
...
qê~åëéçëÉ=çÑ=~=j~íêáñ=
fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå=
íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK===
fÑ= ^= áë= íÜÉ= çêáÖáå~ä= ã~íêáñI= áíë= íê~åëéçëÉ= áë= ÇÉåçíÉÇ= ^ q = çê=
ú
^ K==
=
537
...
fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
(^_ )q = _ q ^ q K=
=
=
113
CHAPTER 5
...
^Çàçáåí=çÑ=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å × å ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ~Çà ^ I=
áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= ` áà =çÑ=^W=
[ ]
~Çà ^ = ` áà K==
=
540
...
fåîÉêëÉ=çÑ=~=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= å × å ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí=
ÇÉí ^ I=íÜÉå=áíë=áåîÉêëÉ= ^ −N =áë=ÖáîÉå=Äó=
~Çà ^
K=
^ −N =
ÇÉí ^
=
542
...
fÑ==^==áë=~=ëèì~êÉ=== å × å ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó=
íÜÉ=Éèì~íáçå=
^u = λu I==
ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë= λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=
^ − λf = M K===
=
=
=
q
5
...
MATRICES AND DETERMINANTS
aÉíÉêãáå~åíëW=aI= añ I= aó I= aò ==
j~íêáÅÉëW=^I=_I=u=
=
=
~ ñ + ÄNó = ÇN
I==
544
...
fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
aó
a
K=
ñ = ñ I= ó =
a
a
fÑ= a = M = ~åÇ= añ ≠ M Eçê= aó ≠ M FI= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = åç==
ëçäìíáçåK=
fÑ= a = añ = aó = M I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó==
ëçäìíáçåëK=
=
~Nñ + ÄNó + ÅNò = ÇN=
546
...
MATRICES AND DETERMINANTS
ïÜÉêÉ==
~N ÄN
a = ~ O ÄO
~ P ÄP
ÅN
ÇN
ÄN
ÅN
Å O I= añ = Ç O
ÄO
Å O I=
ÅP
ÄP
ÅP
ÇP
~N
ÇN
ÅN
~N
ÄN
ÇN
aó = ~ O
~P
ÇO
ÇP
Å O I= aò = ~ O
ÅP
~P
ÄO
ÄP
Ç O K==
ÇP
=
547
...
j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå=================
å=råâåçïåë=
qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë==
~NNñ N + ~ NO ñ O + K + ~Nå ñ å = ÄN
~ ñ + ~ ñ + K + ~ ñ = Ä
ON N OO O
Oå å
O
=
KKKKKKKKKKKK
~ åNñ N + ~ å O ñ O + K + ~ åå ñ å = Äå
Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=
~ NN ~ NO K ~ Nå ñ N ÄN
~ ON ~ OO K ~ Oå ñ O Ä O
⋅
=
I==
M
M
M M M
~
åN ~ å O K ~ åå ñ å Ä å
áKÉK==
^ ⋅ u = _ I==
116
CHAPTER 5
...
pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å × å =
u = ^ −N ⋅ _ I==
ïÜÉêÉ= ^ −N =áë=íÜÉ=áåîÉêëÉ=çÑ=^K=
=
=
117
Chapter 6
Vectors
=
=
=
=
r r r r →
sÉÅíçêëW= ì I= î I= ï I= ê I= ^_ I=£=
r r
sÉÅíçê=äÉåÖíÜW= ì I= î I=£=
r r r
råáí=îÉÅíçêëW= á I= à I= â =
r
kìää=îÉÅíçêW= M =
r
`ççêÇáå~íÉë=çÑ=îÉÅíçê= ì W= uN I vN I wN =
r
`ççêÇáå~íÉë=çÑ=îÉÅíçê= î W= u O I vO I wO =
pÅ~ä~êëW= λ I µ =
aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçë β I= Åçë γ =
^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW= θ =
=
=
6
...
råáí=sÉÅíçêë=
r
á = (NI MI M) I=
r
à = (MI NI M) I=
r
â = (MI MI N) I=
r r r
á = à = â = N K=
=
r
r
r
r →
551
...
VECTORS
=======
=
=
Figure 73
...
(ñ N − ñ M )O + (óN − ó M )O + (òN − ò M )O =
=
→
→
r
r
553
...
r
554
...
VECTORS
=
=====
=
Figure 75
...
fÑ= ê (uI v I w ) = êN (uN I vN I wN ) I=íÜÉå==
u = uN I= v = vN I= w = wN K==
==
=
6
...
ï = ì + î =
=
=
==
=
Figure 76
...
VECTORS
=
==
=
Figure 77
...
ï = ìN + ì O + ìP + K + ì å =
=
=
==
=
Figure 78
...
`çããìí~íáîÉ=i~ï=
r r r r
ì+ î =î+ì=
=
559
...
ì + î = (uN + u O I vN + vO I wN + wO ) =
=
=
=
=
=
=
121
CHAPTER 6
...
3 Vector Subtraction
=
r r r r r r
561
...
=
=
==
=
Figure 80
...
ì − î = ì + (− î ) =
=
r r r
563
...
M = M =
=
r r
565
...
4 Scaling Vectors
=
r
r
566
...
VECTORS
=
=
Figure 81
...
r
r
ï = λ⋅ì=
=
r
568
...
λì = ìλ =
=
r
r
r
570
...
λ(µì ) = µ(λì ) = (λµ )ì =
=
r r
r
r
572
...
5 Scalar Product
=
r
r
573
...
VECTORS
=
=
=
Figure 82
...
pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
r
r
fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå==
r r
ì ⋅ î = uNu O + vNvO + wNwO K=
=
575
...
`çããìí~íáîÉ=mêçéÉêíó=
r r r r
ì⋅î = î ⋅ì=
=
577
...
aáëíêáÄìíáîÉ=mêçéÉêíó=
r r r r r r r
ì ⋅ (î + ï ) = ì ⋅ î + ì ⋅ ï =
=
π
r r
r r
579
...
ì ⋅ î > M =áÑ= M < θ < K=
O
=
124
CHAPTER 6
...
ì ⋅ î < M =áÑ= < θ < π K=
O
=
r r r r
582
...
ì ⋅ î = ì ⋅ î =áÑ= ì I î =~êÉ=é~ê~ääÉä=E θ = M FK=
=
r
584
...
á ⋅ á = à ⋅ à = â ⋅ â = N =
=
r r r r r r
586
...
6 Vector Product
=
r
r
587
...
VECTORS
=
=======
=
Figure 83
...
ï = ì × î = u N
uO
r
à
vN
vO
r
â
wN =
wO
=
u wN uN vN
r r r v wN
=
589
...
p = ì × î = ì ⋅ î ⋅ ëáå θ =EcáÖKUPF=
=
591
...
kçåÅçããìí~íáîÉ=mêçéÉêíó=
r r
r r
ì × î = −(î × ì ) ==
=
593
...
VECTORS
594
...
ì × î = M =áÑ= ì =~åÇ= î =~êÉ=é~ê~ääÉä=E θ = M FK=
=
r r r r r r r
596
...
á × à = â I= à × â = á I= â × á = à =
=
=
=
6
...
599
...
601
...
sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ=
r r r
s = ì ⋅ (î × ï ) =
=
127
CHAPTER 6
...
=
603
...
=
r r r
r r
r
604
...
fÑ== ì ⋅ (î × ï ) ≠ M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó=
áåÇÉéÉåÇÉåíK=
=
128
CHAPTER 6
...
sÉÅíçê=qêáéäÉ=mêçÇìÅí=
r r r
r r r r r r
ì × (î × ï ) = (ì ⋅ ï )î − (ì ⋅ î )ï ==
=
=
=
=
=
=
=
=
129
Chapter 7
Analytic Geometry
=
=
=
=
7
...
aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
Ç = ^_ = ñ O − ñ N = ñ N − ñ O =
=
=
=
Figure 86
...
aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ =
ñ + λñ O
^`
I= λ =
ñM = N
I= λ ≠ −N K=
N+ λ
`_
=
=
========
Figure 87
...
ANALYTIC GEOMETRY
609
...
2 Two-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O =
mçä~ê=ÅççêÇáå~íÉëW= êI ϕ =
oÉ~ä=åìãÄÉêW= λ ==
mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
^êÉ~W=p=
=
=
610
...
131
CHAPTER 7
...
aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ =
ñ + λñ O
ó + λó O
ñM = N
I= ó M = N
I==
N+ λ
N+ λ
^`
λ=
I= λ ≠ −N K=
`_
=
=======
=
=
Figure 89
...
ANALYTIC GEOMETRY
=======
=
=
Figure 90
...
jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
ñ + ñO
ó + óO
I= ó M = N
I= λ = N K=
ñM = N
O
O
=
613
...
ANALYTIC GEOMETRY
=========
=
=
Figure 91
...
fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
~ñ + Äñ O + Åñ P
~ó + Äó O + Åó P
ñM = N
I= ó M = N
I==
~ +Ä+Å
~ +Ä+Å
ïÜÉêÉ= ~ = _` I= Ä = `^ I= Å = ^_ K==
=
========
=
=
Figure 92
...
ANALYTIC GEOMETRY
615
...
=
=
=
=
=
=
=
135
CHAPTER 7
...
lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
O
O
óN ñ O ñ P + óN N
ñN + ó OóP ñN N
ñM =
óO
óP
ñPñN + ó O N
O
O
ñ Nñ O + ó P N
ñN
ñO
ñP
óN N
óO N
óP N
I= ó M =
ñ O + ó P óN
O
O
ñ P + ó Nó O
ñN
ñO
ñP
ñO N
ñP N
óN N
óO N
óP N
=
=
======
=
=
Figure 94
...
^êÉ~=çÑ=~=qêá~åÖäÉ=
ñ N óN N
N
N ñ O − ñN
p = (± ) ñ O ó O N = (± )
O
O ñ P − ñN
ñP óP N
=
=
=
136
ó O − óN
=
ó P − óN
CHAPTER 7
...
^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
N
p = (± ) [(ñ N − ñ O )(ó N + ó O ) + (ñ O − ñ P )(ó O + ó P ) + =
O
+ (ñ P − ñ Q )(ó P + ó Q ) + (ñ Q − ñ N )(ó Q + ó N )] =
=
=
===
=
Figure 95
...
aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë=
Ç = ^_ = êNO + êOO − OêNêO Åçë(ϕ O − ϕN ) =
=
137
CHAPTER 7
...
=
620
...
=
621
...
ANALYTIC GEOMETRY
7
...
dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=
^ñ + _ó + ` = M =
=
623
...
=
624
...
ANALYTIC GEOMETRY
qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= â = í~å α K=
=
=
=
Figure 99
...
dê~ÇáÉåí=çÑ=~=iáåÉ==
ó − óN
=
â = í~å α = O
ñ O − ñN
=
=
=
Figure 100
...
ANALYTIC GEOMETRY
626
...
=
627
...
ANALYTIC GEOMETRY
=
=
Figure 102
...
fåíÉêÅÉéí=cçêã=
ñ ó
+ =N=
~ Ä
=
=
=
Figure 103
...
ANALYTIC GEOMETRY
629
...
=
630
...
ANALYTIC GEOMETRY
=
=
Figure 105
...
sÉêíáÅ~ä=iáåÉ=
ñ =~=
=
632
...
sÉÅíçê=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=
r r r
ê = ~ + íÄ I==
ïÜÉêÉ==
l=áë=íÜÉ=çêáÖáå=çÑ=íÜÉ=ÅççêÇáå~íÉëI=
u=áë=~åó=î~êá~ÄäÉ=éçáåí=çå=íÜÉ=äáåÉI==
r
~ =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~=âåçïå=éçáåí=^=çå=íÜÉ=äáåÉ=I=
r
Ä =áë=~=âåçïå=îÉÅíçê=çÑ=ÇáêÉÅíáçåI=é~ê~ääÉä=íç=íÜÉ=äáåÉI==
í=áë=~=é~ê~ãÉíÉêI==
r →
ê = lu =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~åó=éçáåí=u=çå=íÜÉ=äáåÉK==
=
144
CHAPTER 7
...
=
634
...
ANALYTIC GEOMETRY
=
Figure 107
...
aáëí~åÅÉ=cêçã=~=mçáåí=qç=~=iáåÉ=
qÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=éçáåí= m(~ I Ä) =íç=íÜÉ=äáåÉ=
^ñ + _ó + ` = M =áë==
^~ + _Ä + `
Ç=
K=
^O + _O
=
=
=
Figure 108
...
ANALYTIC GEOMETRY
636
...
=
637
...
ANALYTIC GEOMETRY
=
=
Figure 110
...
^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë=
â − âN
í~å ϕ = O
I==
N + â Nâ O
^N^ O + _N_ O
K=
Åçë ϕ =
O
O
^N + _N ⋅ ^ O + _ O
O
O
=
148
CHAPTER 7
...
=
639
...
4 Circle
=
o~ÇáìëW=o=
`ÉåíÉê=çÑ=ÅáêÅäÉW= (~ I Ä) =
mçáåí=ÅççêÇáå~íÉëW=ñI=óI= ñ N I= ó N I=£=
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í=
149
CHAPTER 7
...
bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ=
cçêãF=
ñ O + ó O = oO =
======
=
=
Figure 112
...
bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~I Ä)
(ñ − ~ )O + (ó − Ä)O = o O
Figure 113
...
ANALYTIC GEOMETRY
642
...
=
643
...
dÉåÉê~ä=cçêã
^ñ O + ^ó O + añ + bó + c = M =E^=åçåòÉêçI= aO + b O > Q ^c FK==
qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë= (~ I Ä) I=ïÜÉêÉ==
a
b
~=−
I= Ä = −
K=
O^
O^
qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë
151
CHAPTER 7
...
5 Ellipse
=
pÉãáã~àçê=~ñáëW=~=
pÉãáãáåçê=~ñáëW=Ä=
cçÅáW= cN (− ÅI M) I= cO (ÅI M) =
aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= =
bÅÅÉåíêáÅáíóW=É==
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
645
...
152
CHAPTER 7
...
êN + êO = O~ I=
ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== m(ñ I ó ) ==çå=
íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK=
=
=
=
Figure 116
...
~ O = ÄO + Å O
=
648
...
bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë
~
~O
ñ=± =± =
É
Å
=
650
...
ANALYTIC GEOMETRY
651
...
dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë
^ñ O + `ó O + añ + bó + c = M I==
ïÜÉêÉ= ^` > M K
=
653
...
^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ
i = π NKR(~ + Ä) − ~Ä I==
(
i = π O(~ O + ÄO ) K=
=
655
...
6 Hyperbola
=
qê~åëîÉêëÉ=~ñáëW=~=
`çåàìÖ~íÉ=~ñáëW=Ä=
cçÅáW= cN (− ÅI M) I= cO (ÅI M) =
aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= =
bÅÅÉåíêáÅáíóW=É==
^ëóãéíçíÉëW=ëI=í=
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=íI=â=
=
=
=
154
CHAPTER 7
...
bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF=
ñO óO
− = N=
~ O ÄO
=
=
=
Figure 117
...
êN − êO = O~ I=
ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== m(ñ I ó ) ==çå=
íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK=
=
155
CHAPTER 7
...
658
...
660
...
=
bèì~íáçåë=çÑ=^ëóãéíçíÉë=
Ä
ó=± ñ=
~
=
Å O = ~ O + ÄO =
=
bÅÅÉåíêáÅáíó
Å
É = > N=
~
=
bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë
~
~O
ñ=± =± =
É
Å
=
=
=
156
CHAPTER 7
...
m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~=
ñ = ~ ÅçëÜ í
I= M ≤ í ≤ Oπ K
ó = Ä ëáåÜ í
=
663
...
dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë
^ñ O + `ó O + añ + bó + c = M I==
ïÜÉêÉ= ^` < M K=
665
...
ANALYTIC GEOMETRY
=
=
Figure 119
...
7 Parabola
=
cçÅ~ä=é~ê~ãÉíÉêW=é=
cçÅìëW=c=
sÉêíÉñW= j(ñ M I ó M ) =
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=éI=~I=ÄI=Å=
=
=
666
...
ANALYTIC GEOMETRY
=
=
Figure 120
...
dÉåÉê~ä=cçêã
^ñ O + _ñó + `ó O + añ + bó + c = M I==
ïÜÉêÉ= _ O − Q ^` = M K=
=
N
668
...
ANALYTIC GEOMETRY
é
ó = − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
é
c MI I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
j(MI M) K=
=
=
=
Figure 121
...
dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë==
^ñ O + añ + bó + c = M =E^I=b=åçåòÉêçFI==
N
ó = ~ñ O + Äñ + Å I= é = K==
O~
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
é
ó = ó M − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
160
CHAPTER 7
...
=
=
=
7
...
ANALYTIC GEOMETRY
670
...
=
671
...
ANALYTIC GEOMETRY
========
=
=
Figure 124
...
163
CHAPTER 7
...
jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
ñ + ñO
ó + óO
ò +ò
I= ò M = N O I= λ = N K=
ñM = N
I= ó M = N
O
O
O
=
673
...
sçäìãÉ=çÑ=~=qÉíê~ÜÉÇêçå=
qÜÉ=îçäìãÉ=çÑ=~=íÉíê~ÜÉÇêçå=ïáíÜ=îÉêíáÅÉë= mN (ñ N I ó N I ò N ) I=
mO (ñ O I ó O I ò O ) I= mP (ñ P I ó P I ò P ) I=~åÇ= mQ (ñ Q I ó Q I ò Q ) =áë=ÖáîÉå=Äó==
ñ N óN òN N
N ñO óO òO N
s=±
I==
S ñ P óP òP N
ñQ óQ òQ N
çê=
ñ N − ñ Q óN − ó Q òN − ò Q
N
s = ± ñ O − ñ Q ó O − ó Q ò O − ò Q K=
S
ñP − ñ Q óP − ó Q òP − òQ
kçíÉW=tÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=
~åëïÉê=Ñçê=îçäìãÉK==
=
164
CHAPTER 7
...
=
=
=
7
...
dÉåÉê~ä=bèì~íáçå=çÑ=~=mä~åÉ=
^ñ + _ó + `ò + a = M =
=
=
165
CHAPTER 7
...
kçêã~ä=sÉÅíçê=íç=~=mä~åÉ=
r
qÜÉ=îÉÅíçê= å (^I _I ` ) =áë=åçêã~ä=íç=íÜÉ=éä~åÉ=
^ñ + _ó + `ò + a = M K=
=
=
===
=
Figure 127
...
m~êíáÅìä~ê=`~ëÉë=çÑ=íÜÉ=bèì~íáçå=çÑ=~=mä~åÉ=
^ñ + _ó + `ò + a = M =
=
fÑ= ^ = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ñ-~ñáëK=
fÑ= _ = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ó-~ñáëK=
fÑ= ` = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ò-~ñáëK=
fÑ= a = M I=íÜÉ=éä~åÉ=äáÉë=çå=íÜÉ=çêáÖáåK==
=
fÑ= ^ = _ = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ñó-éä~åÉK=
fÑ= _ = ` = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=óò-éä~åÉK=
fÑ= ^ = ` = M I=íÜÉ=éä~åÉ=áë=é~ê~ääÉä=íç=íÜÉ=ñò-éä~åÉK=
=
166
CHAPTER 7
...
mçáåí=aáêÉÅíáçå=cçêã=
^(ñ − ñ M ) + _(ó − ó M ) + `(ò − ò M ) = M I==
ïÜÉêÉ=íÜÉ=éçáåí= m(ñ M I ó M I ò M ) =äáÉë=áå=íÜÉ=éä~åÉI=~åÇ=íÜÉ=îÉÅíçê= (^I _I ` ) =áë=åçêã~ä=íç=íÜÉ=éä~åÉK===
=
=
====
=
Figure 128
...
fåíÉêÅÉéí=cçêã=
ñ ó ò
+ + = N=
~ Ä Å
=
167
CHAPTER 7
...
=
680
...
ANALYTIC GEOMETRY
=
=====
=
Figure 130
...
kçêã~ä=cçêã=
ñ Åçë α + ó Åçë β + ò Åçë γ − é = M I==
ïÜÉêÉ==é==áë==íÜÉ==éÉêéÉåÇáÅìä~ê==Çáëí~åÅÉ==Ñêçã==íÜÉ=çêáÖáå=íç=
íÜÉ=éä~åÉ=I=~åÇ= Åçë α I= Åçë β I= Åçë γ =~êÉ=íÜÉ=ÇáêÉÅíáçå=ÅçëáåÉë=
çÑ=~åó=äáåÉ=åçêã~ä=íç=íÜÉ=éä~åÉK==
=
169
CHAPTER 7
...
=
682
...
ANALYTIC GEOMETRY
=
=====
=
Figure 132
...
aáÜÉÇê~ä=^åÖäÉ=_ÉíïÉÉå=qïç=mä~åÉë=
fÑ=íÜÉ=éä~åÉë=~êÉ=ÖáîÉå=Äó==
^Nñ + _Nó + `Nò + aN = M I==
^ O ñ + _ O ó + ` Oò + aO = M I==
íÜÉå=íÜÉ=ÇáÜÉÇê~ä=~åÖäÉ=ÄÉíïÉÉå=íÜÉã=áë==
r r
åN ⋅ å O
^N^ O + _N_ O + `N` O
Åçë ϕ = r r =
K=
O
O
O
åN ⋅ å O
^N + _N + `N ⋅ ^ O + _ O + ` O
O
O
O
=
171
CHAPTER 7
...
=
684
...
mÉêéÉåÇáÅìä~ê=mä~åÉë=
qïç=éä~åÉë= ^Nñ + _Nó + `Nò + aN = M =~åÇ=
^ O ñ + _ O ó + ` Oò + aO = M =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ==
^N^ O + _N_ O + `N` O = M K=
=
686
...
ANALYTIC GEOMETRY
ñ − ñN
~N
~O
ó − óN ò − òN
ÄN
ÅN = M =
ÄO
ÅO
=
687
...
=
688
...
ANALYTIC GEOMETRY
Ç=
^ñ N + _ó N + `òN + a
^O + _O + `O
K=
=
=
======
=
Figure 135
...
fåíÉêëÉÅíáçå=çÑ=qïç=mä~åÉë=
fÑ=íïç=éä~åÉë= ^Nñ + _Nó + `Nò + aN = M =~åÇ=
^ O ñ + _ O ó + ` Oò + aO = M = áåíÉêëÉÅíI= íÜÉ= áåíÉêëÉÅíáçå= ëíê~áÖÜí=
äáåÉ=áë=ÖáîÉå=Äó=
ñ = ñ N + ~í
ó = ó N + Äí I==
ò = ò + Åí
N
çê==
ñ − ñ N ó − óN ò − òN
=
=
I==
Ä
Å
~
ïÜÉêÉ==
174
CHAPTER 7
...
10 Straight Line in Space
=
mçáåí=ÅççêÇáå~íÉëW=ñI=óI=òI= ñ N I= ó N I= ò N I=£=
aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçë β I= Åçë γ =
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=~I=ÄI=ÅI= ~N I= ~ O I=íI=£==
r r r
aáêÉÅíáçå=îÉÅíçêë=çÑ=~=äáåÉW= ë I= ëN I= ëO =
r
kçêã~ä=îÉÅíçê=íç=~=éä~åÉW= å =
^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW= ϕ =
=
690
...
ANALYTIC GEOMETRY
=
=====
=
Figure 136
...
qïç=mçáåí=cçêã=
ñ − ñN
ó − óN
ò − òN
=
=
==
ñ O − ñ N ó O − óN ò O − òN
=
=====
Figure 137
...
ANALYTIC GEOMETRY
692
...
=
693
...
ANALYTIC GEOMETRY
=
=====
=
Figure 139
...
m~ê~ääÉä=iáåÉë=
qïç=äáåÉë=~êÉ=é~ê~ääÉä=áÑ==
r r
ëN öö ëO I==
çê==
~N ÄN ÅN
= = K=
~ O ÄO Å O
=
695
...
fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë=
ñ − ñ N ó − óN ò − òN
qïç=äáåÉë=
=
=
=~åÇ=
~N
ÄN
ÅN
178
CHAPTER 7
...
m~ê~ääÉä=iáåÉ=~åÇ=mä~åÉ==
ñ − ñ N ó − óN ò − òN
qÜÉ=ëíê~áÖÜí=äáåÉ=
=
=
=~åÇ=íÜÉ=éä~åÉ=
Ä
Å
~
^ñ + _ó + `ò + a = M =~êÉ=é~ê~ääÉä=áÑ=
r r
å ⋅ ë = M I==
çê==
^~ + _Ä + `Å = M K=
=
=
=====
=
Figure 140
...
ANALYTIC GEOMETRY
698
...
=
=
7
...
ANALYTIC GEOMETRY
699
...
`ä~ëëáÑáÅ~íáçå=çÑ=nì~ÇêáÅ=pìêÑ~ÅÉë=
=
â=ëáÖåë=
qóéÉ=çÑ=pìêÑ~ÅÉ=
`~ëÉ= o~åâEÉF= o~åâEbF=
∆=
p~ãÉ=
oÉ~ä=bääáéëçáÇ=
N=
P=
Q=
fã~Öáå~êó=bääáéëçáÇ=
O=
P=
Q=
> M=
eóéÉêÄçäçáÇ=çÑ=N=pÜÉÉí=
P=
P=
Q=
> M = aáÑÑÉêÉåí=
eóéÉêÄçäçáÇ=çÑ=O=pÜÉÉíë=
Q=
P=
Q=
< M = aáÑÑÉêÉåí=
R=
S=
T=
U=
V=
NM=
NN=
NO=
NP=
NQ=
NR=
NS=
NT=
P=
P=
O=
O=
O=
O=
O=
O=
O=
N=
N=
N=
N=
P=
P=
Q=
Q=
P=
P=
P=
O=
O=
P=
O=
O=
N=
=
=
=
=
=
=
=
=
=
=
=
aáÑÑÉêÉåí=
p~ãÉ=
p~ãÉ=
aáÑÑÉêÉåí=
p~ãÉ=
p~ãÉ=
aáÑÑÉêÉåí=
aáÑÑÉêÉåí=
p~ãÉ=
=
=
=
=
oÉ~ä=nì~ÇêáÅ=`çåÉ=
fã~Öáå~êó=nì~ÇêáÅ=`çåÉ=
bääáéíáÅ=m~ê~ÄçäçáÇ=
eóéÉêÄçäáÅ=m~ê~ÄçäçáÇ=
oÉ~ä=bääáéíáÅ=`óäáåÇÉê=
fã~Öáå~êó=bääáéíáÅ=`óäáåÇÉê=
eóéÉêÄçäáÅ=`óäáåÇÉê=
oÉ~ä=fåíÉêëÉÅíáåÖ=mä~åÉë=
fã~Öáå~êó=fåíÉêëÉÅíáåÖ=mä~åÉë=
m~ê~ÄçäáÅ=`óäáåÇÉê=
oÉ~ä=m~ê~ääÉä=mä~åÉë=
fã~Öáå~êó=m~ê~ääÉä=mä~åÉë=
`çáåÅáÇÉåí=mä~åÉë=
=
eÉêÉ==
^ e n m
^ e d
e _ c n
I= ∆ = ÇÉí(b ) I==
É = e _ c I= b =
d c ` o
d c `
m n o a
â N I â O I â P =~êÉ=íÜÉ=êççíë=çÑ=íÜÉ=Éèì~íáçåI==
^−ñ
e
d
e
_−ñ
c = M K=
d
c
`−ñ
=
=
181
CHAPTER 7
...
oÉ~ä=bääáéëçáÇ=E`~ëÉ=NF=
ñO óO òO
+ + = N=
~ O ÄO Å O
=
=
=
=====
Figure 142
...
fã~Öáå~êó=bääáéëçáÇ=E`~ëÉ=OF=
ñO óO òO
+ + = −N =
~ O ÄO Å O
=
703
...
ANALYTIC GEOMETRY
=
=====
=
Figure 143
...
eóéÉêÄçäçáÇ=çÑ=O=pÜÉÉíë=E`~ëÉ=QF=
ñO óO òO
+ − = −N =
~ O ÄO Å O
=
=
=
=
Figure 144
...
ANALYTIC GEOMETRY
705
...
=
706
...
bääáéíáÅ=m~ê~ÄçäçáÇ=E`~ëÉ=TF=
ñO óO
+ −ò = M=
~ O ÄO
=
184
CHAPTER 7
...
=
708
...
185
=
CHAPTER 7
...
oÉ~ä=bääáéíáÅ=`óäáåÇÉê=E`~ëÉ=VF=
ñO óO
+ = N=
~ O ÄO
=
=
=
=====
Figure 148
...
fã~Öáå~êó=bääáéíáÅ=`óäáåÇÉê=E`~ëÉ=NMF=
ñO óO
+ = −N =
~ O ÄO
=
711
...
ANALYTIC GEOMETRY
=
======
=
Figure 149
...
oÉ~ä=fåíÉêëÉÅíáåÖ=mä~åÉë=E`~ëÉ=NOF=
ñO óO
− =M=
~ O ÄO
=
713
...
m~ê~ÄçäáÅ=`óäáåÇÉê=E`~ëÉ=NQF=
ñO
−ó =M=
~O
=
187
CHAPTER 7
...
=
715
...
fã~Öáå~êó=m~ê~ääÉä=mä~åÉë=E`~ëÉ=NSF=
ñO
= −N =
~O
=
717
...
ANALYTIC GEOMETRY
7
...
bèì~íáçå=çÑ=~=péÜÉêÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ=
cçêãF=
ñ O + ó O + ò O = oO =
=
=
======
=
Figure 151
...
bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~ I ÄI Å )
(ñ − ~ )O + (ó − Ä)O + (ò − Å )O = o O
720
...
ANALYTIC GEOMETRY
ïÜÉêÉ==
mN (ñ N I ó N I ò N ) I= mO (ñ O I ó O I ò O ) =~êÉ=íÜÉ=ÉåÇë=çÑ=~=Çá~ãÉíÉêK==
=
721
...
dÉåÉê~ä=cçêã
^ñ O + ^ó O + ^ò O + añ + bó + cò + j = M =E^=áë=åçåòÉêçFK==
qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ëéÜÉêÉ=Ü~ë=ÅççêÇáå~íÉë= (~ I ÄI Å ) I=ïÜÉêÉ==
a
b
c
~=−
I= Ä = −
I= Å = −
K=
O^
O^
O^
qÜÉ=ê~Çáìë=çÑ=íÜÉ=ëéÜÉêÉ=áë
aO + b O + c O − Q ^ O j
o=
K
O^
=
=
190
Chapter 8
Differential Calculus
=
=
=
=
cìåÅíáçåëW=ÑI=ÖI=óI=ìI=î=
^êÖìãÉåí=EáåÇÉéÉåÇÉåí=î~êá~ÄäÉFW=ñ=
oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=Ç=
k~íìê~ä=åìãÄÉêW=å=
^åÖäÉW= α =
fåîÉêëÉ=ÑìåÅíáçåW= Ñ −N =
=
=
8
...
724
...
726
...
DIFFERENTIAL CALCULUS
=
=====
=
Figure 152
...
`çãéçëáíÉ=cìåÅíáçå=
ó = Ñ (ì ) I= ì = Ö (ñ ) I= ó = Ñ (Ö (ñ )) =áë=~=ÅçãéçëáíÉ=ÑìåÅíáçåK=
=
728
...
DIFFERENTIAL CALCULUS
=
======
=
Figure 153
...
nì~Çê~íáÅ=cìåÅíáçå==
ó = ñ O I= ñ ∈ o K=
=
=
======
=
Figure 154
...
DIFFERENTIAL CALCULUS
730
...
=
731
...
DIFFERENTIAL CALCULUS
=
=
=
Figure 156
...
ó = ~ñ P + Äñ O + Åñ + Ç I= ñ ∈ o K=
=
195
CHAPTER 8
...
=
733
...
DIFFERENTIAL CALCULUS
======
Figure 158
...
=
197
=
CHAPTER 8
...
pèì~êÉ=oççí=cìåÅíáçå==
ó = ñ I= ñ ∈ [MI ∞ ) K=
=
=======
Figure 160
...
bñéçåÉåíá~ä=cìåÅíáçåë=
ó = ~ ñ I= ~ > M I= ~ ≠ N I=
ó = É ñ =áÑ= ~ = É I= É = OKTNUOUNUOUQSK =
=
=
=
=
Figure 161
...
DIFFERENTIAL CALCULUS
736
...
=
737
...
DIFFERENTIAL CALCULUS
=
====
=
Figure 163
...
eóéÉêÄçäáÅ=`çëáåÉ=cìåÅíáçå==
Éñ + É−ñ
I= ñ ∈ o K=
ó = Åçë Ü ñ I= Åçë Ü ñ =
O
=
======
=
Figure 164
...
DIFFERENTIAL CALCULUS
739
...
=
740
...
DIFFERENTIAL CALCULUS
=
======
=
Figure 166
...
eóéÉêÄçäáÅ=pÉÅ~åí=cìåÅíáçå==
N
O
ó = ëÉÅ Ü ñ I= ó = ëÉÅ Ü ñ =
= ñ − ñ I= ñ ∈ o K=
ÅçëÜ ñ É + É
Figure 167
...
DIFFERENTIAL CALCULUS
742
...
=
743
...
DIFFERENTIAL CALCULUS
=
=====
=
Figure 169
...
fåîÉêëÉ=eóéÉêÄçäáÅ=`çëáåÉ=cìåÅíáçå==
ó = ~êÅÅçëÜ ñ I= ñ ∈ [NI ∞ ) K=
=
=
=====
=
Figure 170
...
fåîÉêëÉ=eóéÉêÄçäáÅ=q~åÖÉåí=cìåÅíáçå==
ó = ~êÅí~åÜ ñ I= ñ ∈ (− NI N) K=
=
204
CHAPTER 8
...
=
746
...
DIFFERENTIAL CALCULUS
=
=====
=
Figure 172
...
fåîÉêëÉ=eóéÉêÄçäáÅ=pÉÅ~åí=cìåÅíáçå==
ó = ~êÅëÉÅÜ ñ I= ñ ∈ (MI N] K=
=
206
CHAPTER 8
...
=
=
748
...
=
207
=
CHAPTER 8
...
2 Limits of Functions
=
cìåÅíáçåëW= Ñ (ñ ) I= Ö(ñ ) =
^êÖìãÉåíW=ñ=
oÉ~ä=Åçåëí~åíëW=~I=â=
=
=
749
...
äáã[Ñ (ñ ) − Ö (ñ )] = äáã Ñ (ñ ) − äáã Ö (ñ ) =
ñ →~
ñ →~
ñ →~
=
751
...
äáã
ñ →~
=
753
...
äáã Ñ (Ö (ñ )) = Ñ äáã Ö (ñ ) =
ñ →~
ñ →~
=
755
...
äáã
=
757
...
äáã
208
CHAPTER 8
...
äáã
=
äå(N + ñ )
= N=
ñ →M
ñ
760
...
äáã N + = É =
ñ →∞
ñ
=
ñ
â
762
...
äáã ~ ñ = N =
ñ →M
=
=
=
8
...
ó′(ñ ) = äáã
==
= äáã
=
∆ñ → M
∆ñ →M ∆ñ
Çñ
Ɩ
=
209
CHAPTER 8
...
=
Çó
= í~å α ==
Çñ
765
...
=
767
...
Ç(ì + î ) Çì Çî
=
+
=
Çñ
Çñ Çñ
Ç(ì − î ) Çì Çî
=
−
=
Çñ
Çñ Çñ
Ç(âì )
Çì
=â
=
Çñ
Çñ
=
769
...
DIFFERENTIAL CALCULUS
770
...
`Ü~áå=oìäÉ=
ó = Ñ (Ö(ñ )) I= ì = Ö(ñ ) I==
Çó Çó Çì
=
⋅
K=
Çñ Çì Çñ
=
772
...
oÉÅáéêçÅ~ä=oìäÉ=
Çó
Ç N
= − Çñ =
Çñ ó
óO
=
774
...
4 Table of Derivatives
=
fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=
oÉ~ä=Åçåëí~åíëW=`I=~I=ÄI=Å=
k~íìê~ä=åìãÄÉêW=å=
211
CHAPTER 8
...
=
Ç
(ñ ) = N =
Çñ
776
...
=
778
...
=
780
...
=
Ç
Çñ
( ñ ) = O Nñ =
Ç
Çñ
782
...
N
å
å å ñ å −N
=
=
Ç
(äå ñ ) = N =
ñ
Çñ
784
...
=
212
CHAPTER 8
...
=
787
...
=
Ç
(Åçë ñ ) = − ëáå ñ =
Çñ
789
...
=
Ç
(Åçí ñ ) = − NO = − ÅëÅ O ñ =
ëáå ñ
Çñ
791
...
=
Ç
(ÅëÅ ñ ) = − Åçí ñ ⋅ ÅëÅ ñ =
Çñ
793
...
=
Ç
(~êÅÅçë ñ ) = − N O =
Çñ
N− ñ
795
...
=
213
CHAPTER 8
...
=
Ç
(~êÅ ëÉÅ ñ ) = NO =
Çñ
ñ ñ −N
798
...
=
Ç
(ëáåÜ ñ ) = ÅçëÜ ñ =
Çñ
800
...
=
Ç
(í~åÜ ñ ) = N O = ëÉÅÜ O ñ =
ÅçëÜ ñ
Çñ
802
...
=
Ç
(ëÉÅÜ ñ ) = −ëÉÅÜ ñ ⋅ í~åÜ ñ =
Çñ
804
...
=
Ç
(~êÅëáåÜ=ñ ) = N =
Çñ
ñO +N
806
...
N
Ç
(~êÅÅçëÜ=ñ ) = O =
Çñ
ñ −N
214
CHAPTER 8
...
=
Ç
(~êÅÅçíÜ=ñ ) = − ON I= ñ > N K=
Çñ
ñ −N
809
...
Ç î
Çî
(ì ) = îì î −N ⋅ Çì + ì î äå ì ⋅ Çñ =
Çñ
Çñ
=
=
=
8
...
812
...
814
...
=
cìåÅíáçåëW=ÑI=óI=ìI=î=
fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=
k~íìê~ä=åìãÄÉêW=å=
=
=
pÉÅçåÇ=ÇÉêáî~íáîÉ=
O
′ = Çó ′ = Ç Çó = Ç ó =
Ñ ′′ = (Ñ ′)
O
Çñ Çñ Çñ Çñ
=
eáÖÜÉê-lêÇÉê=ÇÉêáî~íáîÉ=
Çå ó
(å )
Ñ = å = ó (å ) = (Ñ (å −N) ) ′ =
Çñ
=
(ì + î )(å ) = ì (å ) + î (å ) =
=
(ì − î )(å ) = ì(å ) − î (å ) =
=
iÉáÄåáíò∞ë=cçêãìä~ë=
(ìî )′′ = ì′′î + Oì′î′ + ìî′′ =
215
CHAPTER 8
...
=
=
ã −å
=
(ñ )( ) = å> =
å
817
...
ñ äå ~
=
819
...
=
ñ
821
...
å
å
å ãñ
äå å ~ =
=
(ëáå ñ )(å ) = ëáå ñ + åπ =
823
...
O
=
=
=
216
CHAPTER 8
...
6 Applications of Derivative
=
cìåÅíáçåëW=ÑI=ÖI=ó=
mçëáíáçå=çÑ=~å=çÄàÉÅíW=ë==
sÉäçÅáíóW=î=
^ÅÅÉäÉê~íáçåW=ï=
fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=
qáãÉW=í=
k~íìê~ä=åìãÄÉêW=å=
=
=
825
...
q~åÖÉåí=iáåÉ=
ó − ó M = Ñ ′(ñ M )(ñ − ñ M ) =
=
217
CHAPTER 8
...
=
827
...
fåÅêÉ~ëáåÖ=~åÇ=aÉÅêÉ~ëáåÖ=cìåÅíáçåëK==
fÑ= Ñ ′(ñ M ) > M I=íÜÉå=ÑEñF=áë=áåÅêÉ~ëáåÖ=~í= ñ M K=EcáÖ=NTTI= ñ < ñ N I=
ñ O < ñ FI=
fÑ= Ñ ′(ñ M ) < M I=íÜÉå=ÑEñF=áë=ÇÉÅêÉ~ëáåÖ=~í= ñ M K=EcáÖ=NTTI=
ñ N < ñ < ñ O FI=
fÑ= Ñ ′ (ñ M ) =ÇçÉë=åçí=Éñáëí=çê=áë=òÉêçI=íÜÉå=íÜÉ=íÉëí=Ñ~áäëK==
=
218
CHAPTER 8
...
=
829
...
`êáíáÅ~ä=mçáåíë=
^=ÅêáíáÅ~ä=éçáåí=çå=ÑEñF=çÅÅìêë=~í= ñ M =áÑ=~åÇ=çåäó=áÑ=ÉáíÜÉê=
Ñ ′ (ñ M ) =áë=òÉêç=çê=íÜÉ=ÇÉêáî~íáîÉ=ÇçÉëå∞í=ÉñáëíK=
=
831
...
DIFFERENTIAL CALCULUS
832
...
pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=içÅ~ä=bñíêÉã~K=
fÑ= Ñ ′ (ñ N ) = M =~åÇ= Ñ ′′(ñ N ) < M I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ã~ñáãìã=
~í== ñ N K=
fÑ= Ñ ′ (ñ O ) = M =~åÇ= Ñ ′′(ñ O ) > M I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=
~í= ñ O K=EcáÖKNTTF=
=
834
...
pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=`çåÅ~îáíóK==
fÑ= Ñ ′′(ñ M ) > M I=íÜÉå=ÑEñF=áë=ÅçåÅ~îÉ=ìéï~êÇ=~í= ñ M K==
fÑ= Ñ ′′(ñ M ) < M I=íÜÉå=ÑEñF=áë=ÅçåÅ~îÉ=Ççïåï~êÇ=~í= ñ M K=
fÑ= Ñ ′′(ñ ) =ÇçÉë=åçí=Éñáëí=çê=áë=òÉêçI=íÜÉå=íÜÉ=íÉëí=Ñ~áäëK=
=
836
...
i∞eçéáí~ä∞ë=oìäÉ=
M
Ñ ′(ñ )
Ñ (ñ )
= äáã
=áÑ= äáã Ñ (ñ ) = äáã Ö (ñ ) = K==
äáã
ñ → Å Ö (ñ )
ñ →Å Ö ′(ñ )
ñ →Å
ñ →Å
∞
=
220
CHAPTER 8
...
7 Differential
=
cìåÅíáçåëW=ÑI=ìI=î=
fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=
aÉêáî~íáîÉ=çÑ=~=ÑìåÅíáçåW= ó ′(ñ ) I= Ñ ′(ñ ) =
oÉ~ä=Åçåëí~åíW=`=
aáÑÑÉêÉåíá~ä=çÑ=ÑìåÅíáçå= ó = Ñ (ñ ) W=Çó=
aáÑÑÉêÉåíá~ä=çÑ=ñW=Çñ=
pã~ää=ÅÜ~åÖÉ=áå=ñW= ∆ñ =
pã~ää=ÅÜ~åÖÉ=áå=óW= ∆ó =
=
=
838
...
Ñ (ñ + ∆ñ ) = Ñ (ñ ) + Ñ ′(ñ )∆ñ =
=
=
=
Figure 178
...
DIFFERENTIAL CALCULUS
840
...
Ç(ì + î ) = Çì + Çî =
=
842
...
Ç(`ì ) = `Çì =
=
844
...
Ç =
=
îO
î
=
=
=
8
...
cáêëí=lêÇÉê=m~êíá~ä=aÉêáî~íáîÉë=
qÜÉ=é~êíá~ä=ÇÉêáî~íáîÉ=ïáíÜ=êÉëéÉÅí=íç=ñ=
∂Ñ
∂ò
= Ñ ñ =E~äëç= = ò ñ FI=
∂ñ
∂ñ
qÜÉ=é~êíá~ä=ÇÉêáî~íáîÉ=ïáíÜ=êÉëéÉÅí=íç=ó=
∂Ñ
∂ò
= Ñ ó =E~äëç= = ò ó FK=
∂ó
∂ó
=
=
222
CHAPTER 8
...
pÉÅçåÇ=lêÇÉê=m~êíá~ä=aÉêáî~íáîÉë=
∂ ∂Ñ ∂ OÑ
= Ñ ññ I==
=
∂ñ ∂ñ ∂ñ O
∂ ∂Ñ ∂ OÑ
=
= Ñ óó I==
∂ó ∂ó ∂ó O
∂ ∂Ñ ∂ OÑ
= Ñ ñó I==
=
∂ó ∂ñ ∂ó∂ñ
∂ ∂Ñ ∂ OÑ
=
= Ñ óñ K==
∂ñ ∂ó ∂ñ∂ó
fÑ=íÜÉ=ÇÉêáî~íáîÉë=~êÉ=ÅçåíáåìçìëI=íÜÉå==
∂ OÑ
∂ OÑ
=
K==
∂ó∂ñ ∂ñ∂ó
=
848
...
pã~ää=`Ü~åÖÉë=
∂Ñ
∂Ñ
Ƙ ŠƖ + Ɨ =
∂ó
∂ñ
=
=
223
CHAPTER 8
...
içÅ~ä=j~ñáã~=~åÇ=jáåáã~=
Ñ (ñ I ó ) =Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í= (ñ M I ó M ) =áÑ= Ñ (ñ I ó ) ≤ Ñ (ñ M I ó M ) =
Ñçê=~ää= (ñ I ó ) =ëìÑÑáÅáÉåíäó=ÅäçëÉ=íç= (ñ M I ó M ) K==
=
Ñ (ñ I ó ) =Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= (ñ M I ó M ) =áÑ= Ñ (ñ I ó ) ≥ Ñ (ñ M I ó M ) =
Ñçê=~ää= (ñ I ó ) =ëìÑÑáÅáÉåíäó=ÅäçëÉ=íç= (ñ M I ó M ) K=
=
851
...
p~ÇÇäÉ=mçáåí=
^=ëí~íáçå~êó==éçáåí==ïÜáÅÜ==áë==åÉáíÜÉê==~==äçÅ~ä==ã~ñáãìã=
åçê=~=äçÅ~ä=ãáåáãìã=
=
853
...
q~åÖÉåí=mä~åÉ=
qÜÉ=Éèì~íáçå=çÑ=íÜÉ=í~åÖÉåí=éä~åÉ=íç=íÜÉ=ëìêÑ~ÅÉ= ò = Ñ (ñ I ó ) =
~í= (ñ M I ó M I ò M ) =áë==
ò − ò M = Ñ ñ (ñ M I ó M )(ñ − ñ M ) + Ñ ó (ñ M I ó M )(ó − ó M ) K=
=
224
CHAPTER 8
...
kçêã~ä=íç=pìêÑ~ÅÉ=
qÜÉ=Éèì~íáçå=çÑ=íÜÉ=åçêã~ä=íç=íÜÉ=ëìêÑ~ÅÉ= ò = Ñ (ñ I ó ) =~í=
(ñ M I ó M I ò M ) =áë==
ñ − ñM
ó − óM
ò − òM
=
=
K=
Ñ ñ ( ñ M I ó M ) Ñ ó (ñ M I ó M )
−N
=
=
=
8
...
dê~ÇáÉåí=çÑ=~=pÅ~ä~ê=cìåÅíáçå=
∂Ñ ∂Ñ ∂Ñ
Öê~Ç Ñ = ∇Ñ = I I I==
∂ñ ∂ó ∂ò
∂ì ∂ì
∂ì
K=
Öê~Ç ì = ∇ì =
I
IKI
∂ñ ∂ñ
∂ñ å
O
N
=
857
...
DIFFERENTIAL CALCULUS
ïÜÉêÉ=íÜÉ=ÇáêÉÅíáçå=áë=ÇÉÑáåÉÇ=Äó=íÜÉ=îÉÅíçê=
r
ä (Åçë αI Åçë βI Åçë γ ) I= Åçë O α + Åçë O β + Åçë O γ = N K==
=
858
...
`ìêä=çÑ=~=sÉÅíçê=cáÉäÇ=
r
r
r
á
à
â
r
r ∂
∂
∂
=
Åìêä c = ∇ × c =
∂ñ ∂ñ ∂ñ
m n o
∂o ∂n r ∂m ∂o r ∂n ∂m r
=
∂ó − ∂ò á + ∂ò − ∂ñ à + ∂ñ − ∂ó â =
=
860
...
Çáî Åìêä c = ∇ ⋅ ∇ × c ≡ M =
=
862
...
Çáî (Öê~Ç Ñ ) = ∇ ⋅ (∇Ñ ) = ∇ OÑ =
=
r
r
r
r
r
864
...
1 Indefinite Integral
=
865
...
=
867
...
=
869
...
N
∫ Ñ (~ñ )Çñ = ~ c(~ñ ) + ` =
227
CHAPTER 9
...
=
N
∫ Ñ (ñ )Ñ ′(ñ )Çñ = O Ñ (ñ ) + ` =
O
872
...
=
874
...
fåíÉÖê~íáçå=Äó=m~êíë=
∫ ìÇî = ìî − ∫ îÇì I==
ïÜÉêÉ= ì(ñ ) I= î (ñ ) =~êÉ=ÇáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåëK==
=
=
=
9
...
=
∫ ñÇñ =
877
...
ñ é +N
+ ` I= é ≠ −N K=
é+N
=
879
...
INTEGRAL CALCULUS
(~ñ + Ä)å+N + ` I= å ≠ −N K=
∫ (~ñ + Ä) Çñ = ~(å + N)
å
880
...
Çñ
= äå ñ + ` =
ñ
=
Çñ
N
∫ ~ñ + Ä = ~ äå ~ñ + Ä + ` =
882
...
ÄÅ − ~Ç
äå Åñ + Ç + ` =
ÅO
=
Çñ
ñ+Ä
N
∫ (ñ + ~ )(ñ + Ä) = ~ − Ä äå ñ + ~ + ` I= ~ ≠ Ä K=
884
...
O
=
ñ OÇñ
N N
O
O
∫ ~ + Äñ = ÄP O (~ + Äñ ) − O~(~ + Äñ ) + ~ äå ~ + Äñ + ` =
886
...
~ + Äñ
+`=
ñ
=
888
...
ñÇñ
∫ (~ + Äñ )
O
=
N
~
äå ~ + Äñ +
+`=
O
Ä
~ + Äñ
=
229
CHAPTER 9
...
=
Çñ
∫ ñ (~ + Äñ )
891
...
Çñ
N ñ −N
+`=
= äå
−N O ñ +N
O
=
Çñ
∫ N− ñ
893
...
O
=
895
...
O
= í~å −N ñ + ` =
=
∫~
O
∫ñ
897
...
Çñ
ñ
N
= í~å −N + ` =
O
~
~
+ñ
ñÇñ
N
= äå(ñ O + ~ O ) + ` =
O
+~
O
=
Çñ
∫ ~ + Äñ
899
...
INTEGRAL CALCULUS
ñÇñ
∫ ~ + Äñ
900
...
=
∫~
902
...
∫ O
=
+ ` I=
äå
~ñ + Äñ + Å
ÄO − Q~Å O~ñ + Ä + ÄO − Q~Å
ÄO − Q~Å > M K=
=
O~ñ + Ä
Çñ
O
=
+ ` I=
~êÅí~å
+ Äñ + Å
Q~Å − ÄO
Q~Å − ÄO
ÄO − Q~Å < M K=
∫ ~ñ
904
...
3 Integrals of Irrational Functions
=
∫
Çñ
O
~ñ + Ä + ` =
=
~ñ + Ä ~
∫
~ñ + Ä Çñ =
∫
905
...
=
907
...
INTEGRAL CALCULUS
∫ñ
908
...
=
~ñ + Ä
Çñ
N
~êÅí~å
=
+ ` I==
~Å − Ä
~ñ + Ä
~Å − Ä
Ä − ~Å < M K=
∫ (ñ + Å )
910
...
~ñ + Ä
N
Çñ =
Åñ + Ç
Å
=
912
...
=
∫
914
...
Çñ
N
~ + Äñ − ~
=
äå
+ ` I= ~ > M K=
~ + Äñ
~
~ + Äñ + ~
=
232
CHAPTER 9
...
Çñ
O
~ + Äñ
=
~êÅí~å
+ ` I= ~ < M K=
−~
~ + Äñ
−~
=
∫
~−ñ
Çñ =
Ä+ ñ
∫
~+ñ
Ä−ñ
Çñ = − (~ + ñ )(Ä − ñ ) − (~ + Ä)~êÅëáå
+`=
Ä−ñ
~+Ä
∫
917
...
=
919
...
=
∫
921
...
~ñ O + Äñ + Å
~ > M K=
=
N
äå O~ñ + Ä + O ~ (~ñ O + Äñ + Å ) + ` I==
~
=
∫
923
...
ñ O + ~ O Çñ =
ñ O O ~O
ñ + ~ + äå ñ + ñ O + ~ O + ` =
O
O
=
233
CHAPTER 9
...
ñ O + ~ O Çñ =
N O O PO
(ñ + ~ ) + ` =
P
=
∫ñ
926
...
ñ O + ~O
ñ O + ~O
Çñ = −
+ äå ñ + ñ O + ~ O + ` =
ñO
ñ
=
∫
928
...
ñ O + ~O
ñ
Çñ = ñ O + ~ O + ~ äå
+`=
ñ
~ + ñ O + ~O
=
∫
930
...
ñ OÇñ
ñ O + ~O
=
ñ O O ~O
ñ + ~ − äå ñ + ñ O + ~ O + ` =
O
O
=
∫ñ
932
...
ñ O − ~ O Çñ =
ñ O O ~O
ñ − ~ − äå ñ + ñ O − ~ O + ` =
O
O
=
934
...
INTEGRAL CALCULUS
∫
ñO − ~O
~
Çñ = ñ O − ~ O + ~ ~êÅëáå + ` =
ñ
ñ
∫
935
...
=
∫
937
...
ñÇñ
ñ −~
O
= ñO − ~O + ` =
O
=
∫
939
...
O
O
=
Çñ
∫ (ñ + ~ )
941
...
ñ −~
O
O
=−
N ñ+~
+`=
~ ñ −~
=
∫ñ
943
...
Çñ
O
− ~O ) O
P
=−
ñ
~O ñ O − ~O
+`=
=
235
CHAPTER 9
...
O
− ~ O ) O Çñ = −
P
ñ
(Oñ O − R~ O ) ñ O − ~ O + =
U
P~ Q
+
äå ñ + ñ O − ~ O + ` =
U
=
∫
946
...
N O
(~ − ñ O )P O + ` =
P
=
O
O
O
∫ ñ ~ − ñ Çñ =
948
...
~O − ñ O
ñ
Çñ = ~ O − ñ O + ~ äå
+`=
ñ
~ + ~O − ñ O
~O − ñO
~O − ñO
ñ
Çñ = −
− ~êÅëáå + ` =
O
ñ
ñ
~
=
950
...
Çñ
= ~êÅëáå ñ + ` =
N− ñO
=
∫
952
...
ñÇñ
~ −ñ
O
O
= − ~O − ñ O + ` =
=
954
...
INTEGRAL CALCULUS
Çñ
∫ (ñ + ~ )
955
...
~O − ñ O
=
Çñ
∫ (ñ + Ä )
957
...
~ −ñ
O
O
N
=
~ −Ä
O
O
äå
ñ+Ä
~ −Ä
O
O
~ O − ñ O + ~ O + Äñ
Ä < ~ K=
=
∫ñ
959
...
O
− ñ O ) O Çñ =
P
Q
ñ O
(R~ − Oñ O ) ~ O − ñ O + P~ ~êÅëáå ñ + ` =
U
U
~
=
∫ (~
961
...
4 Integrals of Trigonometric Functions
=
∫ ëáå ñÇñ = − Åçë ñ + ` =
962
...
∫ Åçë ñÇñ = ëáå ñ + ` =
237
+ `I =
CHAPTER 9
...
O
ñ Çñ =
ñ N
− ëáå Oñ + ` =
O Q
ñ Çñ =
ñ N
+ ëáå Oñ + ` =
O Q
=
∫ Åçë
965
...
P
N
N
P
ñ Çñ = ÅçëP ñ − Åçë ñ + ` = Åçë Pñ − Åçë ñ + ` =
NO
Q
P
=
∫ Åçë
967
...
=
π
∫ Åçë ñ = ∫ ëÉÅ ñ Çñ = äå í~å O + Q + ` =
969
...
O
ñ
= ∫ ÅëÅ O ñ Çñ = − Åçí ñ + ` =
ñ
= ∫ ëÉÅ O ñ Çñ = í~å ñ + ` =
ñ
= ∫ ÅëÅ P ñ Çñ = −
ñ
= ∫ ëÉÅ P ñ Çñ =
=
Çñ
∫ Åçë
971
...
P
Åçë ñ
N
ñ
+ äå í~å + ` =
O
O ëáå ñ O
O
=
Çñ
∫ Åçë
973
...
N
∫ ëáå ñ Åçë ñ Çñ = − Q Åçë Oñ + ` =
238
CHAPTER 9
...
O
N
ñ Åçë ñ Çñ = ëáå P ñ + ` =
P
=
∫ ëáå ñ Åçë
976
...
O
ñ Åçë O ñ Çñ =
ñ N
− ëáå Q ñ + ` =
U PO
=
∫ í~å ñÇñ = − äå Åçë ñ + ` =
978
...
=
ëáå O ñ
ñ π
∫ Åçë ñ Çñ = äå í~å O + Q − ëáå ñ + ` =
980
...
O
ñ Çñ = í~å ñ − ñ + ` =
=
∫ Åçí ñÇñ = äå ëáå ñ + ` =
982
...
=
Åçë O ñ
ñ
Çñ = äå í~å + Åçë ñ + ` =
984
...
∫ Åçí O ñ Çñ = − Åçí ñ − ñ + ` =
=
986
...
INTEGRAL CALCULUS
∫ ëáå
987
...
O
ñ
=
N
ñ
+ äå í~å + ` =
Åçë ñ
O
=
∫ ëáå
989
...
+
ëáå(ã − å )ñ
+ ` I=
O(ã − å )
−
Åçë(ã − å )ñ
+ ` I=
O(ã − å )
ã O ≠ å O K=
=
Åçë(ã + å )ñ
∫ ëáå ãñ Åçë åñ Çñ = − O(ã + å)
991
...
ã O ≠ å O K=
=
∫ ëÉÅ ñ í~å ñÇñ = ëÉÅ ñ + ` =
993
...
=
å
∫ ëáå ñ Åçë ñ Çñ = −
995
...
∫ ëáå ñ Åçë ñ Çñ =
+`=
å +N
=
å
240
+
ëáå(ã − å )ñ
+ ` I=
O(ã − å )
CHAPTER 9
...
N− ñO + ` =
=
998
...
O
+ N) + ` =
=
N
1000
...
5 Integrals of Hyperbolic Functions
=
1001
...
∫ ÅçëÜ ñÇñ = ëáåÜ ñ + ` =
=
1003
...
∫ ÅçíÜ ñ Çñ = äå ëáåÜ ñ + ` =
=
1005
...
∫ ÅëÅÜ O ñÇñ = − ÅçíÜ ñ + ` =
=
1007
...
INTEGRAL CALCULUS
1008
...
6 Integrals of Exponential and Logarithmic
Functions
=
1009
...
∫ ~ Çñ =
+`=
äå ~
=
É ~ñ
1011
...
∫ ñÉ ~ñ Çñ = O (~ñ − N) + ` =
~
=
1013
...
=
äå ñ
N
1015
...
∫ É ~ñ ëáå Äñ Çñ =
É +`=
~ O + ÄO
=
242
CHAPTER 9
...
∫ É ~ñ Åçë Äñ Çñ =
~ Åçë Äñ + Ä ëáå Äñ ~ñ
É +`=
~ O + ÄO
=
=
=
9
...
∫ ñ å É ãñ Çñ =
N å ãñ å
ñ É − ∫ ñ å −NÉ ãñ Çñ =
ã
ã
=
É ãñ
É ãñ
ã É ãñ
Çñ = −
+
Çñ I= å ≠ N K=
∫ ñå
(å − N)ñ å−N å − N ∫ ñ å−N
1019
...
∫ ëáåÜ å ñÇñ =
=
Çñ
∫ ëáåÜ
1021
...
∫ ÅçëÜ å ñÇñ =
=
Çñ
∫ ÅçëÜ
1023
...
∫ ëáåÜ ñ ÅçëÜ ñÇñ =
=
å+ã
ã −N
å
ã −O
+
∫ ëáåÜ ñ ÅçëÜ ñÇñ =
å+ã
=
ëáåÜ å −N ñ ÅçëÜ ã +N ñ
1025
...
INTEGRAL CALCULUS
−
å −N
å −O
ã
∫ ëáåÜ ñ ÅçëÜ ñÇñ =
å+ã
=
1026
...
∫ ÅçíÜ å ñÇñ = −
N
ÅçíÜ å −N ñ + ∫ ÅçíÜ å − O ñÇñ I= å ≠ N K=
å −N
=
ëÉÅÜ å −O ñ í~åÜ ñ å − O
å −O
1028
...
∫ ëáå å ñÇñ = − ëáå å −N ñ Åçë ñ +
∫ ëáå ñÇñ =
å
å
=
Çñ
Åçë ñ
å−O
Çñ
1030
...
∫ Åçë å ñÇñ = ëáå ñ Åçë å −N ñ +
∫ Åçë ñÇñ =
å
å
=
Çñ
ëáå ñ
å−O
Çñ
1032
...
∫ ëáå å ñ Åçë ã ñÇñ =
=
å+ã
ã −N
+
ëáå å ñ Åçë ã −O ñÇñ =
å+ã∫
=
ëáå å −N ñ Åçë ã +N ñ
1034
...
INTEGRAL CALCULUS
+
å −N
å −O
ã
∫ ëáå ñ Åçë ñÇñ =
å+ã
=
1035
...
∫ Åçí å ñÇñ = −
N
Åçí å −N ñ − ∫ Åçí å−O ñÇñ I= å ≠ N K=
å −N
=
ëÉÅ å −O ñ í~å ñ å − O
å −O
1037
...
∫ ÅëÅ å ñÇñ = −
+
∫ ÅëÅ ñÇñ I= å ≠ N K=
å −N
å −N
=
ñ å +N äå ã ñ ã
1039
...
∫ å Çñ = −
(å − N)ñ å−N å − N ∫ ñ å
ñ
=
1041
...
∫ ñ å ëáåÜ ñÇñ = ñ å ÅçëÜ ñ − å ∫ ñ å −N ÅçëÜ ñÇñ =
=
1043
...
∫ ñ å ëáå ñÇñ = − ñ å Åçë ñ + å ∫ ñ å −N Åçë ñÇñ =
=
1045
...
INTEGRAL CALCULUS
1046
...
∫ ñ å Åçë −N ñÇñ =
ñ å +N
N
ñ å +N
Åçë −N ñ +
Çñ =
å +N
å + N ∫ N− ñO
=
1048
...
∫ å
= − ∫ å
=
~ñ + Ä ~ ~ ~ñ + Ä
=
Çñ
− O~ñ − Ä
1050
...
Çñ
+~
å ≠ N K=
O
)
O å
=
ñ
O(å − N)~ O (ñ O + ~
)
O å −N
+
Oå − P
O(å − N)~ O
=
∫ (ñ
1052
...
INTEGRAL CALCULUS
9
...
∫ Ñ (ñ )Çñ =
~
å
äáã
∑ Ñ (ξ )∆ñ
å→∞
ã~ñ ∆ñ á →M á =N
á
á
I==
ïÜÉêÉ== ∆ñ á = ñ á − ñ á −N I== ñ á −N ≤ ξ á ≤ ñ á K==
=
=
=
Figure 179
...
INTEGRAL CALCULUS
Ä
1054
...
∫ âÑ (ñ )Çñ = â ∫ Ñ (ñ )Çñ =
=
Ä
Ä
Ä
~
~
~
Ä
Ä
Ä
~
~
~
∫ [Ñ (ñ ) + Ö(ñ )]Çñ = ∫ Ñ (ñ )Çñ + ∫ Ö(ñ )Çñ =
1056
...
=
~
1058
...
∫ Ñ (ñ )Çñ = − ∫ Ñ (ñ )Çñ =
=
Ä
Å
Ä
~
~
Å
1060
...
∫ Ñ (ñ )Çñ ≥ M =áÑ= Ñ (ñ ) ≥ M =çå= [~ I Ä] K=
~
=
Ä
1062
...
INTEGRAL CALCULUS
1063
...
jÉíÜçÇ=çÑ=pìÄëíáíìíáçå==
fÑ= ñ = Ö (í ) I=íÜÉå==
Ä
Ç
~
Å
∫ Ñ (ñ )Çñ = ∫ Ñ (Ö(í ))Ö′(í )Çí I==
ïÜÉêÉ=
Å = Ö −N (~ ) I= Ç = Ö −N (Ä) K=
=
1065
...
qê~éÉòçáÇ~ä=oìäÉ=
Ä
å −N
Ä−~
Ñ (ñ )Çñ =
Ñ (ñ M ) + Ñ (ñ å ) + O∑ Ñ (ñ á ) =
∫
Oå
á =N
~
=
249
CHAPTER 9
...
=
1067
...
INTEGRAL CALCULUS
=
=
Figure 181
...
^êÉ~=råÇÉê=~=`ìêîÉ=
Ä
p = ∫ Ñ (ñ )Çñ = c(Ä) − c(~ ) I==
~
ïÜÉêÉ= c′(ñ ) = Ñ (ñ ) K=
=
251
CHAPTER 9
...
=
1069
...
INTEGRAL CALCULUS
=
=
Figure 183
...
9 Improper Integral
=
Ä
1070
...
fÑ= Ñ (ñ ) =áë=~=Åçåíáåìçìë=ÑìåÅíáçå=çå= [~ I ∞ ) I=íÜÉå==
•
•
∞
å
∫ Ñ (ñ )Çñ = äáã ∫ Ñ (ñ )Çñ K=
~
å →∞
~
=
253
CHAPTER 9
...
=
1072
...
254
CHAPTER 9
...
=
=
=
Figure 186
...
`çãé~êáëçå=qÜÉçêÉãë=
iÉí== Ñ (ñ ) =~åÇ== Ö(ñ ) ==ÄÉ==Åçåíáåìçìë==ÑìåÅíáçåë==çå=íÜÉ=ÅäçëÉÇ=
áåíÉêî~ä= [~ I ∞ ) K= pìééçëÉ= íÜ~í= M ≤ Ö (ñ ) ≤ Ñ (ñ ) = Ñçê= ~ää= ñ= áå=
[~I ∞ ) K=
255
CHAPTER 9
...
^ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=
=
∞
∞
~
~
fÑ= ∫ Ñ (ñ ) Çñ =áë=ÅçåîÉêÖÉåíI=íÜÉå=íÜÉ=áåíÉÖê~ä ∫ Ñ (ñ )Çñ =áë=~ÄëçäìíÉäó=ÅçåîÉêÖÉåíK===
=
1076
...
256
CHAPTER 9
...
iÉí= Ñ (ñ ) =ÄÉ=~=Åçåíáåìçìë=ÑìåÅíáçå=Ñçê=~ää=êÉ~ä=åìãÄÉêë==ñ==áå=
íÜÉ=áåíÉêî~ä== [~ I Ä] ==ÉñÅÉéí==Ñçê==ëçãÉ=éçáåí==Å==áå= (~ I Ä) K=qÜÉå=
Å −ε
Ä
Ä
∫ Ñ (ñ )Çñ = äáã ∫ Ñ (ñ )Çñ + äáã ∫ Ñ (ñ )Çñ K=
~
ε →M +
δ →M +
~
Å +δ
=
=
=
Figure 188
...
10 Double Integral
=
cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= Ñ (ñ I ó ) I= Ñ (ìI î ) I=£=
açìÄäÉ=áåíÉÖê~äëW= ∫∫ Ñ (ñ I ó )ÇñÇó I= ∫∫ Ö (ñ I ó )ÇñÇó I=£=
o
ã
å
o
(
)
oáÉã~åå=ëìãW= ∑∑ Ñ ì á I î à ∆ñ á ∆ó à =
á =N à=N
pã~ää=ÅÜ~åÖÉëW= ∆ñ á I= ∆ó à =
oÉÖáçåë=çÑ=áåíÉÖê~íáçåW=oI=p==
mçä~ê=ÅççêÇáå~íÉëW= ê I= θ =
257
CHAPTER 9
...
aÉÑáåáíáçå=çÑ=açìÄäÉ=fåíÉÖê~ä=
qÜÉ=ÇçìÄäÉ=áåíÉÖê~ä=çîÉê=~=êÉÅí~åÖäÉ= [~I Ä]× [ÅI Ç] =áë=ÇÉÑáåÉÇ=
íç=ÄÉ==
∫∫ Ñ (ñ I ó )Ç^ = äáã
[~ I Ä ]×[Å I Ç ]
(
)
∑∑ Ñ (ì I î )∆ñ ∆ó
ã
å
ã~ñ ∆ñ á →M
ã~ñ ∆ó à → M á =N à=N
á
à
á
à
I==
ïÜÉêÉ= ì á I î à =áë=ëçãÉ=éçáåí=áå=íÜÉ=êÉÅí~åÖäÉ=
(ñ á −N I ñ á )× ó à−N I ó à I=~åÇ= ∆ñ á = ñ á − ñ á −N I= ∆ó à = ó à − ó à−N K=
=
===
(
)
Figure 189
...
INTEGRAL CALCULUS
qÜÉ=ÇçìÄäÉ=áåíÉÖê~ä=çîÉê=~=ÖÉåÉê~ä=êÉÖáçå=o=áë==
∫∫ Ñ (ñ I ó )Ç^ = ∫∫ Ö(ñ I ó )Ç^ I==
o
[~ I Ä ]×[Å I Ç ]
ïÜÉêÉ=êÉÅí~åÖäÉ= [~I Ä]× [ÅI Ç] =Åçåí~áåë=oI==
Ö(ñ I ó ) = Ñ (ñ I ó ) =áÑ= Ñ (ñ I ó ) =áë=áå=o=~åÇ= Ö(ñ I ó ) = M =çíÜÉêïáëÉK=
=
=
=
Figure 190
...
∫∫ [Ñ (ñ I ó ) + Ö(ñ I ó )]Ç^ = ∫∫ Ñ (ñ I ó )Ç^ + ∫∫ Ö(ñ I ó )Ç^ =
o
=
1080
...
o
o
o
∫∫ âÑ (ñ I ó )Ç^ = â ∫∫ Ñ (ñ I ó )Ç^ I==
o
o
ïÜÉêÉ=â=áë=~=Åçåëí~åíK=
=
1082
...
fÑ= Ñ (ñ I ó ) ≥ M =çå=o=~åÇ= p ⊂ o I=íÜÉå=
259
o
CHAPTER 9
...
=
1084
...
=
=
260
CHAPTER 9
...
fíÉê~íÉÇ=fåíÉÖê~äë=~åÇ=cìÄáåá∞ë=qÜÉçêÉã=
Ä è(ñ )
∫∫ Ñ (ñ I ó )Ç^ = ∫ ∫ Ñ) (ñ I ó )ÇóÇñ ==
(
o
~é ñ
Ñçê=~=êÉÖáçå=çÑ=íóéÉ=fI==
o = {(ñ I ó ) ö ~ ≤ ñ ≤ ÄI é(ñ ) ≤ ó ≤ è(ñ )} K=
=
=
=
Figure 193
...
INTEGRAL CALCULUS
=
=
Figure 194
...
açìÄäÉ=fåíÉÖê~äë=çîÉê=oÉÅí~åÖìä~ê=oÉÖáçåë=
=
fÑ=o=áë=íÜÉ=êÉÅí~åÖìä~ê=êÉÖáçå= [~I Ä]× [ÅI Ç] I=íÜÉå==
Ä Ç
Ç Ä
Ñ (ñ I ó )ÇñÇó = ∫ ∫ Ñ (ñ I ó )Çó Çñ = ∫ ∫ Ñ (ñ I ó )Çñ Çó K==
∫∫
o
~Å
Å~
=
få=íÜÉ=ëéÉÅá~ä=Å~ëÉ=ïÜÉêÉ=íÜÉ=áåíÉÖê~åÇ= Ñ (ñ I ó ) =Å~å=ÄÉ=ïêáííÉå=~ë= Ö (ñ )Ü(ó ) =ïÉ=Ü~îÉ==
Ç
Ä
∫ Ö (ñ )Çñ ∫ Ü(ó )Çó K==
Ñ (ñ I ó )ÇñÇó = ∫∫ Ö(ñ )Ü(ó )ÇñÇó =
∫∫
o
o
Å
~
=
1087
...
INTEGRAL CALCULUS
Å~å=ÄÉ=ÅçãéìíÉÇ=Äó= ñ = ñ (ìI î ) I= ó = ó (ìI î ) =áåíç=íÜÉ=ÇÉÑáåáíáçå=çÑ=oK==
=
1088
...
=
1089
...
INTEGRAL CALCULUS
=
=
Figure 196
...
=
=
264
CHAPTER 9
...
^êÉ~=çÑ=~=oÉÖáçå=
Ä Ñ (ñ )
^ = ∫ ∫ ÇóÇñ =EÑçê=~=íóéÉ=f=êÉÖáçåFK=
~ Ö (ñ )
=
=
=
Figure 198
...
=
=
265
CHAPTER 9
...
sçäìãÉ=çÑ=~=pçäáÇ=
s = ∫∫ Ñ (ñ I ó )Ç^ K==
o
=
=
=
Figure 200
...
INTEGRAL CALCULUS
fÑ== Ñ (ñ I ó ) ≥ Ö (ñ I ó ) ==çîÉê==~==êÉÖáçå==oI==íÜÉå==íÜÉ==îçäìãÉ==çÑ=
íÜÉ= ëçäáÇ= ÄÉíïÉÉå= òN = Ñ (ñ I ó ) = ~åÇ= ò O = Ö(ñ I ó ) = çîÉê= o= áë=
ÖáîÉå=Äó=
s = ∫∫ [Ñ (ñ I ó ) − Ö (ñ I ó )]Ç^ K==
o
=
1092
...
=
1093
...
INTEGRAL CALCULUS
1094
...
jçãÉåíë=
qÜÉ=ãçãÉåí=çÑ=íÜÉ=ä~ãáå~=~Äçìí=íÜÉ==ñ-~ñáë==áë=ÖáîÉå=Äó=Ñçêãìä~=
j ñ = ∫∫ óρ(ñ I ó )Ç^ K==
o
=
qÜÉ=ãçãÉåí=çÑ=íÜÉ=ä~ãáå~=~Äçìí=íÜÉ=ó-~ñáë=áë=
j ó = ∫∫ ñρ(ñ I ó )Ç^ K=
o
=
qÜÉ=ãçãÉåí=çÑ=áåÉêíá~=~Äçìí=íÜÉ=ñ-~ñáë=áë=
f ñ = ∫∫ ó Oρ(ñ I ó )Ç^ K==
o
=
qÜÉ=ãçãÉåí=çÑ=áåÉêíá~=~Äçìí=íÜÉ=ó-~ñáë=áë=
f ó = ∫∫ ñ Oρ(ñ I ó )Ç^ K==
o
=
qÜÉ=éçä~ê=ãçãÉåí=çÑ=áåÉêíá~=áë=
fM = ∫∫ (ñ O + ó O )ρ(ñ I ó )Ç^ K==
o
=
1096
...
INTEGRAL CALCULUS
j
N
ó = ñ = ∫∫ óρ(ñ I ó )Ç^ =
ã ã o
∫∫ óρ(ñ I ó )Ç^
o
∫∫ ρ(ñ I ó )Ç^
K==
o
=
1097
...
^îÉê~ÖÉ=çÑ=~=cìåÅíáçå=
N
µ = ∫∫ Ñ (ñ I ó )Ç^ I==
po
ïÜÉêÉ= p = ∫∫ Ç^ K==
o
=
=
=
9
...
INTEGRAL CALCULUS
j~ëë=çÑ=~=ëçäáÇW=ã==
aÉåëáíóW= µ(ñ I ó I ò ) =
`ççêÇáå~íÉë=çÑ=ÅÉåíÉê=çÑ=ã~ëëW= ñ I= ó I= ò =
cáêëí=ãçãÉåíëW= j ñó I= j óò I= j ñò =
jçãÉåíë=çÑ=áåÉêíá~W= f ñó I= f óò I= f ñò I= f ñ I= f ó I= fò I= fM =
=
=
1099
...
∫∫∫ [Ñ (ñ I ó I ò ) + Ö (ñ I ó I ò )]Çs = ∫∫∫ Ñ (ñ I ó I ò )Çs + ∫∫∫ Ö (ñ I ó I ò )Çs
(
d
d
d
=
1101
...
∫∫∫ âÑ (ñ I óI ò )Çs = â ∫∫∫ Ñ (ñ I ó I ò )Çs I==
d
d
ïÜÉêÉ=â=áë=~=Åçåëí~åíK=
=
1103
...
INTEGRAL CALCULUS
1104
...
qêáéäÉ=fåíÉÖê~äë=çîÉê=m~ê~ääÉäÉéáéÉÇ=
fÑ=d=áë=~=é~ê~ääÉäÉéáéÉÇ= [~ I Ä]× [ÅI Ç]× [êI ë] I=íÜÉå=
Ä Ç ë
Ñ (ñ I ó I ò )ÇñÇóÇò = ∫ ∫ ∫ Ñ (ñ I ó I ò )Çò Çó Çñ K==
∫∫∫
d
~ Å ê
=
få==íÜÉ=ëéÉÅá~ä=Å~ëÉ==ïÜÉêÉ=íÜÉ=áåíÉÖê~åÇ== Ñ (ñ I ó I ò ) ==Å~å=ÄÉ=
ïêáííÉå=~ë= Ö (ñ ) Ü(ó ) â (ò ) =ïÉ=Ü~îÉ==
Ä
Ç
ë
Ñ (ñ I ó I ò )ÇñÇóÇò = ∫ Ö (ñ )Çñ ∫ Ü(ó )Çó ∫ â (ò )Çò K==
∫∫∫
d
~
Å
ê
=
1106
...
INTEGRAL CALCULUS
∂ñ ∂ñ ∂ñ
∂ì ∂î ∂ï
∂(ñ I ó I ò ) ∂ó ∂ó ∂ó
ïÜÉêÉ==
=
≠ M ==áë==íÜÉ==à~ÅçÄá~å==çÑ=
∂(ìI î I ï ) ∂ì ∂î ∂ï
∂ò ∂ò ∂ò
∂ì ∂î ∂ï
íÜÉ= íê~åëÑçêã~íáçåë= (ñ I ó I ò ) → (ìI î I ï ) I= ~åÇ= p= áë= íÜÉ= éìääÄ~Åâ= çÑ= d= ïÜáÅÜ= Å~å= ÄÉ= ÅçãéìíÉÇ= Äó= ñ = ñ (ìI î I ï ) I=
ó = ó (ìI î I ï ) =
ò = ò (ìI î I ï ) =áåíç=íÜÉ=ÇÉÑáåáíáçå=çÑ=dK=
=
=
1107
...
qêáéäÉ=fåíÉÖê~äë=áå=péÜÉêáÅ~ä=`ççêÇáå~íÉë=
qÜÉ=aáÑÑÉêÉåíá~ä=ÇñÇóÇò=Ñçê=péÜÉêáÅ~ä=`ççêÇáå~íÉë=áë==
∂ (ñ I ó I ò )
ÇñÇóÇò =
ÇêÇθÇϕ = ê O ëáå θÇêÇθÇϕ ==
∂ (êI θI ϕ)
=
∫∫∫ Ñ (ñ I óI ò )ÇñÇóÇò = =
=
d
272
CHAPTER 9
...
=
1109
...
sçäìãÉ=áå=`óäáåÇêáÅ~ä=`ççêÇáå~íÉë=
s = ∫∫∫ êÇêÇθÇò =
p ( ê I θ Iò )
=
1111
...
INTEGRAL CALCULUS
1112
...
`ÉåíÉê=çÑ=j~ëë=çÑ=~=pçäáÇ=
j óò
j ñó
j
I= ó = ñò I= ò =
I==
ñ=
ã
ã
ã
ïÜÉêÉ==
j óò = ∫∫∫ ñµ(ñ I ó I ò ) Çs I==
d
j ñò = ∫∫∫ óµ(ñ I ó I ò ) Çs I==
d
j ñó = ∫∫∫ òµ(ñ I ó I ò ) Çs =
d
~êÉ==íÜÉ==Ñáêëí==ãçãÉåíë==~Äçìí==íÜÉ==ÅççêÇáå~íÉ=éä~åÉë= ñ = M I=
ó = M I= ò = M I=êÉëéÉÅíáîÉäóI== µ(ñ I ó I ò ) =áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåK==
=
1114
...
jçãÉåíë=çÑ=fåÉêíá~=~Äçìí=íÜÉ=ñ-~ñáëI=ó-~ñáëI=~åÇ=ò-~ñáë=
f ñ = f ñó + f ñò = ∫∫∫ (ò O + ó O )µ(ñ I ó I ò ) Çs I==
d
f ó = f ñó + f óò = ∫∫∫ (ò O + ñ O )µ(ñ I ó I ò ) Çs I==
d
274
CHAPTER 9
...
mçä~ê=jçãÉåí=çÑ=fåÉêíá~=
fM = f ñó + f óò + f ñò = ∫∫∫ (ñ O + ó O + ò O )µ(ñ I ó I ò ) Çs =
d
=
=
9
...
INTEGRAL CALCULUS
1117
...
∫ c Çë = ∫ c Çë + ∫ c Çë =
`N ∪` O
`N
`O
=
=
=
Figure 203
...
fÑ=íÜÉ=ëãççíÜ=ÅìêîÉ=`=áë=é~ê~ãÉíêáòÉÇ=Äó= ê = ê (í ) I=
α ≤ í ≤ β I=íÜÉå==
β
∫ c(ñ I óI ò )Çë = ∫ c(ñ (í )I ó(í )I ò(í )) (ñ ′(í )) + (ó′(í )) + (ò′(í )) Çí K=
O
O
O
α
`
=
1120
...
iáåÉ=fåíÉÖê~ä=çÑ=pÅ~ä~ê=cìåÅíáçå=áå=mçä~ê=`ççêÇáå~íÉë=
276
CHAPTER 9
...
iáåÉ=fåíÉÖê~ä=çÑ=sÉÅíçê=cáÉäÇ=
r r
iÉí=~=ÅìêîÉ=`=ÄÉ=ÇÉÑáåÉÇ=Äó=íÜÉ=îÉÅíçê=ÑìåÅíáçå= ê = ê (ë ) I=
M ≤ ë ≤ p K=qÜÉå==
r
Çê r
= τ = (Åçë αI Åçë βI Åçë γ ) ==
Çë
áë=íÜÉ=ìåáí=îÉÅíçê=çÑ=íÜÉ=í~åÖÉåí=äáåÉ=íç=íÜáë=ÅìêîÉK==
=
O
=
=
Figure 204
...
INTEGRAL CALCULUS
1123
...
fÑ=íÜÉ=ÅìêîÉ=`=áë=é~ê~ãÉíÉêáòÉÇ=Äó= ê (í ) = ñ (í )I ó (í )I ò (í ) I=
α ≤ í ≤ β I=íÜÉå==
∫ mÇñ + nÇó + oÇò = =
`
β
Çó
Çñ
Çò
= ∫ m(ñ (í )I ó (í )I ò (í )) + n(ñ (í )I ó (í )I ò (í )) + o(ñ (í )I ó (í )I ò (í )) Çí
Çí
Çí
Çí
α
=
1125
...
dêÉÉå∞ë=qÜÉçêÉã=
∂n ∂m
∫∫ ∂ñ − ∂ó ÇñÇó = ∫ mÇñ + nÇó I==
o
`
r
r
r
ïÜÉêÉ= c = m(ñ I ó ) á + n(ñ I ó ) à = áë= ~= Åçåíáåìçìë= îÉÅíçê= ÑìåÅ∂m ∂n
íáçå=ïáíÜ==Åçåíáåìçìë==Ñáêëí=é~êíá~ä==ÇÉêáî~íáîÉë=
I=
=áå=~=
∂ó ∂ñ
ëçãÉ= Ççã~áå= oI= ïÜáÅÜ= áë= ÄçìåÇÉÇ= Äó= ~= ÅäçëÉÇI= éáÉÅÉïáëÉ=
ëãççíÜ=ÅìêîÉ=`K==
=
=
278
CHAPTER 9
...
^êÉ~=çÑ=~=oÉÖáçå=o=_çìåÇÉÇ=Äó=íÜÉ=`ìêîÉ=`=
N
p = ∫∫ ÇñÇó = ∫ ñÇó − óÇñ =
O`
o
=
1128
...
qÉëí=Ñçê=~=`çåëÉêî~íáîÉ=cáÉäÇ=
r
^=îÉÅíçê=ÑáÉäÇ=çÑ=íÜÉ=Ñçêã= c = Öê~Ç ì =áë=Å~ääÉÇ=~=ÅçåëÉêî~íáîÉ=
r
r
r
r
ÑáÉäÇK=qÜÉ=äáåÉ=áåíÉÖê~ä=çÑ=~=îÉÅíçê=ÑìåÅíáçå= c = m á + n à + oâ =
áë=é~íÜ=áåÇÉéÉåÇÉåí=áÑ=~åÇ=çåäó=áÑ==
r
r
r
á
à
â
r ∂
∂
∂ r
Åìêä c =
= M K==
∂ñ ∂ó ∂ò
m n o
=
fÑ=íÜÉ=äáåÉ=áåíÉÖê~ä=áë=í~âÉå=áå=ñó-éä~åÉ=ëç=íÜ~í==
∫ mÇñ + nÇó = ì(_) − ì(^ ) I==
`
íÜÉå=íÜÉ=íÉëí=Ñçê=ÇÉíÉêãáåáåÖ=áÑ=~=îÉÅíçê=ÑáÉäÇ=áë=ÅçåëÉêî~íáîÉ=
Å~å=ÄÉ=ïêáííÉå=áå=íÜÉ=Ñçêã==
∂m ∂n
=
K==
∂ó ∂ñ
=
279
CHAPTER 9
...
iÉåÖíÜ=çÑ=~=`ìêîÉ=
r
O
O
O
β
β
Çó
Çê
i = ∫ Çë = ∫
(í ) Çí = ∫ Çñ + + Çò Çí I=
Çí
Çí Çí Çí
`
α
α
ïÜÉêÉ=`=á~=~=éáÉÅÉïáëÉ=ëãççíÜ=ÅìêîÉ=ÇÉëÅêáÄÉÇ=Äó=íÜÉ=éçëár
íáçå=îÉÅíçê= ê (í ) I= α ≤ í ≤ β K=
=
fÑ=íÜÉ=ÅìêîÉ=`=áë=íïç-ÇáãÉåëáçå~äI=íÜÉå=
r
O
O
β
β
Çê
Çñ Çó
(í ) Çí = ∫ + Çí K==
i = ∫ Çë = ∫
Çí
Çí Çí
α
α
`
=
fÑ=íÜÉ=ÅìêîÉ=`=áë=íÜÉ=Öê~éÜ=çÑ=~=ÑìåÅíáçå= ó = Ñ (ñ ) =áå=íÜÉ=ñóéä~åÉ= (~ ≤ ñ ≤ Ä) I=íÜÉå==
Ä
i=∫
~
O
Çó
N + Çñ K==
Çñ
=
1131
...
j~ëë=çÑ=~=táêÉ=
ã = ∫ ρ(ñ I ó I ò )Çë I==
`
ïÜÉêÉ= ρ(ñ I ó I ò ) =áë=íÜÉ=ã~ëë=éÉê=ìåáí=äÉåÖíÜ=çÑ=íÜÉ=ïáêÉK=
=
fÑ=`=áë=~=ÅìêîÉ=é~ê~ãÉíêáòÉÇ=Äó=íÜÉ=îÉÅíçê=ÑìåÅíáçå
r
ê (í ) = ñ (í )I ó (í )I ò (í ) I==íÜÉå==íÜÉ==ã~ëë==Å~å==ÄÉ==ÅçãéìíÉÇ=Äó=
íÜÉ=Ñçêãìä~=
280
CHAPTER 9
...
`ÉåíÉê=çÑ=j~ëë=çÑ=~=táêÉ=
j óò
j ñó
j
I= ó = ñò I= ò =
I=
ñ=
ã
ã
ã
ïÜÉêÉ==
j óò = ∫ ñρ(ñ I ó I ò )Çë I==
`
j ñò = ∫ óρ(ñ I ó I ò )Çë I==
`
j ñó = ∫ òρ(ñ I ó I ò )Çë K=
`
=
1134
...
INTEGRAL CALCULUS
1135
...
=
fÑ=íÜÉ=ÅäçëÉÇ=ÅìêîÉ=`=áë=ÖáîÉå=áå=é~ê~ãÉíêáÅ=Ñçêã=
r
ê (í ) = ñ (í )I ó (í ) I=íÜÉå=íÜÉ=~êÉ~=Å~å=ÄÉ=Å~äÅìä~íÉÇ=Äó=íÜÉ=Ñçêãìä~=
β
β
β
Çó
Çó
Çñ
N
Çñ
p = ∫ ñ (í ) Çí = − ∫ ó (í ) Çí = ∫ ñ (í ) − ó (í ) Çí K==
Çí
Çí
O α
Çí
Çí
α
α
=
1136
...
INTEGRAL CALCULUS
=
=
Figure 206
...
tçêâ=
r
tçêâ=ÇçåÉ=Äó=~=ÑçêÅÉ= c =çå=~å=çÄàÉÅí=ãçîáåÖ=~äçåÖ=~=ÅìêîÉ=
`=áë=ÖáîÉå=Äó=íÜÉ=äáåÉ=áåíÉÖê~ä=
r r
t = ∫ c ⋅ Ç ê I=
`
r
r
ïÜÉêÉ= c =áë=íÜÉ=îÉÅíçê=ÑçêÅÉ=ÑáÉäÇ=~ÅíáåÖ=çå=íÜÉ=çÄàÉÅíI= Ç ê =áë=
íÜÉ=ìåáí=í~åÖÉåí=îÉÅíçêK==
=
=
Figure 207
...
INTEGRAL CALCULUS
fÑ=íÜÉ=çÄàÉÅí=áë=ãçîÉÇ=~äçåÖ=~=ÅìêîÉ=`=áå=íÜÉ=ñó-éä~åÉI=íÜÉå=
r r
t = ∫ c ⋅ Ç ê = ∫ mÇñ + nÇó I==
`
`
=
fÑ= ~= é~íÜ= `= áë= ëéÉÅáÑáÉÇ= Äó= ~= é~ê~ãÉíÉê= í= Eí= çÑíÉå= ãÉ~åë=
íáãÉFI=íÜÉ=Ñçêãìä~=Ñçê=Å~äÅìä~íáåÖ=ïçêâ=ÄÉÅçãÉë=
β
Çñ
Çó
Çò
t = ∫ m(ñ (í )I ó (í )I ò (í )) + n(ñ (í )I ó (í )I ò (í )) + o(ñ (í )I ó (í )I ò (í )) ÇíI
Çí
Çí
Çí
α
ïÜÉêÉ=í=ÖçÉë=Ñêçã= α =íç= β K==
=
r
fÑ= ~= îÉÅíçê= ÑáÉäÇ= c = áë= ÅçåëÉêî~íáîÉ= ~åÇ= ì(ñ I ó I ò ) = áë= ~= ëÅ~ä~ê=
éçíÉåíá~ä= çÑ= íÜÉ= ÑáÉäÇI= íÜÉå= íÜÉ= ïçêâ= çå= ~å= çÄàÉÅí= ãçîáåÖ=
Ñêçã=^=íç=_=Å~å=ÄÉ=ÑçìåÇ=Äó=íÜÉ=Ñçêãìä~=
t = ì(_ ) − ì(^ ) K==
=
1138
...
284
=
CHAPTER 9
...
c~ê~Ç~ó∞ë=i~ï=
r r
Çψ
=
ε = ∫ b ⋅ Çê = −
Çí
`
=
qÜÉ= ÉäÉÅíêçãçíáîÉ= ÑçêÅÉ= EÉãÑF= ε = áåÇìÅÉÇ= ~êçìåÇ= ~= ÅäçëÉÇ=
äççé=`=áë=Éèì~ä=íç=íÜÉ=ê~íÉ=çÑ=íÜÉ=ÅÜ~åÖÉ=çÑ=ã~ÖåÉíáÅ=Ñäìñ= ψ =
é~ëëáåÖ=íÜêçìÖÜ=íÜÉ=äççéK===
=
=
=
Figure 209
...
13 Surface Integral
=
pÅ~ä~ê=ÑìåÅíáçåëW= Ñ (ñ I ó I ò ) I= ò (ñ I ó ) =
r
r
mçëáíáçå=îÉÅíçêëW= ê (ìI î ) I= ê (ñ I ó I ò ) =
r r r
råáí=îÉÅíçêëW= á I= à I= â =
pìêÑ~ÅÉW=p=
r
sÉÅíçê=ÑáÉäÇW= c (mI nI o ) =
r
r
aáîÉêÖÉåÅÉ=çÑ=~=îÉÅíçê=ÑáÉäÇW= Çáî c = ∇ ⋅ c =
285
CHAPTER 9
...
pìêÑ~ÅÉ=fåíÉÖê~ä=çÑ=~=pÅ~ä~ê=cìåÅíáçå=
iÉí=~=ëìêÑ~ÅÉ=p=ÄÉ=ÖáîÉå=Äó=íÜÉ=éçëáíáçå=îÉÅíçê=
r
r
r
r
ê (ìI î ) = ñ (ìI î )á + ó (ìI î ) à + ò (ìI î )â I==
ïÜÉêÉ= (ìI î ) = ê~åÖÉë= çîÉê= ëçãÉ= Ççã~áå= a(ìI î ) = çÑ= íÜÉ= ìîéä~åÉK=
qÜÉ==ëìêÑ~ÅÉ==áåíÉÖê~ä==çÑ==~==ëÅ~ä~ê=ÑìåÅíáçå== Ñ (ñ I ó I ò ) =çîÉê=
íÜÉ=ëìêÑ~ÅÉ=p=áë=ÇÉÑáåÉÇ=~ë==
r
r
∂ê ∂ê
Ñ (ñ I ó I ò )Çp = ∫∫ Ñ (ñ (ìI î )I ó (ìI î )I ò (ìI î )) × ÇìÇî I==
∫∫
∂ì ∂î
p
a(ì I î )
r
r
∂ê
∂ê
ïÜÉêÉ=íÜÉ=é~êíá~ä=ÇÉêáî~íáîÉë= =~åÇ= =~êÉ=ÖáîÉå=Äó==
∂ì
∂î
286
CHAPTER 9
...
fÑ==íÜÉ==ëìêÑ~ÅÉ==p==áë==ÖáîÉå=Äó==íÜÉ=Éèì~íáçå= ò = ò(ñ I ó ) =ïÜÉêÉ=
ò (ñ I ó ) ==áë==~==ÇáÑÑÉêÉåíá~ÄäÉ==ÑìåÅíáçå==áå=íÜÉ=Ççã~áå= a(ñ I ó ) I=
íÜÉå==
∫∫ Ñ (ñ I ó I ò )Çp = (∫∫ ) Ñ (ñ I ó I ò(ñ I ó ))
p
a ñ Ió
O
∂ò ∂ò
N + + ÇñÇó K==
∂ñ ∂ó
O
=
r
1142
...
INTEGRAL CALCULUS
r
r
r ∂ó
r ∂ò
∂ê ∂ñ
= (ìI î ) ⋅ á + (ìI î ) ⋅ à + (ìI î ) ⋅ â I==
∂ì ∂ì
∂ì
∂ì
r
r
r ∂ó
r ∂ò
∂ ê ∂ñ
= (ìI î ) ⋅ á + (ìI î ) ⋅ à + (ìI î ) ⋅ â K==
∂î
∂î
∂î ∂î
=
1143
...
∫∫ (c ⋅ å)Çp = ∫∫ mÇóÇò + nÇòÇñ + oÇñÇó =
r r
p
p
= ∫∫ (m Åçë α + n Åçë β + o Åçë γ )Çp I==
p
ïÜÉêÉ= m(ñ I ó I ò ) I= n(ñ I ó I ò ) I= o(ñ I ó I ò ) =~êÉ=íÜÉ=ÅçãéçåÉåíë=çÑ=
r
íÜÉ=îÉÅíçê= ÑáÉäÇ= c K==
Åçë α I= Åçë β I= Åçë γ ==~êÉ=íÜÉ= ~åÖäÉë= ÄÉíïÉÉå=íÜÉ=çìíÉê=ìåáí=
r
åçêã~ä=îÉÅíçê= å =~åÇ=íÜÉ=ñ-~ñáëI=ó-~ñáëI=~åÇ=ò-~ñáëI=êÉëéÉÅíáîÉäóK=
=
288
CHAPTER 9
...
fÑ=íÜÉ=ëìêÑ~ÅÉ=p=áë=ÖáîÉå=áå=é~ê~ãÉíêáÅ=Ñçêã=Äó=íÜÉ=îÉÅíçê=
r
ê (ñ (ìI î )I ó (ìI î )I ò (ìI î )) I==íÜÉå==íÜÉ==ä~ííÉê=Ñçêãìä~=Å~å=ÄÉ=
ïêáííÉå=~ë==
m n o
∂ñ ∂ó ∂ò
∫∫
∫∫Iî ) ∂ì ∂ì ∂ì ÇìÇîI
p
p
a(ì
∂ñ ∂ó ∂ò
∂î ∂î ∂î
ïÜÉêÉ= (ìI î ) = ê~åÖÉë= çîÉê= ëçãÉ= Ççã~áå= a(ìI î ) = çÑ= íÜÉ= ìîéä~åÉK=
=
1146
...
aáîÉêÖÉåÅÉ=qÜÉçêÉã=áå=`ççêÇáå~íÉ=cçêã=
∂m ∂n ∂o
mÇóÇò + nÇñÇò + oÇñÇó = ∫∫∫
∫∫
∂ñ + ∂ó + ∂ò ÇñÇóÇò K==
p
d
=
1148
...
INTEGRAL CALCULUS
ïÜÉêÉ==
r
c(ñ I ó I ò ) = m(ñ I ó I ò )I n(ñ I ó I ò )I o(ñ I ó I ò ) ===
áë==~=îÉÅíçê==ÑáÉäÇ==ïÜçëÉ==ÅçãéçåÉåíë==mI==nI==~åÇ=o==Ü~îÉ=
Åçåíáåìçìë=é~êíá~ä=ÇÉêáî~íáîÉëI==
r
r
r
á
à
â
r ∂
∂
∂ ∂o ∂n r ∂m ∂o r ∂n ∂m r
á +
∇×c=
=
−
−
− â
à +
∂ñ ∂ñ ∂ñ ∂ó ∂ò ∂ò ∂ñ ∂ñ ∂ó
m n o
r
r
áë=íÜÉ=Åìêä=çÑ= c I=~äëç=ÇÉåçíÉÇ= Åìêä c K==
qÜÉ=ëóãÄçä== ∫ =áåÇáÅ~íÉë=íÜ~í=íÜÉ=äáåÉ=áåíÉÖê~ä=áë=í~âÉå=çîÉê=
~=ÅäçëÉÇ=ÅìêîÉK==
=
1149
...
pìêÑ~ÅÉ=^êÉ~=
^ = ∫∫ Çp =
p
=
1151
...
INTEGRAL CALCULUS
1152
...
j~ëë=çÑ=~=pìêÑ~ÅÉ=
ã = ∫∫ µ(ñ I ó I ò )Çp I==
O
p
ïÜÉêÉ= µ(ñ I ó I ò ) = áë= íÜÉ= ã~ëë= éÉê= ìåáí= ~êÉ~= = EÇÉåëáíó= ÑìåÅíáçåFK=
=
1154
...
jçãÉåíë=çÑ=fåÉêíá~=~Äçìí=íÜÉ=ñó-éä~åÉ=Eçê= ò = M FI==óò-éä~åÉ==
E ñ = M FI=~åÇ=ñò-éä~åÉ=E ó = M F=
f ñó = ∫∫ ò Oµ(ñ I ó I ò )Çp I==
p
f óò = ∫∫ ñ Oµ(ñ I ó I ò )Çp I==
p
291
CHAPTER 9
...
jçãÉåíë=çÑ=fåÉêíá~=~Äçìí=íÜÉ=ñ-~ñáëI=ó-~ñáëI=~åÇ=ò-~ñáë=
f ñ = ∫∫ (ó O + ò O )µ(ñ I ó I ò )Çp I==
p
f ó = ∫∫ (ñ O + ò O )µ(ñ I ó I ò )Çp I==
p
fò = ∫∫ (ñ O + ó O )µ(ñ I ó I ò )Çp K==
p
=
1157
...
dê~îáí~íáçå~ä=cçêÅÉ=
r
r
ê
c = dã ∫∫ µ(ñ I ó I ò ) P Çp I=
ê
p
ïÜÉêÉ=ã=áë=~=ã~ëë=~í=~=éçáåí= ñ M I ó M I ò M =çìíëáÇÉ=íÜÉ=ëìêÑ~ÅÉI==
r
ê = ñ − ñ M I ó − ó M I ò − ò M I==
µ(ñ I ó I ò ) =áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåI==
~åÇ=d=áë=Öê~îáí~íáçå~ä=Åçåëí~åíK=
=
1159
...
cäìáÇ=cäìñ=E~Åêçëë=íÜÉ=ëìêÑ~ÅÉ=pF=
r r r
Φ = ∫∫ î (ê ) ⋅ Çp I==
p
292
CHAPTER 9
...
j~ëë=cäìñ=E~Åêçëë=íÜÉ=ëìêÑ~ÅÉ=pF=
r r r
Φ = ∫∫ ρî (ê ) ⋅ Çp I==
p
r
r
ïÜÉêÉ= c = ρî =áë=íÜÉ=îÉÅíçê=ÑáÉäÇI= ρ =áë=íÜÉ=ÑäìáÇ=ÇÉåëáíóK=
=
1162
...
d~ìëë∞=i~ï=
qÜÉ=ÉäÉÅíêáÅ=Ñäìñ=íÜêçìÖÜ=~åó=ÅäçëÉÇ=ëìêÑ~ÅÉ=áë=éêçéçêíáçå~ä=
íç=íÜÉ=ÅÜ~êÖÉ=n=ÉåÅäçëÉÇ=Äó=íÜÉ=ëìêÑ~ÅÉ=
r r n
Φ = ∫∫ b ⋅ Çp = I==
εM
p
ïÜÉêÉ==
Φ =áë=íÜÉ=ÉäÉÅíêáÅ=ÑäìñI==
r
b =áë=íÜÉ=ã~ÖåáíìÇÉ=çÑ=íÜÉ=ÉäÉÅíêáÅ=ÑáÉäÇ=ëíêÉåÖíÜI=
c
ε M = UIUR × NM −NO =áë=éÉêãáííáîáíó=çÑ=ÑêÉÉ=ëé~ÅÉK==
ã
=
=
293
Chapter 10
Differential Equations
=
=
=
=
cìåÅíáçåë=çÑ=çåÉ=î~êá~ÄäÉW=óI=éI=èI=ìI=ÖI=ÜI=dI=eI=êI=ò==
^êÖìãÉåíë=EáåÇÉéÉåÇÉåí=î~êá~ÄäÉëFW=ñI=ó=
cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= Ñ (ñ I ó ) I= j(ñ I ó ) I= k(ñ I ó ) =
Çó
&
cáêëí=çêÇÉê=ÇÉêáî~íáîÉW= ó ′ I= ì′ I= ó I= I=£=
Çí
Ç Of
pÉÅçåÇ=çêÇÉê=ÇÉêáî~íáîÉëW= ó′′ I= && I= O I=£=
ó
Çí
O
∂ì ∂ ì
m~êíá~ä=ÇÉêáî~íáîÉëW= I= O I=£=
∂í ∂ñ
k~íìê~ä=åìãÄÉêW=å=
m~êíáÅìä~ê=ëçäìíáçåëW= ó N I= ó é =
oÉ~ä=åìãÄÉêëW=âI=íI=`I= `N I= ` O I=éI=èI= α I= β =
oççíë=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåëW= λN I= λ O =
qáãÉW=í=
qÉãéÉê~íìêÉW=qI=p=
mçéìä~íáçå=ÑìåÅíáçåW= m(í ) ==
j~ëë=çÑ=~å=çÄàÉÅíW=ã=
píáÑÑåÉëë=çÑ=~=ëéêáåÖW=â=
aáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãW=ó=
^ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW=^=
cêÉèìÉåÅóW= ω =
a~ãéáåÖ=ÅçÉÑÑáÅáÉåíW= γ =
mÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW= δ =
^åÖìä~ê=Çáëéä~ÅÉãÉåíW= θ =
mÉåÇìäìã=äÉåÖíÜW=i=
294
CHAPTER 10
...
1 First Order Ordinary Differential
Equations
=
1164
...
pÉé~ê~ÄäÉ=bèì~íáçåë=
Çó
= Ñ (ñ I ó ) = Ö (ñ )Ü(ó ) =
Çñ
=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó=
Çó
∫ Ü(ó ) = ∫ Ö(ñ )Çñ + ` I==
çê=
e(ó ) = d(ñ ) + ` K=
=
=
=
295
CHAPTER 10
...
eçãçÖÉåÉçìë=bèì~íáçåë=
Çó
= Ñ (ñ I ó ) = áë= ÜçãçÖÉåÉçìëI= áÑ=
Çñ
íÜÉ=ÑìåÅíáçå= Ñ (ñ I ó ) =áë=ÜçãçÖÉåÉçìëI=íÜ~í=áë==
Ñ (íñ I íó ) = Ñ (ñ I ó ) K==
=
ó
qÜÉ=ëìÄëíáíìíáçå= ò = =EíÜÉå= ó = òñ F=äÉ~Çë=íç=íÜÉ=ëÉé~ê~ÄäÉ=
ñ
Éèì~íáçå=
Çò
ñ
+ ò = Ñ (NI ò ) K==
Çñ
=
1167
...
oáÅÅ~íá=bèì~íáçå=
Çó
= é(ñ ) + è(ñ ) ó + ê(ñ ) ó O =
Çñ
=
fÑ=~=é~êíáÅìä~ê=ëçäìíáçå= ó N =áë=âåçïåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=Å~å=ÄÉ=çÄí~áåÉÇ=ïáíÜ=íÜÉ=ÜÉäé=çÑ=ëìÄëíáíìíáçå=
N
ò=
I=ïÜáÅÜ=äÉ~Çë=íç=íÜÉ=Ñáêëí=çêÇÉê=äáåÉ~ê=Éèì~íáçå==
ó − óN
Çò
= −[è(ñ ) + Oó Nê(ñ )] ò − ê(ñ ) K==
Çñ
=
=
=
qÜÉ= ÇáÑÑÉêÉåíá~ä= Éèì~íáçå=
296
CHAPTER 10
...
bñ~Åí=~åÇ=kçåÉñ~Åí=bèì~íáçåë=
qÜÉ=Éèì~íáçå==
j(ñ I ó )Çñ + k(ñ I ó )Çó = M ==
áë=Å~ääÉÇ=Éñ~Åí=áÑ==
∂j ∂k
=
I==
∂ó ∂ñ
~åÇ=åçåÉñ~Åí=çíÜÉêïáëÉK=
=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==
∫ j(ñ I ó )Çñ + ∫ k(ñ I ó )Çó = ` K==
=
1170
...
kÉïíçå∞ë=i~ï=çÑ=`ççäáåÖ=
Çq
= −â (q − p ) I==
Çí
ïÜÉêÉ= q(í ) =áë=íÜÉ=íÉãéÉê~íìêÉ=çÑ=~å=çÄàÉÅí=~í=íáãÉ=íI=p=áë=íÜÉ=
íÉãéÉê~íìêÉ= çÑ= íÜÉ= ëìêêçìåÇáåÖ= ÉåîáêçåãÉåíI= â= áë= ~= éçëáíáîÉ=Åçåëí~åíK==
=
qÜÉ=ëçäìíáçå=áë=
q(í ) = p + (qM − p) É −âí I==
ïÜÉêÉ= qM = q(M) = áë= íÜÉ= áåáíá~ä= íÉãéÉê~íìêÉ= çÑ= íÜÉ= çÄàÉÅí= ~í=
íáãÉ= í = M K==
=
=
297
CHAPTER 10
...
mçéìä~íáçå=aóå~ãáÅë=EiçÖáëíáÅ=jçÇÉäF=
Çm
m
= âm N − I==
Çí
j
ïÜÉêÉ= m(í ) =áë=éçéìä~íáçå=~í=íáãÉ=íI=â=áë=~=éçëáíáîÉ=Åçåëí~åíI=
j=áë=~=äáãáíáåÖ=ëáòÉ=Ñçê=íÜÉ=éçéìä~íáçåK==
=
qÜÉ=ëçäìíáçå=çÑ=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=áë==
jmM
m(í ) =
I=ïÜÉêÉ= mM = m(M) =áë=íÜÉ=áåáíá~ä=éçéìmM + (j − mM )É − âí
ä~íáçå=~í=íáãÉ= í = M K===
=
=
=
10
...
eçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=`çÉÑÑáÅáÉåíë==
ó′′ + éó′ + èó = M K==
qÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=áë==
λO + éλ + è = M K==
=
fÑ= λN = ~åÇ= λ O = ~êÉ= ÇáëíáåÅí= êÉ~ä= êççíë= çÑ= íÜÉ= ÅÜ~ê~ÅíÉêáëíáÅ=
Éèì~íáçåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==
ó = `NÉ λNñ + ` OÉ λ O ñ I=ïÜÉêÉ==
`N =~åÇ= ` O =~êÉ=áåíÉÖê~íáçå=Åçåëí~åíëK==
==
é
fÑ= λN = λ O = − I=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==
O
é
− ñ
ó = (`N + ` O ñ )É O K==
=
fÑ= λN =~åÇ= λ O =~êÉ=ÅçãéäÉñ=åìãÄÉêëW=
298
CHAPTER 10
...
fåÜçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=======================
`çÉÑÑáÅáÉåíë==
ó′′ + éó′ + èó = Ñ (ñ ) K==
=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó==
ó = ó é + ó Ü I=ïÜÉêÉ==
ó é =áë==~=é~êíáÅìä~ê==ëçäìíáçå=çÑ==íÜÉ=áåÜçãçÖÉåÉçìë=Éèì~íáçå=
~åÇ= ó Ü =áë=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=çÑ=íÜÉ=~ëëçÅá~íÉÇ=ÜçãçÖÉåÉçìë=Éèì~íáçå=EëÉÉ=íÜÉ=éêÉîáçìë=íçéáÅ=NNTPFK=
=
fÑ=íÜÉ=êáÖÜí=ëáÇÉ=Ü~ë=íÜÉ=Ñçêã==
Ñ (ñ ) = É αñ (mN (ñ )Åçë β ñ + mN (ñ )ëáå βñ ) I==
íÜÉå=íÜÉ==é~êíáÅìä~ê=ëçäìíáçå= ó é =áë=ÖáîÉå=Äó==
ó é = ñ â É αñ (oN (ñ )Åçë βñ + o O (ñ )ëáå β ñ ) I==
ïÜÉêÉ= íÜÉ= éçäóåçãá~äë= oN (ñ ) = ~åÇ= o O (ñ ) = Ü~îÉ= íç= ÄÉ= ÑçìåÇ=
Äó=ìëáåÖ=íÜÉ=ãÉíÜçÇ=çÑ=ìåÇÉíÉêãáåÉÇ=ÅçÉÑÑáÅáÉåíëK==
• fÑ= α + β á =áë=åçí=~=êççí=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåI=íÜÉå=
íÜÉ=éçïÉê= â = M I=
• fÑ= α + β á =áë=~=ëáãéäÉ=êççíI=íÜÉå= â = N I=
• fÑ= α + β á =áë=~=ÇçìÄäÉ=êççíI=íÜÉå= â = O K==
=
1175
...
DIFFERENTIAL EQUATIONS
1176
...
cêÉÉ=råÇ~ãéÉÇ=sáÄê~íáçåë=
qÜÉ=ãçíáçå=çÑ=~=j~ëë=çå=~=péêáåÖ=áë=ÇÉëÅêáÄÉÇ=Äó=íÜÉ=Éèì~íáçå==
ã&& + âó = M I==
ó
ïÜÉêÉ==
ã=áë=íÜÉ=ã~ëë=çÑ=íÜÉ=çÄàÉÅíI=
â=áë=íÜÉ=ëíáÑÑåÉëë=çÑ=íÜÉ=ëéêáåÖI=
ó=áë=Çáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãK=
=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==
ó = ^ Åçë(ωM í − δ ) I==
ïÜÉêÉ==
^=áë=íÜÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíI=
Oπ
I=
ωM =áë=íÜÉ=ÑìåÇ~ãÉåí~ä=ÑêÉèìÉåÅóI=íÜÉ=éÉêáçÇ=áë= q =
ωM
δ =áë=éÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíK=
qÜáë=áë=~å=Éñ~ãéäÉ=çÑ=ëáãéäÉ=Ü~êãçåáÅ=ãçíáçåK==
=
1178
...
DIFFERENTIAL EQUATIONS
`~ëÉ=NK γ O > Qâã =EçîÉêÇ~ãéÉÇF=
ó (í ) = ^É λNí + _É λ Oí I==
ïÜÉêÉ==
λN =
− γ − γ O − Q âã
− γ + γ O − Q âã
I= λ O =
K==
Oã
Oã
=
`~ëÉ=OK= γ O = Qâã EÅêáíáÅ~ääó=Ç~ãéÉÇF=
ó (í ) = (^ + _í )É λí I==
ïÜÉêÉ==
γ
λ=−
K=
Oã
=
`~ëÉ=PK= γ O < Qâã =EìåÇÉêÇ~ãéÉÇF==
ó (í ) = É
−
γ
í
Oã
^ Åçë(ωí − δ ) I=ïÜÉêÉ==
ω = Qâã − γ O K==
=
1179
...
oi`=`áêÅìáí=
Ç Of
Çf N
i O + o + f = s′(í ) = ωb M Åçë(ωí ) I=
Çí
Çí `
301
CHAPTER 10
...
3
...
qÜÉ=i~éä~ÅÉ=bèì~íáçå=
∂ Oì ∂ Oì
+
= M=
∂ñ O ∂ó O
~ééäáÉë=íç=éçíÉåíá~ä=ÉåÉêÖó=ÑìåÅíáçå= ì(ñ I ó ) ==Ñçê==~==ÅçåëÉêî~íáîÉ= ÑçêÅÉ= ÑáÉäÇ= áå= íÜÉ= ñó-éä~åÉK= m~êíá~ä= ÇáÑÑÉêÉåíá~ä= Éèì~íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=ÉääáéíáÅK==
=
1182
...
DIFFERENTIAL EQUATIONS
~ééäáÉë= íç= íÜÉ= íÉãéÉê~íìêÉ= ÇáëíêáÄìíáçå= ì(ñ I ó ) = áå= íÜÉ= ñóéä~åÉ=ïÜÉå=ÜÉ~í=áë=~ääçïÉÇ=íç=Ñäçï=Ñêçã=ï~êã=~êÉ~ë=íç=Åççä=
çåÉëK=qÜÉ=Éèì~íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=é~ê~ÄçäáÅK==
=
1183
...
1 Arithmetic Series
=
fåáíá~ä=íÉêãW= ~N =
kíÜ=íÉêãW= ~ å =
aáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=ëìÅÅÉëëáîÉ=íÉêãëW=Ç=
kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=
pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= på =
=
=
1184
...
~N + ~ å = ~ O + ~ å−N = K = ~ á + ~ å +N−á =
=
~ +~
1186
...
på = N å ⋅ å = N
⋅å=
O
O
=
=
=
=
=
304
CHAPTER 11
...
2 Geometric Series
=
fåáíá~ä=íÉêãW= ~N =
kíÜ=íÉêãW= ~ å =
`çããçå=ê~íáçW=è=
kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=
pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= på =
pìã=íç=áåÑáåáíóW=p=
=
=
1188
...
~N ⋅ ~ å = ~ O ⋅ ~ å −N = K = ~ á ⋅ ~ å +N−á =
=
1190
...
på =
=
è −N
è −N
=
~
1192
...
3 Some Finite Series
=
kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=
=
=
305
CHAPTER 11
...
N + O + P + K + å =
å(å + N)
=
O
=
1194
...
N + P + R + K + (Oå − N) = å O =
=
1196
...
NO + OO + PO + K + å O =
å(Oâ + å − N)
=
O
å(å + N)(Oå + N)
=
S
=
å(å + N)
1198
...
N + P + R + K + (Oå − N) =
=
P
=
P
1200
...
N + + + + K + å + K = O =
O Q U
O
=
N
N
N
N
1202
...
N + + + + K +
+K = É =
(å − N)>
N> O> P>
=
=
=
O
306
CHAPTER 11
...
4 Infinite Series
=
pÉèìÉåÅÉW= {~ å }=
cáêëí=íÉêãW= ~N =
kíÜ=íÉêãW= ~ å =
=
=
1204
...
kíÜ=m~êíá~ä=pìã=
å
på = ∑ ~ å = ~N + ~ O + K + ~ å =
å =N
=
1206
...
kíÜ=qÉêã=qÉëí=
∞
•
fÑ=íÜÉ=ëÉêáÉë= ∑ ~ å áë=ÅçåîÉêÖÉåíI=íÜÉå= äáã ~ å = M K==
å→∞
å =N
•
fÑ= äáã ~ å ≠ M I=íÜÉå=íÜÉ=ëÉêáÉë=áë=ÇáîÉêÖÉåíK=
å→∞
=
=
=
11
...
SERIES
∞
∞
∞
å =N
å =N
å =N
1208
...
å =N
∞
å
= Å∑ ~ å = Å^ K==
å =N
=
=
=
11
...
qÜÉ=`çãé~êáëçå=qÉëí=
∞
∞
å =N
∞
å =N
iÉí= ∑ ~ å =~åÇ= ∑ Äå =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= M < ~ å ≤ Äå =Ñçê=~ää=åK==
•
∞
fÑ= ∑ Äå áë=ÅçåîÉêÖÉåí=íÜÉå= ∑ ~ å áë=~äëç=ÅçåîÉêÖÉåíK==
å =N
∞
å =N
•
∞
å =N
å =N
fÑ= ∑ ~ å áë=ÇáîÉêÖÉåí=íÜÉå= ∑ Äå áë=~äëç=ÇáîÉêÖÉåíK=
=
1211
...
SERIES
∞
∞
~å
= ∞ =íÜÉå= ∑ Äå ÇáîÉêÖÉåí=áãéäáÉë=íÜ~í= ∑ ~ å =áë=
å →∞ Ä
å =N
å =N
å
~äëç=ÇáîÉêÖÉåíK=
fÑ= äáã
•
=
1212
...
qÜÉ=fåíÉÖê~ä=qÉëí=
iÉí= Ñ (ñ ) = ÄÉ= ~= ÑìåÅíáçå= ïÜáÅÜ= áë= ÅçåíáåìçìëI= éçëáíáîÉI= ~åÇ=
ÇÉÅêÉ~ëáåÖ=Ñçê=~ää= ñ ≥ N K=qÜÉ=ëÉêáÉë==
é-ëÉêáÉë= ∑
∞
∑ Ñ (å) = Ñ (N) + Ñ (O) + Ñ (P) + K + Ñ (å) + K =
å =N
∞
ÅçåîÉêÖÉë=áÑ= ∫ Ñ (ñ )Çñ ÅçåîÉêÖÉëI=~åÇ=ÇáîÉêÖÉë=áÑ=
N
å
∫ Ñ (ñ )Çñ → ∞ =~ë= å → ∞ K=
N
=
1214
...
SERIES
1215
...
7 Alternating Series
=
1216
...
^ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=
∞
•
^= ëÉêáÉë=
∑~
å =N
å
= áë= ~ÄëçäìíÉäó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë=
∞
∑~
å =N
å
=áë=ÅçåîÉêÖÉåíK==
310
CHAPTER 11
...
`çåÇáíáçå~ä=`çåîÉêÖÉåÅÉ=
∞
^= ëÉêáÉë=
∑~
å =N
å
áë= ÅçåÇáíáçå~ääó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë= áë=
ÅçåîÉêÖÉåí=Äìí=áë=åçí=~ÄëçäìíÉäó=ÅçåîÉêÖÉåíK=
=
=
=
11
...
mçïÉê=pÉêáÉë=áå=ñ=
∞
∑~
å=M
å
ñ å = ~ M + ~Nñ + ~ O ñ O + K + ~ å ñ å + K =
=
1220
...
fåíÉêî~ä=çÑ=`çåîÉêÖÉåÅÉ===
qÜÉ=ëÉí=çÑ=íÜçëÉ=î~äìÉë=çÑ=ñ=Ñçê=ïÜáÅÜ=íÜÉ=ÑìåÅíáçå=
∞
Ñ (ñ ) = ∑ ~ å (ñ − ñ M ) =áë=ÅçåîÉêÖÉåí=áë=Å~ääÉÇ==íÜÉ==áåíÉêî~ä=çÑ=
å
å =M
ÅçåîÉêÖÉåÅÉK=
311
å
CHAPTER 11
...
o~Çáìë=çÑ=`çåîÉêÖÉåÅÉ=
fÑ=íÜÉ=áåíÉêî~ä=çÑ=ÅçåîÉêÖÉåÅÉ=áë== (ñ M − oI ñ M + o ) ==Ñçê==ëçãÉ=
o ≥ M I=íÜÉ=o=áë=Å~ääÉÇ==íÜÉ=ê~Çáìë=çÑ=ÅçåîÉêÖÉåÅÉK==fí=áë=ÖáîÉå=
~ë=
~
N
=çê= o = äáã å K==
o = äáã
å →∞ ~
å →∞ å ~
å +N
å
=
=
=
11
...
aáÑÑÉêÉåíá~íáçå=çÑ=mçïÉê=pÉêáÉë=
∞
iÉí= Ñ (ñ ) = ∑ ~ å ñ å = ~ M + ~ Nñ + ~ O ñ O + K =Ñçê= ñ < o K==
å =M
qÜÉåI= = Ñçê= ñ < o I= Ñ (ñ ) = áë= ÅçåíáåìçìëI= íÜÉ= ÇÉêáî~íáîÉ= Ñ ′(ñ ) =
Éñáëíë=~åÇ=
Ç
Ç
Ç
Ñ ′(ñ ) = ~ M + ~ Nñ + ~ O ñ O + K =
Çñ
Çñ
Çñ
∞
= ~N + O~ O ñ + P~ P ñ O + K = ∑ å~ å ñ å −N K=
å =N
=
=
=
312
CHAPTER 11
...
fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë=
∞
iÉí= Ñ (ñ ) = ∑ ~ å ñ å = ~ M + ~ Nñ + ~ O ñ O + K =Ñçê= ñ < o K==
å =M
qÜÉåI==Ñçê= ñ < o I=íÜÉ=áåÇÉÑáåáíÉ=áåíÉÖê~ä= ∫ Ñ (ñ )Çñ Éñáëíë=~åÇ==
∫ Ñ (ñ )Çñ = ∫ ~ Çñ + ∫ ~ ñÇñ + ∫ ~ ñ Çñ + K =
O
M
= ~ M ñ + ~N
N
O
∞
ñO
ñP
ñ å +N
+ ~O + K = ∑ ~å
+ ` K=
O
P
å +N
å =M
=
=
=
11
...
q~óäçê=pÉêáÉë=
∞
(ñ − ~ )å = Ñ (~ ) + Ñ ′(~ )(ñ − ~ ) + Ñ ′′(~ )(ñ − ~ )O + K
Ñ (ñ ) = ∑ Ñ (å ) (~ )
å>
O>
å=M
==
+
Ñ (å ) (~ )(ñ − ~ )
+ o å K==
å>
å
=
1226
...
j~Åä~ìêáå=pÉêáÉë=
313
CHAPTER 11
...
11 Power Series Expansions for Some
Functions
=
tÜçäÉ=åìãÄÉêW=å=
oÉ~ä=åìãÄÉêW=ñ=
=
=
ñO ñP
ñå
1228
...
~ ñ = N +
+
+
+K+
N>
O>
P>
å>
=
(− N)å ñ å+N ± K I= − N < ñ ≤ N K=
ñ O ñP ñ Q
1230
...
äå
= O ñ + + + + K I= ñ < N K=
N− ñ
P
R
T
=
ñ − N N ñ − N P N ñ − N R
+
1232
...
Åçë ñ = N − + − + K +
(Oå )>
O> Q> S>
=
314
CHAPTER 11
...
ëáå ñ = ñ − + − + K +
(Oå + N)>
P> R> T>
=
π
ñ P Oñ R NT ñ T SOñ V
1235
...
Åçí ñ = − + +
P QR VQR + QTOR + K I= ñ < π K=
ñ
=
ñ P N ⋅ Pñ R
N ⋅ P ⋅ RK (Oå − N)ñ Oå +N
1237
...
~êÅÅçë ñ = − ñ +
+
+K+
+ K I
O
O⋅P O⋅Q ⋅R
O ⋅ Q ⋅ SK (Oå )(Oå + N)
å
ñ < N K=
=
1239
...
ÅçëÜ ñ = N +
ñO ñQ ñS
ñ Oå
+
+
+K+
+ K=
(Oå )>
O> Q> S>
=
1241
...
SERIES
11
...
(N + ñ ) = N + å`Nñ + å` O ñ O + K + ã ` å ñ ã + K + ñ å =
=
å(å − N)K[å − (ã − N)]
1243
...
= N − ñ + ñ O − ñ P + K I= ñ < N K=
N+ ñ
=
N
1245
...
N + ñ = N + −
+
−
+ K I= ñ ≤ N K=
O O⋅Q O⋅Q ⋅S O⋅Q ⋅S⋅U
=
ñ N ⋅ Oñ O N ⋅ O ⋅ Rñ P N ⋅ O ⋅ R ⋅ Uñ Q
1247
...
13 Fourier Series
=
fåíÉÖê~ÄäÉ=ÑìåÅíáçåW= Ñ (ñ ) =
cçìêáÉê=ÅçÉÑÑáÅáÉåíëW= ~ M I= ~ å I= Äå =
tÜçäÉ=åìãÄÉêW=å==
316
CHAPTER 11
...
Ñ (ñ ) = M + ∑ (~ å Åçë åñ + Äå ëáå åñ ) =
O
å =N
=
π
N
1249
...
π −π
=
=
317
Chapter 12
Probability
=
=
=
=
12
...
c~Åíçêá~ä=
å> = N ⋅ O ⋅ PK(å − O)(å − N)å =
M> = N =
=
1252
...
å mã =
=
(å − ã )>
=
1254
...
å ` ã = å ` å −ã =
=
1256
...
PROBABILITY
1257
...
m~ëÅ~ä∞ë=qêá~åÖäÉ=
=
oçï=M=
=
=
=
=
=
=
oçï=N=
=
=
=
=
=
N=
oçï=O=
=
=
=
=
N=
=
oçï=P=
=
=
=
N=
=
P=
oçï=Q=
=
=
N=
=
Q=
=
oçï=R=
=
N=
=
R=
= NM=
oçï=S=
N=
=
S=
= NR= =
=
=
=
N=
=
=
=
N=
=
O=
=
N=
=
P=
=
S=
=
Q=
= NM= =
OM= = NR=
12
...
mêçÄ~Äáäáíó=çÑ=~å=bîÉåí=
ã
m(^ ) = I==
å
ïÜÉêÉ==
ã=áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=éçëáíáîÉ=çìíÅçãÉëI==
å=áë=íÜÉ=íçí~ä=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK=
=
319
=
=
=
N=
=
R=
=
=
=
=
=
N=
=
S=
=
=
=
=
=
N=
=
=
=
=
=
=
=
N=
CHAPTER 12
...
o~åÖÉ=çÑ=mêçÄ~Äáäáíó=s~äìÉë=
M ≤ m(^ ) ≤ N =
=
1261
...
fãéçëëáÄäÉ=bîÉåí=
m( ^ ) = M =
=
1263
...
fåÇÉéÉåÇÉåí=bîÉåíë=
m(^ L _ ) = m(^ ) I==
m(_ L ^ ) = m(_ ) =
=
1265
...
jìäíáéäáÅ~íáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=
m(^ ∩ _ ) = m(^ ) ⋅ m(_ ) =
=
1267
...
`çåÇáíáçå~ä=mêçÄ~Äáäáíó=
m(^ ∩ _ )
m( ^ L _ ) =
=
m(_ )
=
1269
...
PROBABILITY
1270
...
_~óÉë∞=qÜÉçêÉã=
m(^ L _ ) ⋅ m(_ )
m(_ L ^ ) =
=
m(^ )
=
1272
...
i~ï=çÑ=i~êÖÉ=kìãÄÉêë=
p
m å − µ ≥ ε → M =~ë= å → ∞ I==
å
p
m å − µ < ε → N =~ë= å → ∞ I==
å
ïÜÉêÉ==
på =áë=íÜÉ=ëìã=çÑ=ê~åÇçã=î~êá~ÄäÉëI=
å =áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK=
=
1274
...
PROBABILITY
1275
...
pí~åÇ~êÇ=kçêã~ä=aÉåëáíó=cìåÅíáçå=
O
N − òO
ϕ(ò ) =
É =
Oπ
^îÉê~ÖÉ=î~äìÉ= µ = M I=ÇÉîá~íáçå= σ = N K=
=
=
=====
=
Figure 210
...
pí~åÇ~êÇ=w=s~äìÉ=
u−µ
w=
=
σ
=
1278
...
PROBABILITY
ïÜÉêÉ==
ñ=áë=~=é~êíáÅìä~ê=çìíÅçãÉI==
í =áë=~=î~êá~ÄäÉ=çÑ=áåíÉÖê~íáçåK=
=
α −µ β−µ
1279
...
m( u − µ < ε ) = Oc I==
σ
ïÜÉêÉ==
u=áë=åçêã~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=î~êá~ÄäÉI=
c=áë=Åìãìä~íáîÉ=åçêã~ä=ÇáëíêáÄìíáçå=ÑìåÅíáçåK=
=
1281
...
_Éêåçìääá=qêá~äë=mêçÅÉëë=
µ = åé = I= σ O = åéè I==
ïÜÉêÉ==
å =áë=~=ëÉèìÉåÅÉ=çÑ=ÉñéÉêáãÉåíëI==
é =áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=ëìÅÅÉëë=çÑ=É~ÅÜ=ÉñéÉêáãÉåíëI=
è =áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N − é K=
=
1283
...
PROBABILITY
µ = åé I= σ O = åéè I=
Ñ (ñ ) = (è + éÉ ñ ) I==
ïÜÉêÉ==
å=áë=íÜÉ=åìãÄÉê=çÑ=íêá~äë=çÑ=ëÉäÉÅíáçåëI=
é=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=ëìÅÅÉëëI=
è=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N − é K=
=
1284
...
mçáëëçå=aáëíêáÄìíáçå=
λâ −λ
m(u = â ) ≈ É I= λ = åé I==
â>
O
µ = λ I= σ = λ I==
ïÜÉêÉ==
λ =áë=íÜÉ=ê~íÉ=çÑ=çÅÅìêêÉåÅÉI=
â=áë=íÜÉ=åìãÄÉê=çÑ=éçëáíáîÉ=çìíÅçãÉëK=
=
1286
...
`çåíáåìçìë=råáÑçêã=aÉåëáíó=
~+Ä
N
I= µ =
I==
Ñ=
Ä−~
O
324
CHAPTER 12
...
bñéçåÉåíá~ä=aÉåëáíó=cìåÅíáçå=
Ñ (í ) = λÉ −λí I= µ = λ I= σ O = λO =
ïÜÉêÉ=í=áë=íáãÉI= λ =áë=íÜÉ=Ñ~áäìêÉ=ê~íÉK=
=
1289
...
bñéÉÅíÉÇ=s~äìÉ=çÑ=aáëÅêÉíÉ=o~åÇçã=s~êá~ÄäÉë=
å
µ = b(u ) = ∑ ñ á éá I==
á =N
ïÜÉêÉ= ñ á =áë=~=é~êíáÅìä~ê=çìíÅçãÉI= é á =áë=áíë=éêçÄ~ÄáäáíóK=
=
1291
...
mêçéÉêíáÉë=çÑ=bñéÉÅí~íáçåë=
b(u + v ) = b(u ) + b(v ) I==
b(u − v ) = b(u ) − b(v ) I=
b(Åu ) = Åb(u ) I==
b(uv ) = b(u ) ⋅ b(v ) I==
ïÜÉêÉ=Å=áë=~=Åçåëí~åíK=
=
1293
...
PROBABILITY
1294
...
s~êá~åÅÉ=çÑ=aáëÅêÉíÉ=o~åÇçã=s~êá~ÄäÉë=
[
]
å
σ O = s(u ) = b (u − µ ) = ∑ (ñ á − µ ) éá I==
O
O
á =N
ïÜÉêÉ==
ñ á =áë=~=é~êíáÅìä~ê=çìíÅçãÉI=
é á =áë=áíë=éêçÄ~ÄáäáíóK=
=
1296
...
mêçéÉêíáÉë=çÑ=s~êá~åÅÉ=
s(u + v ) = s(u ) + s(v ) I==
s(u − v ) = s(u ) + s(v ) I=
s(u + Å ) = s(u ) I=
s(Åu ) = Å O s(u ) I=
ïÜÉêÉ=Å=áë=~=Åçåëí~åíK=
=
1298
...
`çî~êá~åÅÉ=
Åçî (uI v ) = b[(u − µ(u ))(v − µ(v ))] = b(uv ) − µ(u )µ(v ) I==
ïÜÉêÉ==
u =áë=ê~åÇçã=î~êá~ÄäÉI==
s(u ) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uI==
µ =áë=íÜÉ=ÉñéÉÅíÉÇ=î~äìÉ=çÑ=u=çê=vK=
O
326
CHAPTER 12
...
`çêêÉä~íáçå=
Åçî (uI v )
I==
ρ(uI v ) =
s(u )s(v )
ïÜÉêÉ==
s(u ) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uI==
s(v ) =áë=íÜÉ=î~êá~åÅÉ=çÑ=vK=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
327
=
iççâ=Ñçê=çíÜÉê=Ü~åÇÄççâë=~åÇ=ëçäîÉÇ=éêçÄäÉã=ÖìáÇÉë=~í=
ïïïKã~íÜ-ÉÄççâëKÅçãK=
=
=
=
=
Title: mathes formulas
Description: esy way to learn mathes formulas. a group of mathes formulas.
Description: esy way to learn mathes formulas. a group of mathes formulas.