1. 1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `==OMMQ=^KpK=^=o=oK=

2. = q=~==~==~K=i 3. Preface = = = = q= ~= = ~= = = = = = ~= K= f= ~= = = = = ~==~==~~=~~==I= I=~=I=~=~~K=q== ~= = = ~I= ~I= ~= = = k= pI= ^~I= dI= qI= j~= ~= a~I= sI= ^~= dI= `~I= a~=b~I=pI=~=m~=qK== q= = ~= = I= I= ~= ~= ~= = = ~= ~= = ~= ~I= = = ~===~=~=~===K=== = =ii 4. Contents = = = = 1 krj_bo=pbqp= NKN= p=f==1= NKO= p==k==5= NKP= _~=f==7= NKQ= `=k==8= = 2 ^idb_o^= OKN= c~=c~==12= OKO= m=c~==13= OKP= m==14= OKQ= o==15= OKR= i~==16= OKS= b~==18= OKT= f~==19= OKU= `=f=c~==22= = 3 dbljbqov= PKN= o=q~==24= PKO= f=q~==27= PKP= b~~=q~==28= PKQ= p~=q~==29= PKR= p~==33= PKS= o~==34= PKT= m~~~==35= PKU= o==36= PKV= q~==37= PKNM= f=q~==38= PKNN= f=q~==f=`==40= PKNO= q~==f=`==41=iii 5. PKNP= h==42= PKNQ= `=n~~~==43= PKNR= q~~=n~~~==45= PKNS= d~=n~~~==46= PKNT= o~=e~==47= PKNU= o~=m==48= PKNV= `==50= PKOM= p==~=`==53= PKON= p==~=`==54= PKOO= `==55= PKOP= o~~=m~~==56= PKOQ= m==57= PKOR= o~=q~==58= PKOS= o~=m~==59= PKOT= c==~=o~=m~==61= PKOU= o~~=o=t==62= PKOV= m~=p==63= PKPM= o=`~=`==66= PKPN= o=`~=`==~=l=m~=c~==68= PKPO= o=`~=`==69= PKPP= c==~=o=`~=`==70= PKPQ= p==72= PKPR= p~=`~==72= PKPS= p~=p==73= PKPT= p~=p==74= PKPU= p~=t==75= PKPV= b==76= PKQM= `~=q==78= = = 4 qofdlkljbqov= QKN= o~~=~=a=j~==^==80= QKO= a=~=d~==q=c==81= QKP= p==q=c==86= QKQ= q=c==`=^==87= QKR= j=f~=c~==88=iv 6. QKS= o=c~==89= QKT= m==q=c==90= QKU= o~==q=c==90= QKV= ^=~=p~=c~==91= QKNM= a=^=c~==92= QKNN= j=^=c~==93= QKNO= e~=^=c~==94= QKNP= e~=^=q~=f==94= QKNQ= q~==q=b==m==95= QKNR= q~==q=b==p==97=== QKNS= m==q=c==98= QKNT= d~==f=q=c==99= QKNU= m~=s~==f=q=c==102= QKNV= o~==f=q=c==103= QKOM= q=b~==106= QKON= o~==e=c==106= = = 5 j^qof`bp=^ka=abqbojfk^kqp= RKN= a~==107= RKO= m==a~==109= RKP= j~==110= RKQ= l~==j~==111= RKR= p==i~=b~==114= = = 6 sb`qlop= SKN= s=`~==118= SKO= s=^==120= SKP= s=p~==122= SKQ= p~=s==122= SKR= p~~=m==123= SKS= s=m==125= SKT= q=m=127= = = 7 ^k^ivqf`=dbljbqov= TKN= l=-a~=`~=p==130=v 7. TKO= q=-a~=`~=p==131= TKP= p~=i==m~==139= TKQ= `==149= TKR= b==152= TKS= e~==154= TKT= m~~~==158= TKU= q=-a~=`~=p==161= TKV= m~==165= TKNM= p~=i==p~==175= TKNN= n~=p~==180= TKNO= p==189= = = 8 afccbobkqf^i=`^i`rirp= UKN= c=~=q=d~==191= UKO= i==c==208= UKP= a=~=m===a~==209= UKQ= q~==a~==211= UKR= e=l=a~==215= UKS= ^~==a~==217= UKT= a~==221= UKU= j~~=c==222= UKV= a~=l~==225= = = 9 fkqbdo^i=`^i`rirp= VKN= f=f~==227= VKO= f~==o~~=c==228= VKP= f~==f~~=c==231= VKQ= f~==q=c==237= VKR= f~==e=c==241= VKS= f~==b~=~=i~=c==242= VKT= o=c~==243= VKU= a=f~==247= VKV= f=f~==253= VKNM= a=f~==257= VKNN= q=f~==269=vi 8. VKNO= i=f~==275= VKNP= p~=f~==285= = = 10 afccbobkqf^i=bnr^qflkp= NMKN= c=l=l~=a~=b~==295= NMKO= p=l=l~=a~=b~==298= NMKP= p=m~~=a~=b~==302= = = 11 pbofbp= NNKN= ^=p==304= NNKO= d=p==305= NNKP= p=c=p==305= NNKQ= f=p==307= NNKR= m==`=p==307= NNKS= `=q==308= NNKT= ^~=p==310= NNKU= m=p==311= NNKV= a~=~=f~==m=p==312= NNKNM= q~=~=j~~=p==313= NNKNN= m=p=b~==p=c==314= NNKNO= _~=p==316= NNKNP= c=p==316= = = 12 mol_^_fifqv= NOKN= m~=~=`~==318= NOKO= m~=c~==319= = = = = =vii 9. = q=~==~==~K= =viii 10. Chapter 1Number Sets = = = =1.1 Set Identities = pW=^I=_I=`= r~=W=f= `=W= ^ = m=W= ^ _ == b=W= = r==W= ^ _ = f==W= ^ _ = a==W= ^ y _ = == 1.= 2.= 3. 4. 5.^ f= ^ ^=^ = _ == ^ _ =~= _ ^ .== b=p= ^= = r==p== ` = ^ _ = { ^ _}= =1 11. CHAPTER 1. NUMBER SETS======= Figure 1.6.= 7. = 8.= `~= ^_ = _^= ^~= ^ (_ ` ) = (^ _ ) ` = f==p= ` = ^ _ = { ^ ~ _} = ======== Figure 2.9.= 10.== `~= ^_ = _^= ^~= ^ (_ ` ) = (^ _ ) ` ==2 12. CHAPTER 1. NUMBER SETS11.= 12. = 13. = 14.a= ^ (_ ` ) = (^ _ ) (^ ` ) I= ^ (_ ` ) = (^ _ ) (^ ` ) K= f= ^ ^ = ^ I== ^^ = ^= a~= ^ = I= ^f= f= f= ^ = ^ I== ^f= ^ =15.16.17.18.`= ^ = { f ^} = `==f=~=r ^ ^ = f I== ^ ^ = = = a=j~=i~ (^ _ ) = ^ _ I== (^ _ ) = ^ _ = = a==p ` = _ y ^ = { _ ~ ^} = =3 13. CHAPTER 1. NUMBER SETS======= Figure 3.= 19._ y ^ = _ y (^ _ ) =20._ y ^ = _ ^21.^y^=22.^ y _ = ^ == ^ _ = .= ======== Figure 4.= 23.(^ y _) ` = (^ `) y (_ `)24.^ = f y ^25.`~~=m ` = ^ _ = {( I ) ^ ~ _} = =4= 14. CHAPTER 1. NUMBER SETS1.2 Sets of Numbers =26.27.= 28.= 29.= 30.k~~=W=k= t=W= kM = fW=w= m=W= w + = k~=W= w = o~~=W=n= o~=W=o== `=W=`== = = k~~=k `=W k = {NI OI PI K} K= t=k `==~=W= k M = {MI NI OI PI K} K= f t==~===~=W= 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~~=k o~==~=~W== ~ n = = ~ ~ w ~ w ~ M K= f~~=k k~=~=~=~K=5 15. CHAPTER 1. NUMBER SETS31.o~=k== r==~~=~=~~=W=oK== 32.`=k ` = { + o ~ o}I== ====~~=K= 33.k wn o `= ====== Figure 5.= = = = = =6 16. CHAPTER 1. NUMBER SETS1.3 Basic Identities = o~=W=~I=I== = = 34.^=f= ~+M=~ = =35.^=f= ~ + ( ~ ) = M = =36.`~==^= ~ += +~ =37.^~==^= (~ + ) + = ~ + ( + ) === 38.a==p~= ~ = ~ + ( ) = =39.= 40.41. 42.j~=f= ~ N = ~ = j~=f= N ~ = N I= ~ M ~ = j~=q=M ~ M = M = `~==j~= ~ = ~ = =7 17. CHAPTER 1. NUMBER SETS43.^~==j~= (~ ) = ~ ( ) = a=i~= ~ ( + ) = ~ + ~ =44.= 45.a==a= ~ N = ~ = = = =1.4 Complex Numbers = k~~=W== f~~=W== `=W== o~=~W=~I== f~~=~W=I== j==~==W=I= N I= O = ^==~==W= I= N I= O = = = 46.= 47. = 48.N = = O = N = P = = Q = N=R = = S = N = T = = U = N = Q +N = = Q + O = N = Q + P = = Q = N = = ~ + =`=m~= =8 18. CHAPTER 1. NUMBER SETS======= Figure 6.= 49. = 50. = 51. =(~ + ) ( + ) = (~ ) + ( ) = (~ + )( + ) = (~ ) + (~ + ) = ~ + ~ + ~ = + = + O + O O + O52.= 53.(~ + ) + ( + ) = (~ + ) + ( + ) =`~=`=k= |||||||~ + = ~ = = 54.~ = I= = == =9 19. CHAPTER 1. NUMBER SETS== Figure 7.55.= 56.= m~=m~==`=k= ~ + = ( + ) = j=~=^==~=`=k= f= ~ + ==~==I== = ~ O + O =EFI== = ~~ =E~FK= ~= 57.= 58.m==m~=o~= N O = N ( N + N ) O ( O + O ) = = NO [(N + O ) + (N + O )] = `~=k==m~=o~= |||||||||||||||||||||( + ) = [( ) + ( )] == 59.f==~=`=k==m~=o~= N N = [( ) + ( )] = ( + ) 10 20. CHAPTER 1. NUMBER SETS60.= 61. = 62. = 63.= 64.n==m~=o~= N N ( N + N ) N = [(N O ) + (N O )] = = O O ( O + O ) O m==~=`=k= = [( + )] = [() + ()] = c~=a=j= ( + ) = () + () = k=o==~=`=k= + O + O = ( + ) = + I== == = MI NI OI KI N K== b=c~= = + = = =11 21. Chapter 2Algebra = = = =2.1 Factoring Formulas = o~=W=~I=I=== k~~=W== = = 65. = 66. = 67. = 68. = 69. = 70. = 71.= 72.~ O O = (~ + )(~ ) = ~ P P = (~ )(~ O + ~ + O ) = ~ P + P = (~ + )(~ O ~ + O ) = ~ Q Q = (~ O O )(~ O + O ) = (~ )(~ + )(~ O + O ) = ~ R R = (~ )(~ Q + ~ P + ~ O O + ~P + Q ) = ~ R + R = (~ + )(~ Q ~ P + ~ O O ~P + Q ) = f===I== ~ + = (~ + )(~ N ~ O + ~ P O K ~ O + N ) K== f===I=== ~ = (~ )(~ N + ~ O + ~ P O + K + ~O + N ) I==12 22. CHAPTER 2. ALGEBRA~ + = (~ + )(~ N ~ O + ~ P O K + ~O N ) K= = = =2.2 Product Formulas73. = 74. = 75. = 76. = 77. = 78. = 79.= 80. = 81.o~=W=~I=I=== t=W=I== = = (~ )O = ~ O O~ + O =(~ + )O = ~ O + O~ + O = (~ )P = ~ P P~ O + P~O P = (~ + )P = ~ P + P~ O + P~O + P = (~ )Q = ~ Q Q~ P + S~ O O Q~P + Q = (~ + )Q = ~ Q + Q~ P + S~ OO + Q~P + Q = _~=c~= (~ + ) = ` M~ + `N~ N + ` O~ OO + K + ` N~N + ` I > = ` = =~==~=K= > ( )>(~ + + )O = ~ O + O + O + O~ + O~ + O = (~ + + + K + + )O = ~ O + O + O + K + O + O + = + O(~ + ~ + K + ~ + ~ + + K + + + K + ) =13 23. CHAPTER 2. ALGEBRA2.3 Powers = _~=E=~=FW=~I=== m=E~~=FW=I== = = ~ ~ = ~ + =82. = 83.~ = ~ = ~= 84. =(~) = ~ =85.~ ~ = = = 86. = 87. = 88. =(~ ) = ~ =~ M = N I= ~ M = ~N = N = ~ =89.N = ~= ~ = ~ =90.= = = = =14 24. CHAPTER 2. ALGEBRA2.4 Roots =91. =_~W=~I=== m=E~~=FW=I== ~ I M ====E = O I= k F= = = ~ = ~ =92. =~ = ~ =93.~ ~ = I= M = = 94.= 95. = 96. =~ ~ ~ I= M K= = = (~ ) ( ~) = ~ ==~= 97. =~ =98. =~ =~ =99. = 100. = ~ = ~ =( ~) ~ == ~ =15 25. CHAPTER 2. ALGEBRAN ~ N = I= ~ M K= ~ ~101.= ~ =102.~ + ~O ~ ~O = O O= N ~m = = ~ ~ 103.= = =2.5 Logarithms =104.105. 106. 107. 108. 109.m=~=W=I=I=~I=I== k~~=W=== = = a==i~= = ~ ==~=== = ~ I= ~ > M I= ~ N K= = ~ N = M = = ~ ~ = N = = ~ > N ~ M = = + ~ < N = ~ ( ) = ~ + ~ = = ~ = ~ ~ = 16 26. CHAPTER 2. ALGEBRA110. ~ ( ) = ~ = = N 111. ~ = ~ = = 112. ~ = = ~ I= > M I= N K= ~ = N 113. ~ = = ~ = 114. = ~ ~ = = 115. i~==_~=NM= NM = = = 116. k~~=i~= = I== N = = N + = OKTNUOUNUOUK = = N 117. = = MKQPQOVQ = NM = N 118. = = OKPMORUR = = = = = =17 27. CHAPTER 2. ALGEBRA2.6 Equations = o~=W=~I=I=I=I=I=I== pW= N I= O I= N I= O I= P = = = 119. i~=b~==l=s~~= ~ + = M I= = K== ~ = 120. n~~=b~= O Q~ ~ + + = M I= NI O = K= O~ = 121. a~= a = O Q~ = = 122. s=c~= f= O + + = M I=== N + O = K= N O = = 123. ~ O + = M I= N = M I= O = K= ~ = 124. ~ O + = M I= NI O = K= ~ = 125. `=b~K=`~~=c~K== P + + = M I== O18 28. CHAPTER 2. ALGEBRA N = + I= OI P = N ( + ) P ( + ) I== O O== O=P OOO + + I= = P + K== O O O P O P= =2.7 Inequalities s~~W=I=I== ~ I I I o~=W= I=I== ~N I ~ O I ~ P I KI ~ a~W=aI= a I= a I= a == = = 126. f~I=f~=k~=~=d~== = f~= f~=k~= d~= [~I ]= ~ = ~ < =(~I ] ==~ < =[~I ) ==~ < < =(~I ) == < I= = < < I= ~=( I ] == =( I ) ==[~I ) ==(~I ) ==19 29. CHAPTER 2. ALGEBRA127. = 128. = 129. = 130. = 131. = 132. = 133. =f= ~ > I== < ~ K= f= ~ > I== ~ > M == ~ < M K= f= ~ > I== ~ + > + K= f= ~ > I== ~ > K= f= ~ > =~= > I== ~ + > + K= f= ~ > =~= > I== ~ > K= f= ~ > =~= > M I== ~ > K=134. f= ~ > =~= > M I==~ > K= = 135. f= ~ > =~= < M I== ~ < K= = ~ 136. f= ~ > =~= < M I== < K= = 137. f= M < ~ < =~= > M I== ~ < K= = 138. f= M < ~ < =~= < M I== ~ > K= = 139. f= M < ~ < I== ~ < K= = ~+ I== 140. ~ O = ~ > M =I= > M X=~=~==~=== ~ = K== = N 141. ~ + O I== ~ > M X=~=~=~=~==~= ~ = N K= ~20 30. CHAPTER 2. ALGEBRA142.~N~ O K~ ~N + ~ O + K + ~ I== ~N I ~ O I KI ~ > M K= = 143. f= ~ + > M =~= ~ > M I== > K= ~ = 144. f= ~ + > M =~= ~ < M I== < K== ~ = 145. ~ O + + > M = = = ~ > M= = = = = a>M== = = a=M== = = a O = = N < I= > N = == < ~ I== < ~ =~= > ~ I== ~ > M K= f= O < ~ I== < ~ I== ~ > M K= f= O > ~ I== > ~ I== ~ > M K=151. f== ( ) ( ) > M ( ) > M I== K= ( ) ( ) M ( ) ( ) < M ( ) < M I== K= ( ) ( ) M152.= = =2.8 Compound Interest Formulas = c=~W=^= f~=W=`= ^~=~==W== k==~=W== k=====~W== = = 153. d~=`=f=c~= ^ = ` N + = =22 32. CHAPTER 2. ALGEBRA154. p=`=f=c~= f======~I==== ~==W= ^ = `(N + ) K= = 155. `=`=f= f====~=E FI=== ^ = ` K= = =23 33. Chapter 3Geometry = = = =3.1 Right Triangle = i==~==~W=~I== eW== ^W== j~W= ~ I= I= = ^W= I = o~===W=o= o~===W== ^~W=p= = === Figure 8.= 156. + = VM = =24 34. CHAPTER 3. GEOMETRY157. =~ = = = 158. = = = =159. ~ =~ = = = 160. = = ~ = ~ = 161. = = = = 162. = = = ~ =163. m~~=q= ~ O + O = O == 164. ~ = I= = I== = = ~= = ~= = = = = ~= ~= I= I====K= = O=O====== Figure 9.=25 35. CHAPTER 3. GEOMETRY165. O = I=== ====~====~K== = O O ~ 166. O = O I= O = ~ O I=== ~ Q Q = ~ =~= =~==~====~=~=K== === Figure 10.= 167. = I== O = ===~====K= = 168. o = = = O = ~ + ~ = = 169. = O ~ ++ = 170. ~ = = = =26 36. CHAPTER 3. GEOMETRY171. p =~ = = O O= = =3.2 Isosceles Triangle = _~W=~= iW== _~=~W= = s=~W= = ^===~W== mW=i= ^~W=p= = === Figure 11.= 172. = VM = O =173. O = O O~ = Q27 37. CHAPTER 3. GEOMETRY174. i = ~ + O == 175. p =O~ = = O O= = =3.3 Equilateral Triangle = p==~=~~=~W=~= ^W== o~===W=o= o~===W== mW=i= ^~W=p= = === Figure 12.= 176. =~ P = O =28 38. CHAPTER 3. GEOMETRYO ~ P = 177. o = = P P = N ~ P o = = 178. = = P S O =179. i = P~ == 180. p =O~ ~ P = = O Q= = =3.4 Scalene Triangle E^=~=====~F= = = p==~=~W=~I=I== ~ ++ == pW= = O ^==~=~W= I I = ^====~I=I=W= ~ I I = j~====~I=I=W= ~ I I = _===~= I I W= ~ I I = o~===W=o= o~===W== ^~W=p= = =29 39. CHAPTER 3. GEOMETRY======= Figure 13.= 181. + + = NUM = 182. ~ + > I== + > ~ I== ~ + > K= = 183. ~ < I== < ~ I== ~ < K=== 184. j= ~ = I= ~ K= O ======== Figure 14.=30 40. CHAPTER 3. GEOMETRY185. i~==`= ~ O = O + O O I= O = ~ O + O O~ I= O = ~ O + O O~ K= = 186. i~==p= ~ = = = Oo I== =o===~====K== = ~ ~ ~ ~ = = = = = = 187. o = = O O O O ~ O O Qp = ( ~ )( )( ) I== 188. O = N N N N = + + K= ~ = ( )( ) I= 189. = O ( ~ ) I= = O ~ = O( )( ) K= ( ~ )= O 190. ~ = ( ~ )( )( ) I= ~ O ( ~ )( )( ) I= = O = ( ~ )( )( ) K= 31 41. CHAPTER 3. GEOMETRY191. ~ = = I= = ~ = I= = ~ = K== + ~ I== O Q O O ~ + O O = I== O Q O O ~ + O O = K= O Q192. O = ~OOO======== Figure 15.= O O O 193. ^j = ~ I= _j = I= `j = =EcKNRFK= P P P = Q( ~ ) 194. O = I== ~ ( + )O Q~( ) O = I== (~ + )O Q~( ) O = K= (~ + )O =32 42. CHAPTER 3. GEOMETRY~ ~ = = I== O O O ~ ~ I== p= = = O O O p = ( ~ )( )( ) =Ee=c~FI= p = I== ~ p= I= Qo p = Oo O I= p = O ~ ~ ~ K= O O O195. p == = =3.5 Square p==~=~W=~= a~~W== o~===W=o= o~===W== mW=i= ^~W=p= === Figure 16.33 43. CHAPTER 3. GEOMETRY196. = ~ O === 197. o = ~ O = = O O =~ 198. = = O 199. i = Q~ == =200. p = ~ = = = = O3.6 Rectangle = p==~=~W=~I== a~~W== o~===W=o= mW=i= ^~W=p= = === Figure 17.= 201. = ~ O + O ==34 44. CHAPTER 3. GEOMETRY202. o = = O =203. i = O(~ + ) ==204. p = ~ = = = =3.7 Parallelogram = p==~=~~~W=~I== a~~W= N I O = `=~W= I = ^===~~W= = ^W=== mW=i= ^~W=p= = ========Figure 18.= 205. + = NUM = 206. + = O(~ + ) = O NO OO=O=35 45. CHAPTER 3. GEOMETRY207. = = = 208. i = O(~ + ) = 209. p = ~ = ~ I== N p = N O K= O = = == =3.8 Rhombus = p==~=W=~= a~~W= N I O = `=~W= I = ^W=e= o~===W== mW=i= ^~W=p= = ======== Figure 19.=36 46. CHAPTER 3. GEOMETRY210. + = NUM == 211. + = Q~ = O NO OO= 212. = ~ =N O = O~ = ~ 213. = = N O = = O Q~ O=214. i = Q~ == 215. p = ~ = ~ I== N p = N O K= O = = = O3.9 Trapezoid = _~==~=~W=~I== jW== ^W== ^~W=p= = =37 47. CHAPTER 3. GEOMETRY== Figure 20.= 216. = 217. p =~+ = O ~+ = = O== = =3.10 Isosceles Trapezoid = _~==~=~W=~I== iW== jW== ^W== a~~W== o~===W=o= ^~W=p= = =38 48. CHAPTER 3. GEOMETRY== Figure 21.= 218. =~+ = O =219. = ~ + = = N O 220. = O ( ~ ) = Q O= ~ + O = (O ~ + )(O + ~ ) = ~+ 222. p = = = O = = = = = = 221. o =39 49. CHAPTER 3. GEOMETRY3.11 Isosceles Trapezoid with Inscribed Circle = _~==~=~W=~I== iW== jW== ^W== a~~W== o~===W=o= o~===W== mW=i= ^~W=p= = === Figure 22.= 223. ~ + = O = = ~+ 224. = == O = 225. = + = OOO=40 50. CHAPTER 3. GEOMETRY226. = ~ = = O O = ~+ ~ N+ O + O = = = = +S+ = O Q O ~ O U ~ = 228. i = O(~ + ) = Q = = (~ + ) ~ = = = i == ~+ = 229. p = O O O = = = 227. o =O3.12 Trapezoid with Inscribed Circle = _~==~=~W=~I== i~~=W=I== jW== ^W== a~~W= N I O = ^===~~W= = o~===W== o~===W=o= mW=i= ^~W=p= =41 51. CHAPTER 3. GEOMETRY== Figure 23.= 230. ~ + = + = ~+ + = = 231. = O O 232. i = O(~ + ) = O( + ) === =~+ + = = I== O O N p = N O K= O233. p == = =3.13 Kite = p==~=W=~I== a~~W= N I O = ^W= I I = mW=i= ^~W=p= = =42 52. CHAPTER 3. GEOMETRY== Figure 24.=234. + + O = PSM = 235. i = O(~ + ) == =236. p =N O = O= = =3.14 Cyclic Quadrilateral p==~=~~~W=~I=I=I== a~~W= N I O = ^===~~W= = f~=~W= I I I = o~===W=o= mW=i= pW=== ^~W=p=43 53. CHAPTER 3. GEOMETRY== Figure 25.=237. + = + = NUM == 238. m=q= ~ + = N O = 239. i = ~ + + + === N (~ + )(~ + )(~ + ) I== 240. o = Q ( ~ )( )( )( ) i = = K= O = N 241. p = N O I== O p = ( ~ )( )( )( ) I== i = = K= O = = =44 54. CHAPTER 3. GEOMETRY3.15 Tangential Quadrilateral = p==~=~~~W=~I=I=I== a~~W= N I O = ^===~~W= = o~===W== mW=i= pW=== ^~W=p= = === Figure 26.= 242. ~ + = + = = 243. i = ~ + + + = O(~ + ) = O( + ) = = O N O (~ ) (~ + ) O I== O i = = K== O = OO244. =45 55. CHAPTER 3. GEOMETRYN 245. p = = N O = O = = =3.16 General Quadrilateral = p==~=~~~W=~I=I=I== a~~W= N I O = ^===~~W= = f~=~W= I I I = mW=i= ^~W=p= = ========== Figure 27.=246. + + + = PSM = 247. i = ~ + + + == =46 56. CHAPTER 3. GEOMETRYN 248. p = N O = O = = =3.17 Regular Hexagon = pW=~= f~=~W= = p~=W== o~===W== o~===W=o= mW=i= pW=== ^~W=p= = === Figure 28.= 249. = NOM = = 250. = =~ P = O47 57. CHAPTER 3. GEOMETRY251. o = ~ ==252. i = S~ == O~ P P I== O i = = K= O = = =253. p = =3.18 Regular Polygon = pW=~= k==W== f~=~W= = p~=W== o~===W== o~===W=o= mW=i= pW=== ^~W=p= = =48 58. CHAPTER 3. GEOMETRY== Figure 29.= 254. =255. =O NUM = O =O NUM = O= 256. o =~ O = =257. = =~ O ~ = oO ~O = Q=258. i = ~ == 259. p =o O I== O Op = = o O ~O I== Q49 59. CHAPTER 3. GEOMETRY= =i K== O= = =3.19 Circle = o~W=o= a~W== `W=~= p~=W=I== q~=W== `~=~W= = f=~W= = mW=i= ^~W=p= = = 260. ~ = Oo = O === Figure 30.=50 60. CHAPTER 3. GEOMETRY261. ~N~ O = NO = === Figure 31.=262. N = N = ======== Figure 32.= 263. O = N = =51 61. CHAPTER 3. GEOMETRY======= Figure 33.= 264. = = O=== Figure 34.= 265. i = Oo = = = 266. p = o O =io = == Q O O=52 62. CHAPTER 3. GEOMETRY3.20 Sector of a Circle = o~==~=W=o= ^=W== `~=~=E=~~FW== `~=~=E=FW= = mW=i= ^~W=p= = === Figure 35.=267. = o = 268. ==o = NUM=269. i = + Oo == 270. p =o o o = = == O O PSM OO= =53 63. CHAPTER 3. GEOMETRY3.21 Segment of a Circle = o~==~=W=o= ^=W== `W=~= `~=~=E=~~FW== `~=~=E=FW= = e===W== mW=i= ^~W=p= = === Figure 36.= 271. ~ = O Oo O == N 272. = o Qo O ~ O I= < o = O = 273. i = + ~ = =54 64. CHAPTER 3. GEOMETRYO O N [o ~(o )] = o = o ( ) I== O O NUM O O p ~ K= P274. p == = =3.22 Cube = bW=~== a~~W== o~===W== o~===W== p~=~~W=p= sW=s= = ====== Figure 37.= 275. = ~ P == ~ 276. = = O =55 65. CHAPTER 3. GEOMETRY277. o =~ P = O =278. p = S~ = O= 279. s = ~ == = = = P3.23 Rectangular Parallelepiped = bW=~I=I=== a~~W== p~=~~W=p= sW=s= = ======== Figure 38.= 280. = ~ O + O + O = 281. p = O(~ + ~ + ) = 282. s = ~ === =56 66. CHAPTER 3. GEOMETRY3.24 Prism = i~~=W== eW== i~~=~~W= p i = ^~==~W= p_ = q~=~=~~W=p= sW=s= = ======== Figure 39.= 283. p = p i + Op_ K== = 284. i~~=^~==~=o=m= p i = (~ N + ~ O + ~ P + K + ~ ) = = 285. i~~=^~==~=l=m= p i = I== ========K= =57 67. CHAPTER 3. GEOMETRY286. s = p_ == 287. `~~D=m== d=====~~=~K=f== ~===~~====~=~==~= ~~===I=======~=~K= = = =3.25 Regular Tetrahedron = q~==W=~= eW== ^~==~W= p_ = p~=~~W=p= sW=s= = === Figure 40.= 288. =O ~= P=58 68. CHAPTER 3. GEOMETRY289. p_ =P~ O = Q =290. p = P~ = = N ~P 291. s = p_ = K== P S O = = = O3.26 Regular Pyramid = p==~W=~= i~~=W== eW== p~=W=== k==W=== p==~W== o~=====~W== ^~==~W= p_ = i~~=~=~~W= pi = q~=~=~~W=p= sW=s= = =59 69. CHAPTER 3. GEOMETRY== Figure 41.= 292. = O ~O = Q =293. = O ~ = O QO O= N N 294. p i = ~ = ~ QO ~ O = = O Q = 295. p_ = = = 296. p = p_ + p i = = N N 297. s = p_ = == P P = = =60 70. CHAPTER 3. GEOMETRY3.27 Frustum of a Regular Pyramid = ~N I ~ O I ~ P IKI ~ = _~=~===W= N I O I P IKI eW== p~=W=== ^~==~W= pN I= pO = i~~=~=~~W= p i = m==~W= mN I= mO = p~=~W== q~=~=~~W=p= sW=s= = === Figure 42.= 298.N O P = = =K= = = = ~N ~ O ~ P ~ ~ =61 71. CHAPTER 3. GEOMETRY299.pO = O = pN =(mN + mO ) = 300. p i = O=301. p = p i + pN + pO == 302. s = pN + pNpO + pO = P = O p p 303. s = N N + + = N N + + O = P ~ ~ P = = =()[]3.28 Rectangular Right Wedge = p==~W=~I== q=W== eW== i~~=~=~~W= p i = ^~==~W= p_ = q~=~=~~W=p= sW=s= = =62 72. CHAPTER 3. GEOMETRY== Figure 43.= N (~ + ) QO + O + O + (~ )O = O = 305. p_ = ~ = = 306. p = p_ + p i = = (O~ + ) = 307. s = S = = = 304. p i =3.29 Platonic Solids = bW=~= o~===W== o~===W=o= p~=~~W=p= sW=s= = =63 73. CHAPTER 3. GEOMETRY308. c=m~=p= q= ~= = ~= = ~= = ~= ~=====~=K== = k= k= p= p= k= =s =b= =c~= q~== Q= S= Q= PKOR= `= U= NO= S= PKOO= l~= S= NO= U= PKOT= f~= NO= PM= OM= PKOT= a~= OM= PM= NO= PKOT= = =Octahedron === Figure 44.= 309. =~ S = S= 310. o =~ O = O=64 74. CHAPTER 3. GEOMETRY311. p = O~ O P = = ~P O 312. s = = P = =Icosahedron === Figure 45.= 313. =(= 314. o =)~ P P+ R = NO()~ O R+ R = Q= 315. p = R~ O P = = R~ P P + R 316. s = = NO = =()65 75. CHAPTER 3. GEOMETRYDodecahedron === Figure 46.317. =(~ NM OR + NN R = O= 318. o =)=()~ P N+ R = Q=()319. p = P~ O R R + O R = = ~ P NR + T R 320. s = = Q = = =()3.30 Right Circular Cylinder = o~==~W=o= a~==~W==66 76. CHAPTER 3. GEOMETRYeW=e= i~~=~=~~W= p i = ^~==~W= p_ = q~=~=~~W=p= sW=s= = ======== Figure 47.= 321. p i = Ooe = = 322. p = p i + Op_ = Oo(e + o ) = e + = O = 323. s = p_ e = o O e = = = =67 77. CHAPTER 3. GEOMETRY3.31 Right Circular Cylinder with an Oblique Plane Face = o~==~W=o= q=~===~=W= N = q====~=W= O = i~~=~=~~W= p i = ^~==~==~W= p_ = q~=~=~~W=p= sW=s= = === Figure 48.=324. p i = o(N + O ) = = O O 325. p_ = o + o o + N = O = OO68 78. CHAPTER 3. GEOMETRYO N O O 326. p = p i + p_ = o N + O + o + o + = O = o O (N + O ) = 327. s = O = = =3.32 Right Circular Cone o~==~W=o= a~==~W== eW=e= p~=W== i~~=~=~~W= pi = ^~==~W= p_ = q~=~=~~W=p= sW=s= = === Figure 49.69 79. CHAPTER 3. GEOMETRY328. e = O o O = = 329. p i = o = = O = 330. p_ = o O = =N 331. p = p i + p_ = o ( + o ) = + = O O = N N 332. s = p_ e = o O e = P P = = =3.33 Frustum of a Right Circular Cone = o~==~W=oI== eW=e= p~=W== p~=~W== ^~==~W= pN I= pO = i~~=~=~~W= p i = q~=~=~~W=p= sW=s= = =70 80. CHAPTER 3. GEOMETRY== Figure 50.= 333. e = O (o ) = = o 334. == = p oO 335. O = O = O = pN = 336. p i = (o + ) = = 337. p = pN + pO + p i = o O + O + (o + ) = = 338. s = pN + pNpO + pO = P = O pN o o pN 339. s = N+ + O = N + + = P P = = = O[(])[71] 81. CHAPTER 3. GEOMETRY3.34 Sphere = o~W=o= a~W== p~=~~W=p= sW=s= === Figure 51.= 340. p = Qo O = = Q N N 341. s = o P e = P = po = P S P = = =3.35 Spherical Cap o~==W=o= o~==~W== eW== ^~==~=~W= p_ = ^~==~=~W= p` = q~=~=~~W=p= sW=s=72 82. CHAPTER 3. GEOMETRY== Figure 52.= 342. o = O + O = O= 343. p_ = O = = 344. p` = ( O + O )= = 345. p = p_ + p` = ( O + O O ) = (Oo + O ) = = 346. s = O (Po ) = (P O + O ) = S S = = =3.36 Spherical Sector = o~==W=o= o~==~==~=~W== eW== q~=~=~~W=p= sW=s= =73 83. CHAPTER 3. GEOMETRY===========Figure 53.= 347. p = o(O + ) = = O 348. s = o O = P = kW= q= = ~= ~= = = = = ~= =~=K= = = =3.37 Spherical Segment = o~==W=o= o~==~W= N I= O = eW== ^~==~=~W= pp = ^~==~==~W= pN I= pO = q~=~=~~W=p= sW=s= =74 84. CHAPTER 3. GEOMETRY======= Figure 54.= 349. pp = Oo = = 350. p = pp + pN + pO = (Oo + NO + OO ) = = N 351. s = (PNO + POO + O )= S = = =3.38 Spherical Wedge = o~W=o= a~=~==W== a~=~==~~W= = ^~==~=W= p i = q~=~=~~W=p= sW=s= = =75 85. CHAPTER 3. GEOMETRY== Figure 55.= 352. p i =o O = Oo O = VM= 353. p = o O +o O = o O + Oo O = VM= 354. s =oP O = oP = OTM P= = =3.39 Ellipsoid = p-~W=~I=I== sW=s=76 86. CHAPTER 3. GEOMETRY========= Figure 56.= Q 355. s = ~ = P = = =Prolate Spheroid = p-~W=~I=I==E ~ > F= p~=~~W=p= sW=s= = = ~ ~ 356. p = O + I== = =~ O O K= ~= Q 357. s = O~ = P =77 87. CHAPTER 3. GEOMETRYOblate Spheroid = p-~W=~I=I==E ~ < F= p~=~~W=p= sW=s= = = ~ ~ ~ I== 358. p = O + L ~ = =O ~ O K= = Q 359. s = O~ = P = = =3.40 Circular Torus = j~=~W=o= j=~W== p~=~~W=p= sW=s= =78 88. CHAPTER 3. GEOMETRY==Picture 57.= 360. p = QOo = = 361. s = OOo O = = =79= 89. Chapter 4Trigonometry = = = = ^W= I= = o~==E~==~=FW=I=== t=W== = =4.1 Radian and Degree Measures of Angles = 362. N ~ == 363. N == 364. N D == 365. N ? == 366. = == ==NUM RTNT DQR? = ~ MKMNTQRP ~ = NUM ~ MKMMMOVN ~ = NUM SM ~ MKMMMMMR ~ = NUM PSMM ^= EF= ^= E~~F=M=PM= QR= SM= VM= NUM= OTM= PSM=M= = S = Q80 = P = O=P = OO = 90. CHAPTER 4. TRIGONOMETRY4.2 Definitions and Graphs of Trigonometric Functions ==== Figure 58.= 367. = = = 368. = = = 369. ~ = = = 370. = = =81 91. CHAPTER 4. TRIGONOMETRY371. = = = 372. = = =373. p=c= = I= N N K= == Figure 59.= 374. `=c== = I= N N K=82 92. CHAPTER 4. TRIGONOMETRY== Figure 60.= 375. q~=c= = ~ I= (O + N) I= ~ K = O === Figure 61.=83 93. CHAPTER 4. TRIGONOMETRY376. `~=c== = I= I== K= === Figure 62.= 377. p~=c= = I= (O + N) K= O ==84 94. CHAPTER 4. TRIGONOMETRY== Figure 63.= 378. `~=c== = I= K== Figure 64.85 95. CHAPTER 4. TRIGONOMETRY4.3. Signs of Trigonometric Functions 379. = == = 380. =n~~==f= ff= fff= fs=p = H= H= = =` = H= = = H=q~ = H= = H= =` = H= = H= =p = H= = = H=`= = H= H= = === Figure 65.= = = = = = = = = =86 96. CHAPTER 4. TRIGONOMETRY4.4 Trigonometric Functions of Common Angles 381. = = ~ = M= M= = PM= S = QR= Q = SM= P = VM= O O = NOM= P NUM= = P = OTM= O PSM= O = = = = = = = = = = = = = =O = O P = O = N= P = O O = O N = ON=M=P = O M=N = O N= N= M= = M= N = O~ = M= = N = P= P = N= O = P == O=N=N=P=N = PO=O = PM==N==O=O=M=N P = N=O = P =M==M== N=N=M==N== P=87O= 97. CHAPTER 4. TRIGONOMETRY382. = = ~ = = NR= NO = =~ = =S O = QS+ O = QO P =O+ P =RO R = RR+O R =NU= = NMR N = QNM + O R QPS= = RNM O R QR +N = QRQ=P = NMR +N = QNM O R QTO=O = RNM + O R QR N = QTR=R = NOS+ O = QS O = Q= = =4.5 Most Important Formulas = 383. O + O = N = = 384. O ~ O = N = = 385. O O = N = = = 386. ~ = 88NM O R R +NR +N NM O RR +N NM O R = NM O R R +N =R+O R =RO R R =O+ P =O P = 98. CHAPTER 4. TRIGONOMETRY387. = = = 388. ~ = N = = N 389. = = = N 390. = = = = =4.6 Reduction Formulas = 391. = == = = = == = VM = VM + = NUM NUM + OTM OTM + PSM = PSM + = = + = + = + = = = = = + =89 = + = + = = = = = + = + = + =~ = ~ = + = = ~ = + ~ = + = = ~ = + ~ = = = + ~ = ~ = = + = + ~ = ~ = = + = 99. CHAPTER 4. TRIGONOMETRY4.7 Periodicity of Trigonometric Functions =392. ( O ) = I== O == PSM K= = 393. ( O ) = I== O == PSM K= = 394. ~( ) = ~ I== == NUM K= = 395. ( ) = I== == NUM K= = = =4.8 Relations between Trigonometric Functions = 396. = N O = O = = N + ~ O ON (N O ) = O O N = O O QO ~= =397. = N O = O= = N + ~ O ON (N + O ) = O O N = O ON ~ O= = 398. ~ = O N O = O N = = = N + O O90 100. CHAPTER 4. TRIGONOMETRY N O O = = = N + O O N + ~ O O ~= = N + O O = O N = = = O N O N ~ O N + O O= = = = N O O ~ O399. == N O= 400. = = N + ~ O = N ~ O O = N + ~ O N O= 401. = = N + O = O ~ O = = = N + ~ O4.9 Addition and Subtraction Formulas = 402. ( + ) = + = = 403. ( ) = = = 404. ( + ) = = = 405. ( ) = + =91 101. CHAPTER 4. TRIGONOMETRY406. ~( + ) == 407. ~( ) == 408. ( + ) == 409. ( ) =~ + ~ = N ~ ~ ~ ~ = N + ~ ~ N ~ ~ = ~ + ~ N + ~ ~ = ~ ~ = = =4.10 Double Angle Formulas =410. O = O = = 411. O = O O = N O O = O O N = = O ~ O 412. ~ O = = = O N ~ ~ = O N ~ = = 413. O = O O = = = = = =92 102. CHAPTER 4. TRIGONOMETRY4.11 Multiple Angle Formulas = 414. P = P Q P = P O P = = 415. Q = Q U P = = 416. R = R OM P + NS R = = 417. P = Q P P = P P O = = 418. Q = U Q U O + N = = 419. R = NS R OM P + R = = P ~ ~ P 420. ~ P = = N P ~ O = Q ~ Q ~ P = 421. ~ Q = N S ~ O + ~ Q = ~ R NM ~ P + R ~ = 422. ~ R = N NM ~ O + R ~ Q = P P 423. P = = P O N = N S ~ O + ~ Q == 424. Q = Q ~ Q ~ P =93 103. CHAPTER 4. TRIGONOMETRY425. R =N NM ~ O + R ~ Q = ~ R NM ~ P + R ~ = = =4.12 Half Angle Formulas = 426. N = = O O= 427. N + = = O O= 428. ~ N N = = = = = O N + N + = 429. N + N + = = = = + = O N N = = =4.13 Half Angle Tangent Identities = O = 430. = N + ~ O O = O ~94 104. CHAPTER 4. TRIGONOMETRY O= 431. = O N + ~ O = O ~ O = 432. ~ = N ~ O O = N ~ O O= 433. = O ~ O = = = N ~ O4.14 Transforming of Trigonometric Expressions to Product = 434. + = O = 435. = O + = O O + = O O= 436. + = O + = O O= 437. = O + = O O=95 105. CHAPTER 4. TRIGONOMETRY438. ~ + ~ == 439. ~ ~ == 440. + == 441. =( + ) = ( ) = ( + ) = ( ) = = 442. + = O = O + = Q Q = 443. = O = O + = Q Q = ( ) = 444. ~ + = = ( + ) = 445. ~ = = 446. N + = O O = O = 447. N = O O = O =96 106. CHAPTER 4. TRIGONOMETRY 448. N + = O O = Q O = 449. N = O O = Q O = = =4.15 Transforming of Trigonometric Expressions to Sum = 450. =( ) ( + ) = O= 451. == 452. == 453. ~ ~ == 454. == 455. ~ =( ) + ( + ) = O ( ) + ( + ) = O ~ + ~ = + + = ~ + ~ ~ + = + ~ = = =97 107. CHAPTER 4. TRIGONOMETRY4.16 Powers of Trigonometric Functions = 456. O == 457. P == 458. Q == 459. R == 460. S == 461. O == 462. P == 463. Q == 464. R == 465. S =N O = O P P = Q Q Q O + P = U NM R P + R = NS NM NR O + S Q S = PO N + O = O P + P = Q Q + Q O + P = U NM + R P + R = NS NM + NR O + S Q + S = PO=98 108. CHAPTER 4. TRIGONOMETRY4.17 Graphs of Inverse Trigonometric Functions = 466. f=p=c== = ~ I= N N I= ~ K= O O=== Figure 66.= 467. f=`=c== = ~ I= N N I= M ~ K= =99 109. CHAPTER 4. TRIGONOMETRY== Figure 67.= 468. f=q~=c== = ~~ I= I= < ~~ < K= O O======== Figure 68.100 110. CHAPTER 4. TRIGONOMETRY469. f=`~=c== = ~ I= I= M < ~ < K======= Figure 69.= 470. f=p~=c== = ~= I ( I N] [NI )I ~ MI I K O O = Figure 70.101 111. CHAPTER 4. TRIGONOMETRY471. f=`~=c== = ~ I ( I N] [NI )I ~ I M MI K O O== Figure 71.= =4.18 Principal Values of Inverse Trigonometric Functions 472.=M=N = O PM = SM = O O~ = M = ~ = VM N = O PM ~ = QR = NOM ~ = NPR = =O = O QR = QR = P OP O SM PMVM M = N== VM = NUM NRM = = SM102N== = 112. CHAPTER 4. TRIGONOMETRY473.=M=P PN=~~ =M =PMQRSM~ = VMSMQRPMP= P P4.19 Relations between Inverse Trigonometric Functions =474. ~( ) = ~ = = 475. ~ = ~ = O = 476. ~ = ~ N O I= M N K= = 477. ~ = ~ N O I= N M K= = O I= < N K= 478. ~ = ~~ O N = N O I= M < N K= = 480. ~ = ~ N O I= N < M K= = 481. ~( ) = ~ =103 P= QR SM = = NPR NOM = NRM = = PM= = =479. ~ = ~ N= 113. CHAPTER 4. TRIGONOMETRY482. ~ = ~ = O= 483. ~ = ~ N O I= M N K= = 484. ~ = ~ N O I= N M K= = 485. ~ = ~~N O I= M < N K= = N O I= N < M K= 486. ~ = + ~~= 487. ~ = ~ N OI= N N K==488. ~~( ) = ~~ = = 489. ~~ = ~ = O = = 490. ~~ = ~ N+ O = N I= M K= 491. ~~ = ~ N+ O = N I= M K= 492. ~~ = ~ N+ O =104 114. CHAPTER 4. TRIGONOMETRY493. ~~ = N ~~ I= > M K= O = N 494. ~~ = ~~ I= < M K= O = N 495. ~~ = ~ I= > M K= = N 496. ~~ = ~ I= < M K= = 497. ~ ( ) = ~ = = 498. ~ = ~~ = O = N I= > M K= 499. ~ = ~ N+ O = N I= < M K= 500. ~ = ~ N+ O = = 501. ~ = ~ N+ O = N 502. ~ = ~~ I= > M K= = N 503. ~ = + ~~ I= < M K= = =105 115. CHAPTER 4. TRIGONOMETRY4.20 Trigonometric Equations504. 505. 506. 507.= t=W== = = = ~ I= = ( N) ~ ~ + = = = ~ I= = ~ ~ + O = = ~ = ~ I= = ~~ ~ + = = = ~ I= = ~ ~ + = = = =4.21 Relations to Hyperbolic Functions508. 509. 510. 511. 512.= f~~=W== = = ( ) = = = ~( ) = ~ = = ( ) = = = ( ) = = = ( ) = = = = =106 116. Chapter 5Matrices and Determinants = = = = j~W=^I=_I=`= b==~=~W= ~ I= I= ~ I= I= = a~==~=~W= ^ = j==~== ~ W= j = `~==~== ~ W= ` = q~==~=~W= ^ q I= ^ = ^==~=~W= ~ ^ = q~==~=~W= ^ = f==~=~W= ^ N = o~=W== o~=~~W= = k~~=W=I==== = =5.1 Determinants = 513. p=l=a~= ~ N ^ = N = ~ N O ~ O N = ~ O O = = = = =107 117. CHAPTER 5. MATRICES AND DETERMINANTS514. q=l=a~= ~NN ~NO ~NP ^ = ~ ON ~ OO~ OP = ~NN~ OO~ PP + ~NO~ OP~ PN + ~ NP~ ON~ PO =~ PN ~ PO ~ PP ~NN~ OP~ PO ~NO~ ON~ PP ~ NP~ OO~ PN == 515. p~=o=E^=oF=== Figure 72.= 516. k-=l=a~= ~NN ~NO K ~N ~ ON ~ OO K ~ O K K K K ^ = ~ N ~ O K ~ K K K K ~ N ~ O K ~ K ~N K ~ O K K K ~ =K K K ~ = 517. j= q== j =~~==== ~ ==-==~= ^= = = ( N) -= = ~= = = =~=^=====-==~=-=K=== =108 118. CHAPTER 5. MATRICES AND DETERMINANTS518. `~= + ` = ( N) j == 519. i~~=b~==-=l=a~= i~~=~=====-== ^ = ~ ` I= = NI OI KI K= =Ni~~=~=====-== ^ = ~ ` I= = NI OI KI K== =N= = =5.2 Properties of Determinants = 520. q==~===~=~=~==~===~= ~===~===K= ~ ~ O ~N N = == = N N O ~ O O = 521. f=====E==F=~==~I==== =~==~K= ~N N ~ O = O = ~ O O ~N N = 522. f====E==F=~==~I==~=== ~==K= ~N ~N = M= ~O ~O =109 119. CHAPTER 5. MATRICES AND DETERMINANTS523. f=========~===E=F=~======= ~====~I====~========~= ~K= ~ N N ~ N = N = ~ O O ~ O O = 524. f========~====E==F=~=~=E= ~F=~====== =~====E=FI===~===~= =~K= ~N + N N ~N N = = ~ O + O O ~ O O = = =5.3 Matrices = 525. a= ^= =~=^==~=~~=~~===E==F====~==K== ~ NN ~ NO K ~ N ~ ~ OO K ~ O == ON ^ = ~ = M M M ~ N ~ O K ~ = 526. p~=~==~=~=== K== = 527. ^=~=~== ~ ======== ~ = ~ I==KK=====[ ][ ]=~==~=~~K== = 528. ^=~=~= ~ ==-== ~ = ~ K== =[ ]110 120. CHAPTER 5. MATRICES AND DETERMINANTS529. a~~=~====~=~==~==~===== ====~=~~K== = 530. r=~====~=~~==~======= =~=~~=~=~=K=q==~============ ==fK== = 531. ^==~=====~=~=K= = = =5.4 Operations with Matrices = 532. q=~=^=~=_=~=~=I=~==I==~== ====~==~== ==~===~= ~K= = 533. q=~==^=~=_==~==~=E=~F=I=~= =I==~==~=~= K=f== ~NN ~NO K ~N ~ ~ OO K ~ O I== ^ = ~ = ON M M M ~ N ~ O K ~ NN NO K N OO K O I== _ = = ON M M M N O K = = = = =[ ][ ]111 121. CHAPTER 5. MATRICES AND DETERMINANTS==~NO + NO K ~N + N ~NN + NN ~ + ~ OO + OO K ~ O + O K= ON ON ^+_= M M M ~ N + N ~ O + O K ~ + = 534. f===~=~~I=~= ^ = ~ ==~=~I==[ ] ~NN ~NO K ~N ~ ~ OO K ~ O K= ON ^ = ~ = M M M ~ N ~ O K ~ = 535. j~==q=j~= q= ~= ~= = = = = = = = = = = = = = ~= = = = = ===K== = f= ~NN ~NO K ~N ~ ~ OO K ~ O I== ^ = ~ = ON M M M ~ N ~ O K ~ NN NO K N OO K O I= _ = = ON M M M N O K = = = = =[ ][ ][ ]112 122. CHAPTER 5. MATRICES AND DETERMINANTS== NN NO K N OO K O I== ON ^_ = ` = M M M N O K == = ~ NN + ~ O O + K + ~ = ~ = E = NI OI KI X = NI OI KI FK== = q==[ ]~ NN ^ = ~ = ~ ON~ NO ~ OO =N N ~ NP I= _ = [ ] = O I== ~ OP P == ~ NN ~ NO ^_ = ~ ON ~ OO ~ NP N ~ NNN = ~ OP O ~ ONN P~ NO O ~ OO O~ NP P K== ~ OP P = 536. q~==~=j~= f===~===~=~=~=~I== ==~==~==~===~=~K=== f= ^= = = ~= ~I= = ~= = = ^ q = = ^ K== = 537. q=~=^==~== ^^ q = f K== = 538. f==~==^_==I=== (^_ )q = _ q ^ q K= = =113 123. CHAPTER 5. MATRICES AND DETERMINANTS539. ^==j~= f=^==~=~= ~I==~I=== ~ ^ I= ==~===~==~= ` ==^W=[ ]~ ^ = ` K== = 540. q~==~=j~= f= ^= = ~= ~= ~I= = ~I= = = ^ I= = ============~=~~W= ^ = ~NN + ~ OO + K + ~ K= = 541. f==~=j~= f=^==~=~= ~==~=~=~= ^ I==== ^ N ==== ~ ^ ^ N = K= ^ = 542. f==~==^_==I=== (^_)N = _ N^ N K= = 543. f==^===~=~=== ==~I======u===~= =~= ^u = u I== ==~= =~==~~=~= ^ f = M K=== = = = q5.5 Systems of Linear Equations = = s~~W=I=I=I= N I= O I K = o~=W= ~ N I ~ O I ~ P I N I ~ NN I ~ NO I K =114 124. CHAPTER 5. MATRICES AND DETERMINANTSa~W=aI= a I= a I= a == j~W=^I=_I=u= = =~ + N = N I== 544. N ~ O + O = O a a =E`~=FI== = I= = a a == ~ N a= N = ~NO ~ O N I== ~ O O N a = N = NO O N I== O O ~ N a = N = ~N O ~ ON K== ~ O O = 545. f= a M I====~=~==W== a a K= = I= = a a f= a = M = ~= a M E= a M FI= = = = ~= = == K= f= a = a = a = M I= = = = ~= = = = ~== K= = ~N + N + N = N= 546. ~ O + O + O = O I== ~ + + = P P P P =a a a I= = I= = =E`~=FI== a a a=115 125. CHAPTER 5. MATRICES AND DETERMINANTS== ~N N a = ~ O O ~ P PNNNN O I= a = OO O I=PPPP~NNN~NNNa = ~ O ~PO P O I= a = ~ O P ~PO P O K== P= 547. f= a M I====~=~==W== a a a I= = K= = I= = a a a f= a = M =~= a M E= a M == a M FI==== ~==K= f= a = a = a = a = M I= = = = ~= = ~=K= = 548. j~=c==~=p===i~=b~================== =r= q===~=~== ~NN N + ~ NO O + K + ~ N = N ~ + ~ + K + ~ = ON N OO O O O = KKKKKKKKKKKK ~ N N + ~ O O + K + ~ = ~====~== ~ NN ~ NO K ~ N N N ~ ON ~ OO K ~ O O O I== = M M M M M ~ N ~ O K ~ KK== ^ u = _ I==116 126. CHAPTER 5. MATRICES AND DETERMINANTS== ~ NN ~ ^ = ON M ~ N~ NO K ~ N N N ~ OO K ~ O O I= u = I= _ = O K== M M M M ~ O K ~ = 549. p==~=p==i~=b~= = u = ^ N _ I== = ^ N =====^K= = =117 127. Chapter 6Vectors = = = = r r r r sW= I= I= I= I= ^_ I== r r s=W= I= I== r r r r=W= I= I= = r k=W= M = r `~=== W= uN I vN I wN = r `~=== W= u O I vO I wO = p~~W= I = a=W= I= I= = ^===W= = = =6.1 Vector Coordinates = 550. r=s= r = (NI MI M) I= r = (MI NI M) I= r = (MI MI N) I= r r r = = = N K= =r r r r 551. = ^_ = ( N M ) + ( N M ) + ( N M ) = =118 128. CHAPTER 6. VECTORS======= == Figure 73.= r = ^_ =552.( N M )O + (N M )O + (N M )O == r r 553. f= ^_ = I== _^ = K= === Figure 74.r 554. u = I= r v = I= r w = K==119 129. CHAPTER 6. VECTORS======= Figure 75.= r r 555. f= (uI v I w ) = N (uN I vN I wN ) I=== u = uN I= v = vN I= w = wN K== == =6.2 Vector Addition =r r r 556. = + = ===== Figure 76.120 130. CHAPTER 6. VECTORS==== Figure 77.=r r r r r 557. = N + O + P + K + = ===== Figure 78.= 558. `~=i~= r r r r + =+= = 559. ^~=i~= r r r r r r ( + ) + = + ( + ) = = r r 560. + = (uN + u O I vN + vO I wN + wO ) = = = = = = =121 131. CHAPTER 6. VECTORS6.3 Vector Subtraction =r r r r r r 561. = == + = K= === Figure 79.===== Figure 80.=r r r r 562. = + ( ) = = r r r 563. = M = (MI MI M ) = = r 564. M = M == r r 565. = (uN u O I vN vO I wN w O ) I== = = =6.4 Scaling Vectors = r r 566. = =122 132. CHAPTER 6. VECTORS== Figure 81.= 567.r r = ==r 568. = (uI v I w ) = = r r 569. = = = r r r 570. ( + ) = + = = r r r 571. ( ) = ( ) = ( ) = = r r r r 572. ( + ) = + = = = =6.5 Scalar Product = r r 573. p~~=m==s= =~ = r r r r = I== r r = ===~=== =~ K==== =123 133. CHAPTER 6. VECTORS=== Figure 82.= 574. p~~=m==`~=c= r r f= = (uN I vN I wN ) I= = (u O I vO I w O ) I=== r r = uNu O + vNvO + wNwO K= = 575. ^=_=q=s== r r f= = (uN I vN I wN ) I= = (u O I vO I w O ) I=== uNu O + vNvO + wNw O K= = O O O O O O uN + vN + wN u O + vO + w O = 576. `~=m= r r r r = = = 577. ^~=m= r r r r ( ) ( ) = = = 578. a=m= r r r r r r r ( + ) = + = = r r r r 579. = M == I =~=~=E = FK= O = r r 580. > M == M < < K= O =124 134. CHAPTER 6. VECTORS r r 581. < M == < < K= O = r r r r 582. = = r r r r r r 583. = == I =~=~~=E = M FK= = r 584. f= = (uN I vN I wN ) I=== r r r rO O O O = O = = uN + vN + wN K= = r r r r r r 585. = = = N = = r r r r r r 586. = = = M = = = =6.6 Vector Product = r r 587. s=m==s= =~ = r r r = I=== r r r = I== M X= O r r r r = ~= X= r r r =s= I= I= ==~=-~=K= =125 135. CHAPTER 6. VECTORS========= Figure 83.=r r r r 588. = = u N uOr vN vOr wN = wO= uN wN uN vN r r r v wN = 589. = = N I I v w u O w O u O vO O O = r r r r 590. p = = =EcKUPF= = 591. ^=_=q=s=EcKUPF= r r = r r = = 592. k~=m= r r r r = ( ) == = 593. ^~=m= r r r r ( ) ( ) = = = =126 136. CHAPTER 6. VECTORS594. a=m= r r r r r r r ( + ) = + = = r r r r r 595. = M == =~= =~=~~=E = M FK= = r r r r r r r 596. = = = M = = r r r r r r r r r 597. = I= = I= = = = = =6.7 Triple Product 598.599. 600. 601.= p~~=q=m= rr r r r r r r r r r r [ ] = ( ) = ( ) = ( ) = = rr r r rr rr r rr r r rr rrr [ ] = [ ] = [] = [ ] = [] = [ ] = = r r r rr r ( ) = [ ] = = p~~=q=m==`~=c= uN vN wN r r r ( ) = u O vO w O I== uP vP wP== r r r = (uN I vN I wN ) I= = (u O I vO I w O ) I= = (uP I vP I wP ) K== = 602. s==m~~= r r r s = ( ) = =127 137. CHAPTER 6. VECTORS============== Figure 84.= 603. s==m~= Nr r r s = ( ) = S === Figure 85.= r r r r r r 604. f== ( ) = M I===== I= I=~= =~=~= r r r =I== = + ===~~= =~= K== = r r r r r r 605. f== ( ) M I===== I= I=~= =~=~= K= =128 138. CHAPTER 6. VECTORS606. s=q=m= r r r r r r r r r ( ) = ( ) ( ) == = = = = = = = =129 139. Chapter 7Analytic Geometry = = = =7.1 One-Dimensional Coordinate System = m=~W= M I= N I= O I= M I= N I= O = o~=W= == a~===W== = = 607. a~=_=q=m= = ^_ = O N = N O = === Figure 86.= 608. a=~=i=p===o~= = + O ^` I= = M = N I= N K= N+ `_ ========== Figure 87.130= 140. CHAPTER 7. ANALYTIC GEOMETRY609. j==~=i=p= + O M = N I= = N K= O = = =7.2 Two-Dimensional Coordinate System = m=~W= M I= N I= O I= M I= N I= O = m~=~W= I = o~=W= == m=~=W=~I=I=I== a~===W== ^~W=p= = = 610. a~=_=q=m= = ^_ = =( O N )O + ( O N )O === Figure 88.131 141. CHAPTER 7. ANALYTIC GEOMETRY611. a=~=i=p===o~= = + O + O M = N I= M = N I== N+ N+ ^` = I= N K= `_ ======== == Figure 89.= =132 142. CHAPTER 7. ANALYTIC GEOMETRY======= == Figure 90.= 612. j==~=i=p= + O + O I= M = N I= = N K= M = N O O = 613. `=Ef==j~F==~=q~= + O + P + O + P I= M = N M = N I== P P == ^( N I N ) I== _( O I O ) I==~== `( P I P ) ==~=== =~= ^_` K= = =133 143. CHAPTER 7. ANALYTIC GEOMETRY========= == Figure 91.= 614. f=Ef==^=_F==~=q~= ~ + O + P ~ + O + P I= M = N M = N I== ~ ++ ~ ++ = ~ = _` I= = `^ I= = ^_ K== ========= == Figure 92.134 144. CHAPTER 7. ANALYTIC GEOMETRY615. `=Ef===p=m~====================== _F==~=q~= O O O O N + N N N N N + N N O + O O N O O + O N O O O O O O O O P + P P N P P + P N M = I= M = = N N N N N NO O PO N P NO O PO N P N========== == Figure 93.= = = = = = =135 145. CHAPTER 7. ANALYTIC GEOMETRY616. l=Ef==^F==~=q~= O O N O P + N N N + OP N N O PN + O N O + P N O N O O O O P N O + P N P + N O P N I= M = = M = N N N N N N O O N O O N P P N P P N ======== = Figure 94.= 617. ^~==~=q~= N N N N N O N p = ( ) O O N = ( ) O O P N P P N = = =136 O N P N= 146. CHAPTER 7. ANALYTIC GEOMETRY618. ^~==~=n~~~= N p = ( ) [( N O )( N + O ) + ( O P )( O + P ) + = O + ( P Q )( P + Q ) + ( Q N )( Q + N )] = ==== == Figure 95.= kW=f=~=SNTI=SNU=====EHF==EF== ~===~==~==~~K== = 619. a~=_=q=m==m~=`~= = ^_ = NO + OO ONO ( O N ) = =137 147. CHAPTER 7. ANALYTIC GEOMETRY== Figure 96.= 620. `=o~~=`~==m~=`~= = I= = K= === Figure 97.= 621. `=m~=`~==o~~=`~= = O + O I= ~ = K= 138 148. CHAPTER 7. ANALYTIC GEOMETRY7.3 Straight Line in Plane = m=~W=uI=vI=I= M I= N I== M I= N I= ~N I= ~ O I=== o~=W=I=~I=I=I=I=^I=_I=`I= ^N I= ^ O I== ^W= I= = ^===W= = r k~=W= = r r r m=W= I= ~ I= = = = 622. d~=b~==~=p~=i= ^ + _ + ` = M = = 623. k~=s==~=p~=i= r q== (^I _ ) ==~==== ^ + _ + ` = M K= === Figure 98.= 624. b=b~==~=p~=i=Ep-f=cF= = + K==139 149. CHAPTER 7. ANALYTIC GEOMETRYq=~===== = ~ K= === Figure 99.= 625. d~==~=i== N = ~ = O = O N === Figure 100.140 150. CHAPTER 7. ANALYTIC GEOMETRY626. b~==~=i=d=~=m=~==d~= = M + ( M ) I== ====~I= m( M I M ) ==~====K= === Figure 101.= 627. b~==~=i=q~=m~=q=q=m= N N = == O N O N = N N N N = M K= O O N =141 151. CHAPTER 7. ANALYTIC GEOMETRY== Figure 102.= 628. f=c= + =N= ~ === Figure 103.= =142 152. CHAPTER 7. ANALYTIC GEOMETRY629. k~=c= + = M = === Figure 104.= 630. m=a=c= N N = I== u v = (uI v ) = = = = = = = ~= mN ( N I N ) = = ==K= =143 153. CHAPTER 7. ANALYTIC GEOMETRY== Figure 105.= 631. s~=i= =~= = 632. e~=i= == = 633. s=b~==~=p~=i= r r r = ~ + I== == l======~I= u==~=~~====I== r ~ ======~===^====I= r ==~====I=~~===I== ==~=~~I== r = lu ======~==u===K== =144 154. CHAPTER 7. ANALYTIC GEOMETRY== Figure 106.= 634. p~=i==m~~=c= = ~N + N I== = ~ O + O == ( I ) ~==~==~=====I== (~N I ~ O ) =~==~==~=====I== (N I O ) =~==~==~==~~===I== ==~=~~K= =145 155. CHAPTER 7. ANALYTIC GEOMETRY=Figure 107.= 635. a~=c=~=m=q=~=i= q=~==== m(~ I ) ==== ^ + _ + ` = M === ^~ + _ + ` K= = ^ O + _O === Figure 108.146 156. CHAPTER 7. ANALYTIC GEOMETRY636. m~~=i= q== = N + N =~= = O + O =~=~~=== N = O K= q= = ^N + _N + `N = M = ~= ^ O + _O + ` O = M = ~= ~~== ^N _N = K= ^ O _O === Figure 109.= 637. m~=i= q== = N + N =~= = O + O =~=~=== N O = =I=~I= N O = N K= N q= = ^N + _N + `N = M = ~= ^ O + _ O + ` O = M = ~= ~== ^N^ O + _N_ O = M K= =147 157. CHAPTER 7. ANALYTIC GEOMETRY== Figure 110.= 638. ^=_=q=i= N ~ = O I== N + N O ^N^ O + _N_ O = K= O O ^N + _N ^ O + _ O O O =148 158. CHAPTER 7. ANALYTIC GEOMETRY== Figure 111.= 639. f==q=i= f=== ^N + _N + `N = M =~= ^ O + _ O + ` O = M =I====~=~= `N_ O + ` O_N ^N` O + ^ O`N M = I= M = K= ^N_ O ^ O_N ^N_ O ^ O_N = = =7.4 Circle = o~W=o= `==W= (~ I ) = m=~W=I=I= N I= N I== o~=W=^I=_I=`I=aI=bI=cI==149 159. CHAPTER 7. ANALYTIC GEOMETRY640. b~==~=`=`=~==l=Ep~~= cF= O + O = oO =========Figure 112.= 641. b~==~=`=`=~=^=m= (~I )( ~ )O + ( )O = o OFigure 113.150 160. CHAPTER 7. ANALYTIC GEOMETRY642. q=m=c O + O N O O N + N N N N =M O + O O O N O O O O P + P P P N === Figure 114.= 643. m~~=c = o I= M O K = o = 644. d~=c ^ O + ^ O + a + b + c = M =E^=I= aO + b O > Q ^c FK== q=====~=~= (~ I ) I=== a b ~= I= = K= O^ O^ q=~====151 161. CHAPTER 7. ANALYTIC GEOMETRYo=aO + b O Q ^c K O^= = =7.5 Ellipse = p~=~W=~= p=~W== cW= cN ( I M) I= cO (I M) = a~===W=O= = bW=== o~=W=^I=_I=`I=aI=bI=cI== mW=i= ^~W=p= = = 645. b~==~=b=Ep~~=cF O O + =N ~ O O== Figure 115.152 162. CHAPTER 7. ANALYTIC GEOMETRY646. N + O = O~ I= == N I== O ==~==~====~==== m( I ) === =====K= === Figure 116.= 647. ~ O = O + O = 648. b = M K = 653. ` i = Q~b( ) I== =====b========~=== ==K== = 654. ^~=c~===` i = NKR(~ + ) ~ I==(i = O(~ O + O ) K= = 655. p = ~ = = = =)7.6 Hyperbola = q~=~W=~= `~=~W== cW= cN ( I M) I= cO (I M) = a~===W=O= = bW=== ^W=I== o~=W=^I=_I=`I=aI=bI=cI=I== = = =154 164. CHAPTER 7. ANALYTIC GEOMETRY656. b~==~=e~=Ep~~=cF= O O = N= ~ O O === Figure 117.=657.N O = O~ I= == N I== O ==~==~====~=== m( I ) === =~====K= =155 165. CHAPTER 7. ANALYTIC GEOMETRY== Figure 118.658.659. 660.661.= b~==^= = = ~ = O = ~ O + O = = b = > N= ~ = b~==a ~ ~O = = = = = =156 166. CHAPTER 7. ANALYTIC GEOMETRY662. m~~=b~===o=_~==~=e~= = ~ I= M O K = = 663. d~=c ^ O + _ + ` O + a + b + c = M I== = _ O Q ^` > M K= = 664. d~=c==^=m~~===`~=^ ^ O + ` O + a + b + c = M I== = ^` < M K= 665. ^=c= O = I== Q == O = I== = K= Q f= = ~= I= = ~= ~= ~= = M = ~= = M K== =157 167. CHAPTER 7. ANALYTIC GEOMETRY== Figure 119.= = =7.7 Parabola = c~=~~W== cW=c= sW= j( M I M ) = o~=W=^I=_I=`I=aI=bI=cI=I=~I=I== = = 666. b~==~=m~~~=Ep~~=cF O = O =158 168. CHAPTER 7. ANALYTIC GEOMETRY== Figure 120.= b~=== = I= O `~==== c I M I= O `~==== j(MI M) K= = 667. d~=c ^ O + _ + ` O + a + b + c = M I== = _ O Q ^` = M K= = N 668. = ~ O I= = K= O~ b~===159 169. CHAPTER 7. ANALYTIC GEOMETRY = I= O `~==== c MI I= O `~==== j(MI M) K= === Figure 121.= 669. d~=cI=^=m~~===-~== ^ O + a + b + c = M =E^I=b=FI== N = ~ O + + I= = K== O~ b~=== = M I= O `~====160 170. CHAPTER 7. ANALYTIC GEOMETRY c M I M + I= O `~==== Q~ O K= M = I= M = ~ O + M + = M O~ Q~ === Figure 122.= = =7.8 Three-Dimensional Coordinate System = m=~W= M I= M I= M I= N I= N I= N I== o~=W= == a~===W== ^~W=p= sW=s= =161 171. CHAPTER 7. ANALYTIC GEOMETRY670. a~=_=q=m= = ^_ = ==( O N )O + ( O N )O + ( O N )O ===== Figure 123.= 671. a=~=i=p===o~= = + O + O + O M = N I= M = N I= M = N I== N+ N+ N+ = ^` = I= N K= `_ =162 172. CHAPTER 7. ANALYTIC GEOMETRY======== == Figure 124.== Figure 125.163 173. CHAPTER 7. ANALYTIC GEOMETRY672. j==~=i=p= + O + O + M = N I= M = N I= M = N O I= = N K= O O O = 673. ^~==~=q~= q=~~==~=~=== mN ( N I N I N ) I= mO ( O I O I O ) I=~= mP ( P I P I P ) ===== p=N OOOO N + OO N + O O N K=PPPPPN N NN PN NON N NNONN= 674. s==~=q~= q===~=~=== mN ( N I N I N ) I= mO ( O I O I O ) I= mP ( P I P I P ) I=~= mQ ( Q I Q I Q ) ===== N N N N N O O O N s= I== S P P P N Q Q Q N = N Q N Q N Q N s = O Q O Q O Q K= S P Q P Q P Q kW=t====EHF==EF==~===~== ~==K== =164 174. CHAPTER 7. ANALYTIC GEOMETRY==== == Figure 126.= = =7.9 Plane = m=~W=I=I=I= M I= M I= M I= N I= N I= N I== o~=W=^I=_I=`I=aI= ^N I= ^ O I=~I=I=I= ~N I= ~ O I= I=I=I=== r r r k~=W= I= N I= O = a=W= I= I= = a~====~W== = = 675. d~=b~==~=m~= ^ + _ + ` + a = M = = =165 175. CHAPTER 7. ANALYTIC GEOMETRY676. k~=s==~=m~= r q== (^I _I ` ) ==~===~= ^ + _ + ` + a = M K= ====== Figure 127.= 677. m~~=`~===b~==~=m~= ^ + _ + ` + a = M = = f= ^ = M I==~==~~===-~K= f= _ = M I==~==~~===-~K= f= ` = M I==~==~~===-~K= f= a = M I==~====K== = f= ^ = _ = M I==~==~~===-~K= f= _ = ` = M I==~==~~===-~K= f= ^ = ` = M I==~==~~===-~K= =166 176. CHAPTER 7. ANALYTIC GEOMETRY678. m=a=c= ^( M ) + _( M ) + `( M ) = M I== === m( M I M I M ) ====~I=~=== (^I _I ` ) ==~===~K=== ======= Figure 128.= 679. f=c= + + = N= ~ =167 177. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 129.= 680. q=m=c= P P P N P N P N P = M I== O P O P O P == N N N N N = M K= O O O N P P P N =168 178. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 130.= 681. k~=c= + + = M I== ========~==~======= =~=I=~= I= I= =~==== =~==~===~K== =169 179. CHAPTER 7. ANALYTIC GEOMETRY======== Figure 131.= 682. m~~=c= = N + ~N + ~ O = N + N + O I== = + + N N O = ( I I ) =~==~==~==== ==I=== m( N I N I N ) ====~I=== (~N I N I N ) =~= (~ O I O I O ) =~=~~===~K= =170 180. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 132.= 683. a~=^=_=q=m~= f==~=~==== ^N + _N + `N + aN = M I== ^ O + _ O + ` O + aO = M I== ==~=~===== r r N O ^N^ O + _N_ O + `N` O = r r = K= O O O N O ^N + _N + `N ^ O + _ O + ` O O O O =171 181. CHAPTER 7. ANALYTIC GEOMETRY======== Figure 133.= 684. m~~=m~= q=~= ^N + _N + `N + aN = M =~= ^ O + _ O + ` O + aO = M =~=~~=== ^N _N `N = = K= ^ O _O `O = 685. m~=m~= q=~= ^N + _N + `N + aN = M =~= ^ O + _ O + ` O + aO = M =~=~=== ^N^ O + _N_ O + `N` O = M K= = 686. b~==~=m~=q= m( N I N I N ) =~=m~~=q= =s= (~N I N I N ) =~= (~ O I O I O ) =EcKNPOF=172 182. CHAPTER 7. ANALYTIC GEOMETRY N ~N ~O N N N N = M = O O= 687. b~==~=m~=q= mN ( N I N I N ) =~= mO ( O I O I O ) I= ~=m~~=q==s= (~ I I ) = N N N O N ~ O N O N = M = == Figure 134.= 688. a~=c=~=m=q=~=m~= q=~==== mN ( N I N I N ) ===~= ^ + _ + ` + a = M ===173 183. CHAPTER 7. ANALYTIC GEOMETRY=^ N + _ N + `N + a ^O + _O + `OK= ========= Figure 135.= 689. f==q=m~= f==~= ^N + _N + `N + aN = M =~= ^ O + _ O + ` O + aO = M = I= = = ~= ==== = N + ~ = N + I== = + N == N N N = = I== ~ ==174 184. CHAPTER 7. ANALYTIC GEOMETRY~=_N _O`N ` I= = N `O `OaN aOaN aO~aN aON =N =N =^N ^ I= = N ^O ^O_N I== _O`N a _N N `O aO _ O I== O O ~ + + O ^N a `N ~ N ^O aO ` O I== O O ~ + + O a ^N _N N _O aO ^ O K== O O ~ + + O= = =7.10 Straight Line in Space = m=~W=I=I=I= N I= N I= N I== a=W= I= I= = o~=W=^I=_I=`I=aI=~I=I=I= ~N I= ~ O I=I=== r r r a===~=W= I= N I= O = r k~===~=~W= = ^===W= = = 690. m=a=c===b~==~=i== N N N = = I== ~ === mN ( N I N I N ) ====I=~= (~ I I ) == =====K== =175 185. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 136.= 691. q=m=c= N N N = = == O N O N O N ======Figure 137.176= 186. CHAPTER 7. ANALYTIC GEOMETRY692. m~~=c== = N + = N + I== = + N === mN ( N I N I N ) ====~=I= I= I= =~======= ===I==~~===~=~=K== ======== Figure 138.= 693. ^=_=q=p~=i= r r ~N~ O + NO + N O = r N rO = = O O O N O ~N + N + N ~ O + O + O O O O =177 187. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 139.= 694. m~~=i= q==~=~~=== r r N O I== == ~ N N N = = K= ~ O O O = 695. m~=i= q==~=~~=== r r N O = M I== == ~N~ O + NO + N O = M K= = 696. f==q=i= N N N = = =~= q== ~N N N178 188. CHAPTER 7. ANALYTIC GEOMETRY O O O = = ==== ~O O O O N O N O N ~N N N = M K= ~O O O = 697. m~~=i=~=m~== N N N = = =~==~= q=~== ~ ^ + _ + ` + a = M =~=~~== r r = M I== == ^~ + _ + ` = M K= ======= = Figure 140.= =179 189. CHAPTER 7. ANALYTIC GEOMETRY698. m~=i=~=m~== N N N = = =~==~= q=~== ~ ^ + _ + ` + a = M =~=~== r r I== == ^ _ ` = = K= ~ ===== == Figure 141.= =7.11 Quadric Surfaces = m=~===~=~W=I=I== o~=W=^I=_I=`I=~I=I=I= N I O I P I==180 190. CHAPTER 7. ANALYTIC GEOMETRY699. d~=n~~=b~= ^ O + _ O + ` O + Oc + Od + Oe + Om + On + Oo + a = M = 700. `~~==n~=p~= = `~= o~EF= o~EbF= == q==p~= = N= P= Q= p~= o~=b= M= P= P= Q= e==N=p= > M = a= Q= P= Q= e==O=p= < M = a= 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== = M= = = = = = = = = =a= p~= p~= a= p~= p~= a= a= p~= = = = =o~=n~=`= f~~=n~=`= b=m~~= e=m~~= o~=b=`= f~~=b=`= e=`= o~=f=m~= f~~=f=m~= m~~=`= o~=m~~=m~= f~~=m~~=m~= `=m~== e== ^ e n m ^ e d e _ c n = e _ c I= b = I= = (b ) I== d c ` o d c ` m n o a N I O I P =~=====~I== ^ e d e _ c = M K= d c ` = =181 191. CHAPTER 7. ANALYTIC GEOMETRY701. o~=b=E`~=NF= O O O + + = N= ~ O O O ======== Figure 142.= 702. f~~=b=E`~=OF= O O O + + = N = ~ O O O = 703. e==N=p=E`~=PF= O O O + = N= ~ O O O =182 192. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 143.= 704. e==O=p=E`~=QF= O O O + = N = ~ O O O ==== Figure 144.183 193. CHAPTER 7. ANALYTIC GEOMETRY705. o~=n~=`=E`~=RF= O O O + =M= ~ O O O ===== Figure 145.= 706. f~~=n~=`=E`~=SF= O O O + + =M= ~ O O O = 707. b=m~~=E`~=TF= O O + = M= ~ O O =184 194. CHAPTER 7. ANALYTIC GEOMETRY======Figure 146.=708. e=m~~=E`~=UF= O O = M= ~ O O====Figure 147.185= 195. CHAPTER 7. ANALYTIC GEOMETRY709. o~=b=`=E`~=VF= O O + = N= ~ O O ======== Figure 148.= 710. f~~=b=`=E`~=NMF= O O + = N = ~ O O = 711. e=`=E`~=NNF= O O = N= ~ O O =186 196. CHAPTER 7. ANALYTIC GEOMETRY======== Figure 149.= 712. o~=f=m~=E`~=NOF= O O =M= ~ O O = 713. f~~=f=m~=E`~=NPF= O O + = M= ~ O O = 714. m~~=`=E`~=NQF= O =M= ~O =187 197. CHAPTER 7. ANALYTIC GEOMETRY======= Figure 150.= 715. o~=m~~=m~=E`~=NRF= O = N= ~O = 716. f~~=m~~=m~=E`~=NSF= O = N = ~O = 717. `=m~=E`~=NTF= O = M = = = = = =188 198. CHAPTER 7. ANALYTIC GEOMETRY7.12 Sphere = o~==~=W=o= m=~W=I=I=I= N I= N I= N I== `==~=W= (~ I I ) = o~=W=^I=aI=bI=cI=j= = = 718. b~==~=p=`=~==l=Ep~~= cF= O + O + O = oO = ========= Figure 151.= 719. b~==~=`=`=~=^=m= (~ I I )( ~ )O + ( )O + ( )O = o O 720. a~=c ( N )( O ) + ( N )( O ) + ( N )( O ) = M I==189 199. CHAPTER 7. ANALYTIC GEOMETRY== mN ( N I N I N ) I= mO ( O I O I O ) =~====~=~K== = 721. c=m=c= O + O + O N O O O N + N + N N N N N O + O + O O O O N = M = O O O O O O P + P + P P P P N O + O + O Q Q Q N Q Q Q = 722. d~=c ^ O + ^ O + ^ O + a + b + c + j = M =E^==FK== q=====~=~= (~ I I ) I=== a b c ~= I= = I= = K= O^ O^ O^ q=~==== aO + b O + c O Q ^ O j K o= O^ = =190 200. Chapter 8Differential Calculus = = = = cW=I=I=I=I== ^=E=~~FW== o~=W=~I=I=I== k~~=W== ^W= = f=W= N = = =8.1 Functions and Their Graphs 723.724.725.726.= b=c= ( ) = ( ) = = l=c= ( ) = ( ) = = m=c= ( + q ) = ( ) = = f=c= = ( ) ==~=I= = ( ) == = N ( ) ==== K== =191 201. CHAPTER 8. DIFFERENTIAL CALCULUS======= Figure 152.= 727. `=c= = ( ) I= = ( ) I= = (( )) ==~==K= = 728. i~=c= = ~ + I== o I== ~ = ~ =======I===== =-K= =192 202. CHAPTER 8. DIFFERENTIAL CALCULUS======== Figure 153.= 729. n~~=c== = O I= o K= ========= Figure 154.=193 203. CHAPTER 8. DIFFERENTIAL CALCULUS730. = ~ O + + I= o K= ====== Figure 155.= 731. `=c== = P I= o K= =194 204. CHAPTER 8. DIFFERENTIAL CALCULUS=== Figure 156.= 732. = ~ P + O + + I= o K= =195 205. CHAPTER 8. DIFFERENTIAL CALCULUS==== Figure 157.= 733. m=c== = I= k K= =196 206. CHAPTER 8. DIFFERENTIAL CALCULUS======Figure 158.==Figure 159.=197= 207. CHAPTER 8. DIFFERENTIAL CALCULUS734. p~=o=c== = I= [MI ) K= ========Figure 160.== 735. b~=c= = ~ I= ~ > M I= ~ N I= = == ~ = I= = OKTNUOUNUOUQSK = ==== Figure 161.198 208. CHAPTER 8. DIFFERENTIAL CALCULUS736. i~=c= = ~ I= (MI ) I= ~ > M I= ~ N I= = == ~ = I= > M K= == Figure 162.= 737. e=p=c== = I= = I= o K= O =199= 209. CHAPTER 8. DIFFERENTIAL CALCULUS====== Figure 163.= 738. e=`=c== + = I= = I= o K= O======== Figure 164.200 210. CHAPTER 8. DIFFERENTIAL CALCULUS739. e=q~=c== = ~ I= = ~ = = I= o K= + ========= Figure 165.= 740. e=`~=c== + = I= = = = I= o I= M K= =201 211. CHAPTER 8. DIFFERENTIAL CALCULUS======== Figure 166.= 741. e=p~=c== N O = I= = = = I= o K= + Figure 167.202= 212. CHAPTER 8. DIFFERENTIAL CALCULUS742. e=`~=c== N O = I= o I= M K= = I= = = ========= Figure 168.= 743. f=e=p=c== = ~ I= o K= =203 213. CHAPTER 8. DIFFERENTIAL CALCULUS======= Figure 169.= 744. f=e=`=c== = ~ I= [NI ) K= ======== Figure 170.= 745. f=e=q~=c== = ~~ I= ( NI N) K= =204 214. CHAPTER 8. DIFFERENTIAL CALCULUS======= Figure 171.= 746. f=e=`~=c== = ~ I= ( I N) (NI ) K== =205 215. CHAPTER 8. DIFFERENTIAL CALCULUS======= Figure 172.= 747. f=e=p~=c== = ~ I= (MI N] K= =206 216. CHAPTER 8. DIFFERENTIAL CALCULUSFigure 173.== 748. f=e=`~=c== = ~ I= o I= M K==Figure 174.=207= 217. CHAPTER 8. DIFFERENTIAL CALCULUS8.2 Limits of Functions = cW= ( ) I= ( ) = ^W== o~=~W=~I== = = 749. [ ( ) + ( )] = ( ) + ( ) = ~ ~ ~= 750. [ ( ) ( )] = ( ) ( ) = ~ ~ ~= 751. [ ( ) ( )] = ( ) ( ) = ~= ~ ~ ( ) ( ) = ~ I== ( ) M K= ~ ~ ( ) ( )752. ~=753. [ ( )] = ( ) = ~ ~(=)754. ( ( )) = ( ) = ~ ~=755. ( ) = (~ ) I==== ( ) ===~= = ~ K= ~= = N= M 756. = 757. M~ =N= = N = N= M 758. 208 218. CHAPTER 8. DIFFERENTIAL CALCULUS~ N = N= M 759. =(N + ) = N= M 760. = N 761. N + = = = 762. N + = = = 763. ~ = N = M= = =8.3 Definition and Properties of the Derivative = cW=I=I=I=I== f=~~W== o~=~W== ^W= = = = ( + ) ( ) 764. ( ) = == = = M M =209 219. CHAPTER 8. DIFFERENTIAL CALCULUS==== Figure 175.= = ~ == 765.= 766.= 767.= 768.( + ) = + = ( ) = = ( ) = = = 769. m=o= ( ) = + == = =210 220. CHAPTER 8. DIFFERENTIAL CALCULUS770. n=o= = = O = 771. `~=o= = (( )) I= = ( ) I== = K= = 772. a~==f=c= N = I== = ( ) ===== ( ) K== = 773. o~=o= N = = O = 774. i~=a~= = ( ) I= = ( ) I== = ( ) [ ( )] K= = =8.4 Table of Derivatives = f=~~W== o~=~W=`I=~I=I== k~~=W==211 221. CHAPTER 8. DIFFERENTIAL CALCULUS (` ) = M = 775.= ( ) = N = 776.= (~ + ) = ~ = 777.= 778.= 779.= 780. (~ O + + ) = ~ + = ( ) = N = ( ) = +N = = N N = O = 781.= ( ) = O N = 782.( )== 783.N N== ( ) = N = 784.= ( ~ ) = N I= ~ > M I= ~ N K= ~ 785.=212 222. CHAPTER 8. DIFFERENTIAL CALCULUS (~ ) = ~ ~ I= ~ > M I= ~ N K= 786.= 787. ( ) = = = ( ) = = 788.= ( ) = = 789.= (~ ) = NO = O = 790.= ( ) = NO = O = 791.= ( ) = ~ = 792.= ( ) = = 793.= (~ ) = N O = N 794.= (~ ) = N O = N 795.= (~~ ) = N O = N+ 796.=213 223. CHAPTER 8. DIFFERENTIAL CALCULUS (~ ) = N O = N+ 797.= (~ ) = NO = N798.= (~ ) = NO = N799.= ( ) = = 800.= ( ) = = 801.= (~ ) = N O = O = 802.= ( ) = N O = O = 803.= ( ) = ~ = 804.= ( ) = = 805.= (~= ) = N = O +N806.= 807.N (~= ) = O = N214 224. CHAPTER 8. DIFFERENTIAL CALCULUS (~~= ) = N O I= < N K= N 808.= (~= ) = ON I= > N K= N809.= 810. ( ) = N + = = = =8.5 Higher Order Derivatives811.812.813. 814. 815.= cW=I=I=I== f=~~W== k~~=W== = = p=~= O = = = = = ( ) O = e-l=~= ( ) = = ( ) = ( ( N) ) = = ( + )( ) = ( ) + ( ) = = ( )( ) = ( ) ( ) = = i=c~= ( ) = + O + =215 225. CHAPTER 8. DIFFERENTIAL CALCULUS( ) = + P + P + = ( )( ) = ( ) + (N) + ( N) (O) + K + ( ) = N O=( )( ) = (>)> 816.= = =( )( ) = > = 817.( N)N ( N)> = ( ~ ) = ( )818. ~= 819. N ( )( ) = ( N) ( N)> = =(~ )( ) = ~ ~ =( )( ) = =820.=821.=(~ )( ) = ~ 822. ~ ==( )( ) = + = 823.O =( )( ) = + = 824.O = = =216 226. CHAPTER 8. DIFFERENTIAL CALCULUS8.6 Applications of Derivative = cW=I=I== m==~=W=== sW== ^~W== f=~~W== qW== k~~=W== = = 825. s=~=^~= = ( ) =====~==~==~== ~==~=~==I== = = ( ) ===~~====I= = = = ( ) ===~~=~~== =K== = 826. q~=i= M = ( M )( M ) = =217 227. CHAPTER 8. DIFFERENTIAL CALCULUS== Figure 176.= 827. k~=i= N ( M ) =Ec=NTSF= M = ( M ) = 828. f~=~=a~=cK== f= ( M ) > M I==EF==~=~= M K=Ec=NTTI= < N I= O < FI= f= ( M ) < M I==EF==~=~= M K=Ec=NTTI= N < < O FI= f= ( M ) ======I====~K== =218 228. CHAPTER 8. DIFFERENTIAL CALCULUS== Figure 177.= 829. i~=~= ^==EF=~=~=~=~=~= N ==~=== ===~=~= N ==~= ( N ) ( ) ==~====~=EcKNTTFK== = ^==EF=~=~=~==~= O ==~=== ===~=~= O ==~= ( O ) ( ) ==~====~=EcKNTTFK= = 830. `~=m= ^=~===EF==~= M ==~==== ( M ) =====~==K= = 831. c=a~=q==i~=b~K= f=EF===~==E ( ) > M F===~========~= (~I N ] ==~==EF====~==E ( ) < M F===~===== ~= = [ N I ) I= = = EF= ~= ~= = ~= ~= = ~= = N = EcKNTTFK==219 229. CHAPTER 8. DIFFERENTIAL CALCULUS832. f=EF==~=E ( ) < M F==~====~= (~I O ]=~=EF==~=E ( ) > M F==~==== ~= [ O I ) I==EF=~=~=~==~= O K== EcKNTTFK= = 833. p=a~=q==i~=b~K= f= ( N ) = M =~= ( N ) < M I==EF=~=~=~=~= ~== N K= f= ( O ) = M =~= ( O ) > M I==EF=~=~=~== ~= O K=EcKNTTF= = 834. `~K== EF= = = ~= ~= ~= = M = = = = ~= = = = = = ( ) = ============ ~=~= M =EcKNTTI= P < FK=== EF===~==~=~== M ===~==== ( ) ================= ~=~= M K=EcKNTTI= < P FK=== = 835. p=a~=q==`~K== f= ( M ) > M I==EF==~=~=~= M K== f= ( M ) < M I==EF==~=~=~= M K= f= ( ) ======I====~K= = 836. f=m= f== ( P ) ====~== ( ) ==~==~= = P I=== = = ( P I ( P )) = = ~= = = = = ~= = ( ) K=f= ( P ) ==~===I== ( P ) = M = EcKNTTFK= = 837. ie~=o= M ( ) ( ) = == ( ) = ( ) = K== ( ) ( ) =220 230. CHAPTER 8. DIFFERENTIAL CALCULUS8.7 Differential = cW=I=I== f=~~W== a~==~=W= ( ) I= ( ) = o~=~W=`= a~=== = ( ) W== a~==W== p~=~==W= = p~=~==W= = = = 838. = = = 839. ( + ) = ( ) + ( ) = === Figure 178.221 231. CHAPTER 8. DIFFERENTIAL CALCULUS840. p~=`~=== = ( + ) ( ) = = 841. ( + ) = + = = 842. ( ) = = = 843. (` ) = ` = = 844. ( ) = + = = 845. = = O = = =8.8 Multivariable Functions = c===~~W= ( I ) I= ( I ) I= ( I ) I= ( I ) == ^W=I=I== p~=~==I=I=I=W= I= I= K= = = 846. c=l=m~~=a~= q=~~=~===== = =E~= = FI= q=~~=~===== = =E~= = FK= = =222 232. CHAPTER 8. DIFFERENTIAL CALCULUS847. p=l=m~~=a~= O = I== = O O = = I== O O = I== = O = = K== f==~=~=I=== O O = K== = 848. `~=o== f= ( I ) = (( I )) =E==~====~~=FI=== = (( I )) I= = (( I )) K== = + K== f= ( ) = ( ( )I ( )) I== ( ) = = f= = ( (I )I (I )) I=== + K== + = I= = = 849. p~=`~= + = = =223 233. CHAPTER 8. DIFFERENTIAL CALCULUS850. i~=j~~=~=j~= ( I ) =~=~=~=~=~= ( M I M ) == ( I ) ( M I M ) = =~= ( I ) ==== ( M I M ) K== = ( I ) =~=~=~==~= ( M I M ) == ( I ) ( M I M ) = =~= ( I ) ==== ( M I M ) K= = 851. p~~=m= = = M K= i~=~~=~=~=~==~=~~=K= == 852. p~=m= ^=~~==========~==~==~= =~=~== = 853. p=a~=q==p~~=m= = = M FK== i= ( M I M ) ==~=~~==E ( M I M ) ( M I M ) K== a= ( M I M ) ( M I M )= f= a > M I= ( M I M ) > M I== ( M I M ) ===~===~=~K= f= a > M I= ( M I M ) < M I== ( M I M ) ===~===~=~~K= f= a < M I= ( M I M ) ==~=~=K= f= a = M I===~K= = 854. q~=m~= q=~===~=~===~= = ( I ) = ~= ( M I M I M ) === M = ( M I M )( M ) + ( M I M )( M ) K= =224 234. CHAPTER 8. DIFFERENTIAL CALCULUS855. k~==p~= q=~===~===~= = ( I ) =~= ( M I M I M ) === M M M = = K= ( M I M ) ( M I M ) N= = =8.9 Differential Operators = r r r r==~==~=~W= I= I= = p~~==E~~=FW= ( I I ) I= ( N I O I KI ) = d~==~=~~=W= ~ I= = a~=~W= = r s==E=FW= c (mI nI o ) = r r a==~==W= c I= c = r r `==~==W= c I= c = i~~~=~W= O = = = 856. d~==~=p~~=c= ~ = = I I I== K= ~ = = I IKI O N = 857. a~=a~= = + + I== 225 235. CHAPTER 8. DIFFERENTIAL CALCULUS======== r ( I I ) I= O + O + O = N K== = 858. a==~=s=c= r r m n o + + = c = c = = 859. `==~=s=c= r r r r r = c = c = m n o o n r m o r n m r = + + = = 860. i~~~=l~= O O O O = O + O + O = = r r 861. c = c M = = 862. (~ ) = ( ) M = = 863. (~ ) = ( ) = O = = r r r r r 864. c = ~ c Oc = c Oc = = =()()()()226( ) 236. Chapter 9Integral Calculus = = = = cW=I=I=I== f=~~W=I=I= = f=~==~=W= ( ) I= ( ) I==a~==~=W= ( ) I= ( ) I= c( ) I== o~=~W=`I=~I=I=I=I== k~~=W=I=I=I== = =9.1 Indefinite Integral = 865.= 866.= 867.= 868.= 869. ( ) = c( ) + ` == c( ) = ( ) K=( ( ) ) = ( ) = ( ) = ( ) = [ ( ) + ( )] = ( ) + ( ) = [ ( ) ( )] = ( ) ( ) == 870.N (~ ) = ~ c(~ ) + ` = 227 237. CHAPTER 9. INTEGRAL CALCULUSN (~ + ) = ~ c(~ + ) + ` =871.= N ( ) ( ) = O ( ) + ` = O872.= ( ) ( ) = ( ) + ` =873.= 874. j==p= ( ) = (( ))( ) == = ( ) K= = 875. f~==m~= = I=== ( ) I= ( ) =~=~=K== = = =9.2 Integrals of Rational Functions = ~ = ~ + ` =876.= =877.O +`= O= O =P +`= P =878. +N + ` I= N K= +N= 879.=228 238. CHAPTER 9. INTEGRAL CALCULUS(~ + )+N + ` I= N K= (~ + ) = ~( + N) 880.=881. = + ` = = N ~ + = ~ ~ + + ` =882.=~ + ~ + = +883. ~ + + ` = O= +N ( + ~ )( + ) = ~ + ~ + ` I= ~ K=884.= N ~ + = (~ + ~ ~ + ) + ` =885.O= O N N O O ~ + = P O (~ + ) O~(~ + ) + ~ ~ + + ` = 886.= N (~ + ) = ~ 887.~ + +`= = 888.N (~ + ) = ~ + ~ OO~ + +`= = 889. (~ + )O=N ~ ~ + + +`= O ~ + =229 239. CHAPTER 9. INTEGRAL CALCULUS O N ~O +`= = P ~ + O~ ~ + (~ + )O ~ + 890.= (~ + )891.O=N N ~ + + O +`= ~ (~ + ) ~ =892. N N +`= = O N O +N= N 893.ON N+ = +`= O N = N ~+ +`= = O O~ ~ ~O894.O= 895.~ N +`= = O O~ + ~ ~= N+ 896.O= ~ N + ` ==~O897.O= 898.N = ~ N + ` = O ~ ~ +N = ( O + ~ O ) + ` = O +~ O= ~ + 899.O= N ~~ ~ + ` I= ~ > M K= ~ =230 240. CHAPTER 9. INTEGRAL CALCULUS ~ + 900.O=~ N O + + ` = O= N O = (~ + O ) O~ ~ + O + ` =901.=~902.O N ~ + = +`= O O O~ ~ = N O~ + O Q~ 903. O = + ` I= ~ + + O Q~ O~ + + O Q~O Q~ > M K= = O O~ + ~~ + ` I= = + + Q~ O Q~ O O Q~ < M K= ~904.O= = =9.3 Integrals of Irrational Functions = O ~ + + ` = = ~ + ~~ + =905. O(~ O) = ~ + + ` = P~ O ~ + = 906.= 907.P O (~ + ) O + ` = P~=231 241. CHAPTER 9. INTEGRAL CALCULUS908.~ + =P O(P~ O) (~ + ) O + ` = O NR~= ~ + ~ N = + ` I== ~ + + ~ ~ + ~ ~ > M K= ( + )909.= ~ + N = ~~ + ` I== ~ ~ + ~ ~ < M K= ( + )910.=N ~ + (~ + )( + ) = = + ~ ~ ( + ) + (~ + ) + ` I= ~ > M K= ~911.~ + N = + = 912.~ ( + ) ~ ~~ + ` I=E ~ < M I= > M FK== (~ + ) ~= O ~ + =913.=914.(~ + )( + ) =O(U~ O NO~ + NRO O ) NMRP(~ + )P + ` = O O(U~ O Q~ + PO O ) = ~ + + ` = NRP ~ + =915. N ~ + ~ = + ` I= ~ > M K= ~ + ~ ~ + + ~=232 242. CHAPTER 9. INTEGRAL CALCULUS916. O ~ + = ~~ + ` I= ~ < M K= ~ ~ + ~=~ = + ~+ = (~ + )( ) (~ + )~ +`= ~+917.N+ = N O + ~ + ` = N (~ )( + ) + (~ + )~+ +`= ~+= 918.= 919.= ( ~ )( ~ ) = O ~920.=921.O ~ + O + = Q O Q~ O + ~ +`= U P O + Q~~ + O == 922.~ O + + ~ > M K==N O~ + + O ~ (~ O + + ) + ` I== ~=923. ~ +` = ~ ~ + + O=N O~ + O ~ Q~ + ` I= ~ < M K= Q~ ~=924. O + ~ O = O O ~O + ~ + + O + ~ O + ` = O O=233 243. CHAPTER 9. INTEGRAL CALCULUS925. O + ~ O =N O O PO ( + ~ ) + ` = P=926. O + ~ O =O (O O + ~ O ) O + ~ O = U~Q + O + ~ O + ` = U=927.O + ~O O + ~O = + + O + ~ O + ` = O =928. +~ OO= + O + ~ O + ` == 929. O + ~O = O + ~ O + ~ +`= ~ + O + ~O = 930. +~ OO= O + ~O + ` ==931. O O + ~O= O O ~O + ~ + O + ~ O + ` = O O=932.N = +`= O + ~O ~ ~ + O + ~O=933. O ~ O = O O ~O ~ + O ~ O + ` = O O= 934. O ~ O =N O O PO ( ~ ) + ` = P234 244. CHAPTER 9. INTEGRAL CALCULUSO ~O ~ = O ~ O + ~ ~ + ` = 935.O ~O O ~O = + + O ~ O + ` = O = 936.=937. ~ O= + O ~ O + ` =O=938. ~ O= O ~O + ` =O=939. O O O ~O = ~ + + O ~ O + ` = O O O O ~= N ~ = ~ + ` = ~ ~940.OO= ( + ~ )941. O ~O=N ~ +`= ~ +~= ( ~ )942. ~ OO=N +~ +`= ~ ~=943. O O ~O= O ~O +`= ~O= (944.O ~O ) O P= ~O O ~O+`==235 245. CHAPTER 9. INTEGRAL CALCULUS (945.O ~ O ) O = P (O O R~ O ) O ~ O + = UP~ Q + + O ~ O + ` = U =946.~ O O = ~O O ~ O + ~ + ` = ~ O O= O O ~ = 947.N O (~ O )P O + ` = P= O O O ~ =948.Q (O O ~ O ) ~ O O + ~U ~ + ` = U ~= 949.~O O = ~ O O + ~ +`= ~ + ~O O ~O O ~O O = ~ + ` = O ~ = 950.=951.= ~ + ` =N O=952. ~O O= +`= ~=953. ~ OO= ~O O + ` == 954. O ~O O= O ~O ~ O + ~ + ` = O O ~236 246. CHAPTER 9. INTEGRAL CALCULUS ( + ~ )955.~O O=N ~ +`= O ~+=N ~+ +`= O ~= ( ~ )956.~O O= ( + )957.~O ON=O ~ O~ + ~ O + ` I= > ~ K= ~ ( + )= ( + )958.~ OON=~ OO+ ~ OO~ O O + ~ O + < ~ K==959. O~O O=~O O +`= ~O= (~960.O O ) O = PQ O (R~ O O ) ~ O O + P~ ~ + ` = U U ~= (~961. OO)P= O ~O ~O O+`== = =9.4 Integrals of Trigonometric Functions = = + ` =962.= 963. = + ` = 237+ `I = 247. CHAPTER 9. INTEGRAL CALCULUS 964.O = N O + ` = O Q = N + O + ` = O Q= 965.O= 966.PN N P = P + ` = P + ` = P NO Q= 967.PN N P = P + ` = P + + ` = P NO Q= = = ~ O + ` =968.= = = ~ O + Q + ` = 969.= 970.O= O = + ` == O = ~ + ` == P = = P == 971.O= 972.P N + ~ + ` = O O O O= 973.P N + ~ + + ` = O O O O Q= 974.N = Q O + ` = 238 248. CHAPTER 9. INTEGRAL CALCULUS 975.ON = P + ` = P= 976.ON = P + ` = P= 977.O O = N Q + ` = U PO= ~ = + ` =978.= N = + ` = + ` = O 979.= O = ~ O + Q + ` = 980.= ~981.O = ~ + ` == = + ` =982.= N = + ` = + ` = O 983.= O = ~ + + ` = 984. O = 985. O = + ` == 986. = ~ + ` = 239 249. CHAPTER 9. INTEGRAL CALCULUS 987.O N = + ~ + + ` = O Q= 988.O=N + ~ + ` = O= 989.O = ~ + ` = O =( + ) = O( + )990.+( ) + ` I= O( )( ) + ` I= O( ) O O K= =( + ) = O( + )991. O O K= =( + ) = O( + )992. O O K= = ~ = + ` =993.= = + ` =994.= = 995. +N +`= +N= +N 996. = +`= +N = 240+( ) + ` I= O( ) 250. CHAPTER 9. INTEGRAL CALCULUS ~ = ~ +N O + ` = ~ = ~ 997.N O + ` == 998.= ~~ = ~~ O ( N999.O+ N) + ` ==N 1000. ~ = ~ + ( O + N) + ` = O = = =9.5 Integrals of Hyperbolic Functions = 1001. = + ` == 1002. = + ` == 1003. ~ = + ` == 1004. = + ` == 1005. O = ~ + ` == 1006. O = + ` == 1007. ~ = + ` ==241 251. CHAPTER 9. INTEGRAL CALCULUS1008. = + ` == = =9.6 Integrals of Exponential and Logarithmic Functions = 1009. = + ` == ~ 1010. ~ = +`= ~ = ~ 1011. ~ = +`= ~ = ~ 1012. ~ = O (~ N) + ` = ~ = 1013. = + ` = = = + ` =1014.= N 1015. = +N +`= O + N ( + N) = ~ ~ 1016. ~ = +`= ~ O + O =242 252. CHAPTER 9. INTEGRAL CALCULUS1017. ~ =~ + ~ +`= ~ O + O= = =9.7 Reduction Formulas = 1018. =N N = = = + I= N K= ( N) N N N1019.= 1020. == 1021.== 1022. == 1023.=N N O N = O N O I= N K= ( N) N N N O N + = O + N O I= N K= ( N) N= +N N 1024. = = + N O + = + = N +N 1025. = = + 243 253. CHAPTER 9. INTEGRAL CALCULUS N O = += 1026. ~ = N ~ N + ~ O I= N K= N= 1027. = N N + O I= N K= N= 1028. = O ~ O O + I= N K= N N=N N O 1029. = N + = = O 1030. = + N O I= N K= ( N) N = N N O 1031. = N + = = O 1032. = + N O I= N K= ( N) N = +N N 1033. = = + N + O = + = N +N 1034. = = +244 254. CHAPTER 9. INTEGRAL CALCULUS+ N O = += 1035. ~ =N ~ N ~ O I= N K= N= 1036. = N N O I= N K= N= 1037. = O ~ O O + I= N K= N N= 1038. = O O O + I= N K= N N= 1039. = +N N = +N +N = N = + I= N K= ( N) N N 1040.= 1041. = N == 1042. = N == 1043. = N == 1044. = + N == 1045. = N =245 255. CHAPTER 9. INTEGRAL CALCULUS1046. N =N +N +N = N + N N O +N= 1047. N = +N N +N N + = +N + N N O= 1048. ~ N = +N N +N ~ N = +N + N N+ O= 1049. = = ~ + ~ ~ ~ + = O~ 1050. = = O O (~ + + ) ( N)( Q~ )(~ O + + )N O(O P)~ O (~ O + + )N I= N K= ( N)( Q~ ) = (1051.+~ N K= O)O =O( N)~ O ( O + ~)O N+O P O( N)~ O= (1052.O== N O( N)~ O ( O ~ O ) O P I= N K= O O( N)~ ( O ~ O )N ~)O = = = = =246 (O+ ~O ) NI 256. CHAPTER 9. INTEGRAL CALCULUS9.8 Definite Integral = ~~a=~==~=W= ( ) I= ( ) I== o~=W= ( ) == =Np~=~W= == ^~W= c( ) I= d( ) == i==~W=~I=I=I== = = 1053. ( ) = ~ ( ) ~ M =NI==== = N I== N K== === Figure 179.=247 257. CHAPTER 9. INTEGRAL CALCULUS1054. N = ~ = ~= ~~1055. ( ) = ( ) == ~~~~~~ [ ( ) + ( )] = ( ) + ( ) =1056.= [ ( ) ( )] = ( ) ( ) =1057.= ~1058. ( ) = M = ~= ~~1059. ( ) = ( ) == ~~1060. ( ) = ( ) + ( ) == ~ < < K== 1061. ( ) M == ( ) M == [~ I ] K= ~= 1062. ( ) M == ( ) M == [~ I ] K= ~= = = =248 258. CHAPTER 9. INTEGRAL CALCULUS1063. c~~=q==`~= ( ) = c( ) ~= c() c(~ ) == c( ) = ( ) K=~= 1064. j==p== f= = ( ) I=== ~ ( ) = (( ))( ) I== = = N (~ ) I= = N () K= = 1065. f~==m~= = ( ) ~ = ~~= 1066. q~~=o= N ~ ( ) = ( M ) + ( ) + O ( ) = O =N ~ =249 259. CHAPTER 9. INTEGRAL CALCULUS== Figure 180.= 1067. p=o== ~ ( ) = P [ ( M ) + Q ( N ) + O ( O ) + Q ( P ) + = ~ + O ( Q ) + K + Q ( N ) + ( )] I== == ~ = ~ + I= = MI NI OI KI K== =250 260. CHAPTER 9. INTEGRAL CALCULUS== Figure 181.= 1068. ^~=r=~=`= p = ( ) = c() c(~ ) I== ~= c( ) = ( ) K= =251 261. CHAPTER 9. INTEGRAL CALCULUS== Figure 182.= 1069. ^~=_=q=`= p = [ ( ) ( )] = c() d() c(~ ) + d(~ ) I== ~= c( ) = ( ) I= d( ) = ( ) K= =252 262. CHAPTER 9. INTEGRAL CALCULUS== Figure 183.= = =9.9 Improper Integral = 1070. q==~== ( ) ==~=~==~= ~== ~====I= ( ) ==~=========== =======~= [~ I ] K= = 1071. f= ( ) ==~==== [~ I ) I=== ( ) = ( ) K= ~ ~=253 263. CHAPTER 9. INTEGRAL CALCULUS== Figure 184.= 1072. f= ( ) ==~==== ( I ] I=== ( ) = ( ) K= === Figure 185.254 264. CHAPTER 9. INTEGRAL CALCULUSk=W=q==~==NMTNI=NMTO=~== ====~=~=X===~=~= K= = ( ) = ( ) + ( ) =1073.=== Figure 186.= f===~==I====~==== = ~= I= = = ~= ( ) = = ~=====X====K= = 1074. `~=q= i== ( ) =~== ( ) =========== ~= [~ I ) K= p= ~= M ( ) ( ) = = ~= = = [~I ) K=255 265. CHAPTER 9. INTEGRAL CALCULUS~~f= ( ) ==I== ( ) ==~======I= ~~f= ( ) ==I== ( ) ==~=K== 1075. ^=`= = ~~f= ( ) ==I===~ ( ) ==~=K=== = 1076. a=f~= i= ( ) ==~=======~== [~I ) ====~= = K=q== ( ) = ~ M + ( ) = ~=== Figure 187.256 266. CHAPTER 9. INTEGRAL CALCULUS1077. i= ( ) ==~====~=~====== =~== [~ I ] ============ (~ I ) K=q= ( ) = ( ) + ( ) K= ~ M + M +~ +=== Figure 188.= = =9.10 Double Integral = c===~~W= ( I ) I= (I ) I== a=~W= ( I ) I= ( I ) I== oo()o~=W= I = =N =Np~=~W= I= = o==~W=oI=p== m~=~W= I= =257 267. CHAPTER 9. INTEGRAL CALCULUS^~W=^= p~=~~W=p= s==~=W=s= j~==~=~~W== aW= ( I ) = c=W= j I= j = j==~W= f I= f I= fM = `~==~=~W=n= `~=W= ( I ) = `~====~W= I= = ^~==~=W= = = 1078. a==a=f~= q==~==~=~= [~I ] [I ] === === ( I )^ = [~ I ][ I ]() ( I ) ~ M ~ M =N =NI=== I ======~= ( N I ) N I I=~= = N I= = N K= ====()Figure 189.258= 268. CHAPTER 9. INTEGRAL CALCULUSq==~==~=~==o=== ( I )^ = ( I )^ I== o[~ I ][ I ]=~= [~I ] [I ] =~=oI== ( I ) = ( I ) == ( I ) ===o=~= ( I ) = M =K= === Figure 190.= 1079. [ ( I ) + ( I )]^ = ( I )^ + ( I )^ = o= 1080.o [ ( I ) ( I )]^ = ( I )^ ( I )^ = o= 1081.ooo ( I )^ = ( I )^ I== oo===~=~K= = 1082. f= ( I ) ( I ) ==oI== ( I )^ ( I )^ K= o= 1083. f= ( I ) M ==o=~= p o I==259o 269. CHAPTER 9. INTEGRAL CALCULUS= ( I )^ ( I )^ K= po=== Figure 191.= 1084. f= ( I ) M ==o=~=o=~=p=~=-~= I== ( I )^ = ( I )^ + ( I )^ K==o pope= o p =======o=~=pK= === Figure 192.= =260 270. CHAPTER 9. INTEGRAL CALCULUS1085. f~=f~=~=c=q= ( ) ( I )^ = ) ( I ) == ( o~ =~====fI== o = {( I ) ~ I ( ) ( )} K= === Figure 193.= ( ) ( I )^ = ( I ) == o ( )=~====ffI= o = {( I ) ( ) ( )I } K= =261 271. CHAPTER 9. INTEGRAL CALCULUS== Figure 194.= 1086. a=f~==o~~=o= = f=o===~~== [~I ] [I ] I=== ( I ) = ( I ) = ( I ) K== o ~ ~ = f==~=~===~= ( I ) =~===~= ( )( ) ==~== ( ) ( ) K== ( I ) = ( )( ) = o o ~ = 1087. `~==s~~= ( I ) ( I ) = [ (I )I (I )] (I ) I== op ( I ) = = M = = ~~= = = ~ (I ) ~= ( I ) (I ) I=~=p===~==o==262 272. CHAPTER 9. INTEGRAL CALCULUS~==== = (I ) I= = (I ) =====oK== = 1088. m~=`~= = I= = K== === Figure 195.= 1089. a=f~==m~=`~= = q=a~===m~=`~=== ( I ) = = K== (I ) = i===o===~=W= M () () I= I== O K== q== ( ) ( I ) = ( I ) K= o ( )=263 273. CHAPTER 9. INTEGRAL CALCULUS== Figure 196.= f===o===~=~==== M ~ I= I== O I=== == ( I ) = ( I ) K== o ~=== Figure 197.= =264 274. CHAPTER 9. INTEGRAL CALCULUS1090. ^~==~=o= ( )^ = =E=~==f=FK= ~ ( )=== Figure 198.= ( )^ = =E=~==ff=FK= ( )=== Figure 199.= =265 275. CHAPTER 9. INTEGRAL CALCULUS1091. s==~=p= s = ( I )^ K== o=== Figure 200.= f=o==~==f==== = ~ I= = I= = ( ) I= = ( ) I=== ( )s = ( I )^ = ( I ) K== o~ ( )= f=o==~==ff==== = I= = I= = ( ) I= = ( ) I== ( )s = ( I )^ = ( I ) K== o ( )=266 276. CHAPTER 9. INTEGRAL CALCULUSf== ( I ) ( I ) ====~====oI========= = = = N = ( I ) = ~= O = ( I ) = = o= = == s = [ ( I ) ( I )]^ K== o= 1092. ^~=~=s==m~=`~= f=p==~====-~=== = I= = I= = () I= = () I== == ( )^ = ^ = I== p ( )s = (I ) K== p=== Figure 201.= 1093. p~=^~= p = oO N + + = O=267 277. CHAPTER 9. INTEGRAL CALCULUS1094. j~==~=i~~= = ( I )^ I== o====~~====~====o=~======~= ~==EIF== ( I ) K=== = 1095. j= q====~~=~===-~=====~= j = ( I )^ K== o= q====~~=~==-~== j = ( I )^ K= o= q===~=~==-~== f = O( I )^ K== o= q===~=~==-~== f = O( I )^ K== o= q=~===~== fM = ( O + O )( I )^ K== o= 1096. `==j~= jN = = ( I )^ = o ( I )^ o ( I )^ o268I== 278. CHAPTER 9. INTEGRAL CALCULUSj N = = ( I )^ = o ( I )^ o ( I )^K==o= 1097. `~==~=m~= n = ( I )^ I== o=~=~====~==o=~== ~==~=~==EIF== ( I ) K=== = 1098. ^~==~=c= N = ( I )^ I== po = p = ^ K== o= = =9.11 Triple Integral = c===~~W= ( I I ) I= ( I I ) I== q=~W= ( I I )s I= ( I I )s I== dd()o~=W= I I = =N =N =Np~=~W= I= I= = i==~W=~I=I=I=I=I== o==~W=dI=qI=p== `~=~W= I= I== p~=~W= I= I= = s==~=W=s=269 279. CHAPTER 9. INTEGRAL CALCULUSj~==~=W=== aW= ( I I ) = `~====~W= I= I= = c=W= j I= j I= j = j==~W= f I= f I= f I= f I= f I= f I= fM = = = 1099. a==q=f~= q==~==~=~~= [~I ] [I ] [I ] = ===== ( I I )s =[~ I ][ I ][ I ]() ) ( I I ) ~ M ~ M =N =N =N ~ MI= I I ======~~= ( N I ) N I ( N I ) I=~= = N I= = N I= = N K= = 1100. [ ( I I ) + ( I I )]s = ( I I )s + ( I I )s(ddd= 1101. [ ( I I ) ( I I )]s = ( I I )s ( I I )s ddd= 1102. ( I I )s = ( I I )s I== dd===~=~K= = 1103. f== ( I I ) M =~=d=~=q=~=~=~= I=== ( I I )s = ( I I )s + ( I I )s K== dqdqe= d q =======d=~=qK= =270 280. CHAPTER 9. INTEGRAL CALCULUS1104. b~~==q=f~==o~=f~= f===d====== ( I I ) ==~= = ( I ) oI N ( I ) O ( I ) I=== O ( I ) = ( I I ) = ( I I ) I== d o N ( I ) =o====d===-~K= = f===d====== ( I I ) ==~= ~ I N ( ) O ( )I N ( I ) O ( I ) I=== d O ( ) O ( I ) ( I I ) = ( I I ) == ~ N ( ) N ( I ) = 1105. q=f~==m~~= f=d==~=~~= [~ I ] [I ] [I ] I== ( I I ) = ( I I ) K== d ~ = f===~=~====~== ( I I ) ==~== =~= ( ) ( ) ( ) ==~== ( I I ) = ( ) ( ) ( ) K== d ~ = 1106. `~==s~~= ( I I ) = = d= [ (I I )I (I I )I (I I )] p271 ( I I ) I = (I I ) 281. CHAPTER 9. INTEGRAL CALCULUS ( I I ) = M ======~~=== == (I I ) = ~~= ( I I ) (I I ) I= ~= p= = = ~= = d= = ~= = = = = (I I ) I= = (I I ) = = (I I ) =====dK= = = 1107. q=f~==`~=`~= q=~===~=~=== ( I I ) = = K== (I I ) = i===d===~=W= ( I ) oI N ( I ) O ( I ) I= =o====d===-~K=q== ( I I ) = ( I I ) = dp O ( I ) ( I I ) K= o ( I ) N ( I ) e=p===~==d==~=~K= = 1108. q=f~==p~=`~= q=a~===p~=`~=== ( I I ) = = O == (I I ) = ( I I ) = = =d272 282. CHAPTER 9. INTEGRAL CALCULUS========= = ( I I ) O I== p===p===~==d==~=~K=q=~== ==~====M=== O I====~== = ~==M== K== ==========Figure 202.= 1109. s==~=p= s = = d= 1110. s==`~=`~= s = = p ( I I )= 1111. s==p~=`~= s = O = p ( I I )=273 283. CHAPTER 9. INTEGRAL CALCULUS1112. j~==~=p= = ( I I )s I== d= = = = ~= = d= ~= = = ~============ ~== ( I I ) == ( I I ) K=== = 1113. `==j~==~=p= j j j = I= = I= = I== == j = ( I I ) s I== dj = ( I I ) s I== dj = ( I I ) s = d~========~====~=~= = M I= = M I= = M I=I== ( I I ) ====K== = 1114. j==f~=~====-~=E= = M FI=-~== E = M FI=~=-~=E = M F= f = O( I I ) s I== df = O( I I ) s I== df = O( I I ) s K== d= 1115. j==f~=~==-~I=-~I=~=-~= f = f + f = ( O + O )( I I ) s I== df = f + f = ( O + O )( I I ) s I== d274 284. CHAPTER 9. INTEGRAL CALCULUSf = f + f = ( O + O )( I I ) s K== d= 1116. m~=j==f~= fM = f + f + f = ( O + O + O )( I I ) s = d= =9.12 Line Integral = p~~=W= c( I I ) I= c( I ) I= ( ) = p~~=~W= ( I I ) = `W=`I= `N I= ` O = i==~W=~I=I= I= = m~~W=I== m~=~W= I= = r s=W= c (mI nI o ) = r m=W= ( ) = r r r r r=W= I= I= I= = ^~==W=p= i==~=W=i= j~==~=W== aW= ( I I ) I= ( I ) = `~====~W= I= I= = c=W= j I= j I= j = j==~W= f I= f I= f = s==~=W=s= tW=t= r j~=W= _ = `W=f= b=W= = j~=W= =275 285. CHAPTER 9. INTEGRAL CALCULUS1117. i=f~==~=p~~=c= r r i=~==`======= = ( ) I= M p I=~=~=~~==c======`K== q== p r c( ( )) = c( I I ) = c I== M``====~==~K== = 1118. c = c + c = `N ` O`N`O=== Figure 203.= r r 1119. f====`==~~== = ( ) I= I=== c( I I ) = c( ( )I ( )I ( )) ( ( )) + (( )) + (( )) K= OOO`= 1120. f=`==~=====-~====~= = ( ) I= ~ I=== c( I ) = c( I ( )) N + ( ( )) K== O`~= 1121. i=f~==p~~=c==m~=`~=276 286. CHAPTER 9. INTEGRAL CALCULUSO c( I ) = c( I ) + I== ` ===`=====~== EF K= = 1122. i=f~==s=c= r r i=~==`======= = ( ) I= M p K=q== r r = = ( I I ) == ======~====K== = O== Figure 204.= r i= ~= = = c (mI nI o ) = = = = = = `K= r q===~===== c =~=== `=== p m + n + o = (m + n + o ) K= `M= =277 287. CHAPTER 9. INTEGRAL CALCULUS1123. m==i=f~==s=c= r r r r c = c I== ()`()`=-`========~K= = r r r r r r r r c = c = c + c I==()() (`N ` O`) (`N)`O=`======= `N =~= ` O K== = r 1124. f===`==~~== ( ) = ( )I ( )I ( ) I= I=== m + n + o = = ` = m( ( )I ( )I ( )) + n( ( )I ( )I ( )) + o( ( )I ( )I ( )) = 1125. f=`====-~=~====~= = ( ) I= == m + n = m( I ( )) + n( I ( )) K== ` ~= 1126. d=q= n m = m + n I== o ` r r r = c = m( I ) + n( I ) = = ~= = = m n ======~~==~= I= ==~= = ~= oI= = = = = ~= I= = ==`K== = =278 288. CHAPTER 9. INTEGRAL CALCULUS1127. ^~==~=o=o=_===`=`= N p = = = O` o = 1128. m~=f==i=f~= r r r r q= = ~= = ~= = = c = m + n + o = = ~===~=I==~===mI=nI=~=o=~= = = ~= ~= aI= ~= = = = = ~~= == = ( I I ) ==E~==~~==~F====a===~= r = n I= c = ~ I== = o K== = m I= q== rr r c( ) = m + n + o = (_) (^ ) K== ``= 1129. q==~=`~=c= r ^====== c = ~ ==~=~=~= r r r r K=q==~==~=== c = m + n + o = =~===~==== r r r r r c = = M K== m n o = f===~==~==-~==~== m + n = (_) (^ ) I== `======~====~= ~======= m n = K== =279 289. CHAPTER 9. INTEGRAL CALCULUS1130. i==~=`= r O O O ( ) = + + I= i = = ` =`=~=~=======r == ( ) I= K= = f===`==-~I== r O O ( ) = + K== i = = ` = f===`===~==~== = ( ) ===~= (~ ) I=== i= ~O N + K== = 1131. i==~=`==m~=`~= O i = + O I== ===`=====~= = () I= ==~=~K=== = 1132. j~==~=t= = ( I I ) I== `= ( I I ) ===~======K= = f=`==~==~~==== r ( ) = ( )I ( )I ( ) I======~==~====== =~=280 290. CHAPTER 9. INTEGRAL CALCULUSOOO = ( ( )I ( )I ( )) + + K== = f=`==~===-~I===~====== == = ( I ) I= `= = ( ( )I ( )) OO + =E=~~=FK= = 1133. `==j~==~=t= j j j = I= = I= = I= == j = ( I I ) I== `j = ( I I ) I== `j = ( I I ) K= `= 1134. j==f~= q===~=~==-~I=-~I=~=-~= ~====~= f = ( O + O )( I I ) I== `f = ( O + O )( I I ) I== `f = ( O + O )( I I ) K== `=281 291. CHAPTER 9. INTEGRAL CALCULUS1135. ^~==~=o=_==~=`=`= N p = = = K== O` ` ` === Figure 205.= f====`====~~== r ( ) = ( )I ( ) I===~~=~==~~===~= N p = ( ) = ( ) = ( ) ( ) K== O = 1136. s==~=p=c==o~=~=`=`= ~==-~= s = O = O = O + O == O` ` ` =282 292. CHAPTER 9. INTEGRAL CALCULUS== Figure 206.= 1137. t= r t===~== c ==~===~=~== `======~= r r t = c I= `r r = c ======~===I= == ==~=K==== Figure 207.283 293. CHAPTER 9. INTEGRAL CALCULUSf=====~=~==`===-~I== r r t = c = m + n I== ``= f= ~= ~= `= = = = ~= ~~= = E= = ~= FI==~==~~=== t = m( ( )I ( )I ( )) + n( ( )I ( )I ( )) + o( ( )I ( )I ( )) I ==== == K== = r f= ~= = = c = = ~= ~= ( I I ) = = ~= ~~= ~= = = I= = = = = ~= = = =^==_=~=====~= t = (_ ) (^ ) K== = 1138. ^=i~= r r _ = M f K== `r q==~==~=~== _ =~=~==~= `= = ~= = = ~= = f= = = = ~~= ===~K== =Figure 208.284= 294. CHAPTER 9. INTEGRAL CALCULUS1139. c~~~=i~= r r = = b = ` = q= = = EF= = = ~= ~= = =`==~===~===~==~== = ~===K=== === Figure 209.= = =9.13 Surface Integral = p~~=W= ( I I ) I= ( I ) = r r m=W= (I ) I= ( I I ) = r r r r=W= I= I= = p~W=p= r s=W= c (mI nI o ) = r r a==~==W= c = c =285 295. CHAPTER 9. INTEGRAL CALCULUSr r `==~==W= c = c == r s===~=~W= p = r k~==~W= = p~=~~W=^= j~==~=~W== aW= ( I I ) = `~====~W= I= I= = c=W= j I= j I= j = j==~W= f I= f I= f I= f I= f I= f = s==~=W=s= r cW= c = d~~~=~W=d= rr c=W= ( ) = c=W= = r mW= ( ) = j~=I==W= = p~=~W=n= `~=W= ( I ) = r j~====W= b = = = 1140. p~=f~==~=p~~=c= i=~=~=p======= r r r r (I ) = (I ) + (I ) + (I ) I== = (I ) = ~= = = ~= a(I ) = = = ~K= q==~==~====~==~~=== ( I I ) == =~=p===~== r r ( I I )p = ( (I )I (I )I (I )) I== p a( I ) r r ==~~=~= =~= =~==== 286 296. CHAPTER 9. INTEGRAL CALCULUSr r r r = (I ) + (I ) + (I ) I== r r r r = (I ) + (I ) + (I ) = r r ~= ====K== = 1141. f====~==p========~= = ( I ) == ( I ) ====~==~======~= a( I ) I= == ( I I )p = ( ) ( I I ( I )) pa IO N + + K== O= r 1142. p~=f~===s=c= c ===p~=p= f=p===~I=== r r r r ===== c( I I ) p = c( I I ) p = p======pr r r = c( (I )I (I )I (I )) K== a( I )= f=p===~I=== r r r r ===== c( I I ) p = c( I I ) p = p====== =pr r r = c( (I )I (I )I (I )) K== a( I )r r p = p ====~========~K==a= ~= = = = ~~= = = = = = = = ~~= = K= r r q=~~=~= =~== =~==== 287 297. CHAPTER 9. INTEGRAL CALCULUSr r r r = (I ) + (I ) + (I ) I== r r r r = (I ) + (I ) + (I ) K== = 1143. f====~==p=======~= = ( I ) I== ( I ) ====~==~=======~= a( I ) I= == f= p= = = ~I= KK= = -= = = = ~===I==== r r r r ===== c( I I ) p = c( I I ) p = pp=======r r r r c( I I ) + I== a( I )= f=p===~I=KK==-==== ~===~I==== r r r r ===== c( I I ) p = c( I I ) p ppr r r r = c( I I ) + K== a( I )====== = 1144. (c )p = m + n + o = r rpp= (m + n + o )p I== p= m( I I ) I= n( I I ) I= o( I I ) =~==== r == = c K== I= I= ==~== ~= ==== r ~== =~==-~I=-~I=~=-~I=K= =288 298. CHAPTER 9. INTEGRAL CALCULUS1145. f==~=p====~~===== r ( (I )I (I )I (I )) I======~=~=~== =~==m n o I ) I p p a( = (I ) = ~= = = ~= a(I ) = = = ~K= = 1146. a=q= r r r c p = c s I== ( )r r c p = m + n + o =(p)d== r c( I I ) = m( I I )I n( I I )I o( I I ) === ==~==========mI==nI==~==o==~== =~~=~I== r m n o c = + + == r r ======== c I==~==== c K==q=== =~==~==~=~==~==~== ~K== = 1