1300 mathformulas

8/3/2019 1300 mathformulas

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1300 Math Formulas

fp_k VVQVNMTTQN

` OMMQ ^KpK ^ o oK


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ii

Preface

q ~ ~ - ~ K f ~ ~ ~ ~~ ~~ I

I ~ I ~ ~~K q ~ ~I ~I ~ k pI ^~I dI qI j~~ a~I sI ^~ dI `~Ia~ b~I pI ~ m~ qKq ~ I I ~ ~ ~ ~ ~ ~ ~I ~ ~ ~ ~ K


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iii

Contents

1 krj_bo pbqp NKN p f 1 NKO p k 5

NKP _~ f 7 NKQ ` k 82 ^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~ 223 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


5/334

iv

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 784 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


6/334

v

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~ 1146 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


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vi

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 1898 afccbobkqfî `î`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~ 2259 fkqbdoî î`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


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vii

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~ 30211 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 31612 mol_^_fifqv NOKN m~ ~ `~ 318

NOKO m~ c~ 319


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1

Chapter 1

Number Sets

1.1 Set Identities

pW ^I _I `r~ W f` W ^m W _^ b W r W _^

f W _^ a W _y^

1. f^

2. ^^

3. _^ = _^ ~ ^_ .

4. b p^

5. r p{ }_^_^` ==


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CHAPTER 1. NUMBER SETS

2

Figure 1.

6. `~

^__^ =

7. ^~( ) ( ) `_^`_^ =

8. f p

{ }_~^_^` ==

Figure 2.

9. `~^__^ =

10. ^~( ) ( ) `_^`_^ =


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3

11. a( ) ( ) ( )`^_^`_^ = I

( ) ( ) ( )`^_^`_^ = K

12. f^^^ = I^^^ =

13. a~

=^ Iff^ =

14. f^^ = I

^f^ =

15. `

{ }^f^ =

16. ` f ~ rf^^ = I= ^^

17. a j~ i~

( ) _^_^ =

I( ) _^_^ =

18. a p{ }^~_^y_` ==


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4

Figure 3.

19. ( )_^y_^y_ =

20. ^_^y_ =

21. =^y^

22. ^_y^ = = _^ .

Figure 4.

23. ( ) ( ) ( )`_y`^`_y^ =

24. ^yf^ =

25. `~~ m

( ){ }_~^I_^` ==


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5

1.2 Sets of Numbers

k~~ W kt W Mk

fW w

m W +w

k~ W wo~~ W n

o~ W o` W `

26. k~~ k` W { }KIPIOINk = K

27. t k

` ~ W { }KIPIOINIMkM = K

28. ft ~ ~ W

{ }KIPIOINkw ==+ I

{ }NIOIPIw = K I

{ } { }KK IPIOINIMINIOIPIwMww == + K

29. o~~ ko~ ~ ~W

== M~w~w~~

~n K

30. f~~ k

k~ ~ ~ ~K


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6

31. o~ kr ~~ ~ ~~ W oK

32. ` k{ }o~o` += I

~~ K

33. `onwk

Figure 5.


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7

1.3 Basic Identities

o~ W ~I I

34. ^ f~M~ =+

35. ^ f

( ) M~~ =+

36. `~ ^~~ +=+

37. ^~ ^( ) ( )~~ ++=++

38. a p~( )~~ +=

39. j~ f~N~ =

40. j~ f

N~

N~ = I M~

41. j~ q MMM~ =

42. `~ j~

~~ =


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8

43. ^~ j~( ) ( )~~ =

44. a i~( ) ~~~ +=+

45. a aN

~~

=

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

N

= R

= NQ

=+

NO = NS = N OQ =+

P = T = PQ =+

46.

NQ = NU = N Q =

47. ~ +=

48. ` m~


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9

Figure 6.

49. ( ) ( ) ( ) ( )~~ +++=+++

50. ( ) ( ) ( ) ( )~~ +=++

51. ( )( ) ( ) ( )~~~ ++=++

52.

~

~

~OOOO

+

+++

=++

53. `~ ` k

~~|||||||

=+

54. = ~ I =


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10

Figure 7.

55. m~ m~ ` k( )+=+ ~

56. j ~ ^ ~ ` kf ~ + ~ I OO ~ += EFI

~

~~= E~FK

57. m m~ o~

( ) ( )OOONNNON ++= ( ) ( )[ ]ONONON +++=

58. `~ k m~ o~

( ) ( ) ( )[ ]+=+ |||||||||||||||||||||

59. f ~ ` k m~ o~

( )( ) ( )[ ]+=

+

N

N


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11

60. n m~ o~( )

( ) ( ) ( )[ ]ONONON

OOO

NNN

O

N

+=+

+

=

61. m ~ ` k

( )[ ] ( ) ( )[ ]+=+=

62. c~ a j

( ) ( ) ( )+=+

63. k o ~ ` k

( )

++

+=+=

O

O I

NIIOINIM = K K

64. b c~

+=


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12

Chapter 2

Algebra

2.1 Factoring Formulas

o~ W ~I I k~~ W

65. ( )( )~~~ OO +=

66. ( )( )OOPP ~~~~ ++=

67. ( )( )OOPP ~~~~ ++=+

68. ( )( ) ( )( )( )OOOOOOQQ ~~~~~~ ++=+=

69.( )( )

QPOOPQRR

~~~~~~++++=

70. ( )( )QPOOPQRR ~~~~~~ +++=+

71. f I

( )( )NOOPON ~~~~~~ +++=+ K K

72. f I ( )( )NOOPON ~~~~~~ +++++= K I


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CHAPTER 2. ALGEBRA

13

( )( )NOOPON ~~~~~~ +++=+ K K

2.2 Product Formulas

o~ W ~I I t W I

73. ( ) OOO ~O~~ +=

74. ( ) OOO ~O~~ ++=+

75. ( ) POOPP ~P~P~~ +=

76. ( ) POOPP ~P~P~~ +++=+

77. ( ) QPOOPQQ ~Q~S~Q~~ ++=

78. ( ) QPOOPQQ ~Q~S~Q~~ ++++=+

79. _~ c~

( ) I`~`~`~`~`~ N

NOO

ON

N

M +++++=+ K

( )>>

>`

= ~ ~ K

80. ( ) O~O~O~~ OOOO +++++=++

81. ( ) ++++++=+++++OOOOOO

~~ KK( )~~~~O +++++++++++ KKK


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CHAPTER 2. ALGEBRA

14

2.3 Powers

_~ E ~ FW ~I m E~~ FW I

82. ~~~ +=

83.

~~

~=

84. ( ) ~~ =

85.

~

~=

86. ( ) ~~ =

87. N~M = I M~

88. N~N =

89.

~N~ =

90.

~~ =


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CHAPTER 2. ALGEBRA

15

2.4 Roots

_~W ~I m E~~ FW I

MI~ E O = I k F

91. ~~ =

92. ~~ =

93.

~

~= I M

94.

~

~

~

== I M K

95. ( ) ~~ =

96. ( ) ~~ =

97. ~~ =

98.

~~ =

99. ~~ =

100. ( ) ~~ =


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CHAPTER 2. ALGEBRA

16

101.~

~

~

N N

= I M~ K

102.O

~~

O

~~~

OO

+=

103.~

~

~

N

=

m

2.5 Logarithms

m ~ W I I ~I I k~~ W

104. a i~

~= ~ ~= I M~ > I N~ K

105. MN~ =

106. N~~ =

107.

=

N~

N~M~

108. ( ) ~~~ +=

109.

~~~ =


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CHAPTER 2. ALGEBRA

17

110. ( ) ~~ =

111.

N ~

~ =

112. ~

~

~ == I M > I N K

113. ~

N

~ =

114. ~~=

115. i~ _~ NMNM =

116. k~~ i~ = I

KTNUOUNUOUKO

NN

=

+=

117. QPQOVQKM

NM

N ==

118. PMORURKO

N ==


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CHAPTER 2. ALGEBRA

18

2.6 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~

M~O =++ I~O

~Q

O

OIN

= K

121. a~

~Qa

O

=122. s c~

f MO =++ I

=

=+

ON

ON K

123. M~O =+ I MN = I~O = K

124. M~O =+ I~

OIN = K

125. ` b~K `~~ c~K

MP

=++ I

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CHAPTER 2. ALGEBRA

19

N += I ( ) ( ) O

P

O

N PIO ++= I

P

OO

P

O

O

+

+= I P

OO

P

O

O

+

= K

2.7 Inequalities

s~~W I I

o~ W

PON ~II~I~I~

III~

KI I

a~W aI a I a I a

126. f~I f~ k~ ~ d~

f~ f~ k~ d~~ [ ]I~

~ < ( ]I~

~

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CHAPTER 2. ALGEBRA

20

127. f ~ > I ~ < K

128. f ~ > I M~ > M~ I ~ +>+ K

130. f ~ > I ~ > K

131. f ~ > ~ > I ~ +>+ K

132. f ~ > ~ > I ~ > K

133. f ~ > ~ M > I ~ > K

134. f ~ > ~ M > I

~> K

135. f ~ > ~ M ~ M X ~ ~ ~ ~ = K

141. O~

N~ + I M~ > X ~ ~ ~ ~ ~ N~ = K

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CHAPTER 2. ALGEBRA

21

142.

~~~~~~ ON ON

+++

KK I M~II~I~ ON >K K

143. f M~ >+ ~ M~ > I ~

> K

144. f M~ >+ ~ M~ ++

M~ > M~

N ON

Ma I M~ > K

149. f ~O K

150. f ~O

> I ~ > I M~ > K

151. f( )( )

M

> I

( ) ( )

( )

>

M

MK

152.( )( )

M

< I

( ) ( )

( )


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CHAPTER 2. ALGEBRA

23

154. p ` f c~f ~I

~ W( )N`^ += K

155. ` ` ff ~ E FI

`^ = K


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24

Chapter 3

Geometry

3.1 Right Triangle

i ~ ~W ~I eW ^W j~W ~ I I

^W Io~ W oo~ W ^~W p

Figure 8.

156. =+ VM


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CHAPTER 3. GEOMETRY

25

157. ==

~

158. ==

159. == ~

~

160. == ~~

161. ==

162. == ~

163. m~~ qOOO ~ =+

164. ~O = I O = I ~ ~ ~ ~ I -

I K

Figure 9.


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CHAPTER 3. GEOMETRY

26

165. O = I ~ ~K

166.Q

~

OOO

~ = IQ

~

OOO

= I

~ ~ ~ ~ ~ ~ K

Figure 10.

167.O

= I

~ K

168. Oo ==

169.~

~

O

~

++=

+=

170. ~ =


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CHAPTER 3. GEOMETRY

27

171.O

O

~p ==

3.2 Isosceles Triangle

_~W ~iW _~ ~W s ~W ^ ~W mW i^~W p

Figure 11.

172.O

VM

=

173.Q~

OOO =


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CHAPTER 3. GEOMETRY

28

174. O~i +=

175. == O

O

~pO

3.3 Equilateral Triangle

p ~ ~~ ~W ~^W o~ W oo~ W mW i^~W p

Figure 12.

176.O

P~ =


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CHAPTER 3. GEOMETRY

29

177.P

P~

P

Oo ==

178.O

o

S

P~

P

N ===

179. ~Pi =

180. Q

P~

O

~p

O

==

3.4 Scalene Triangle

E^ ~ ~F

p ~ ~W ~I I

pWO

~

++=

^ ~ ~W II^ ~I I W ~ II

j~ ~I I W ~ II_ ~ II W ~ IIo~ W oo~ W ^~W p


38/334

CHAPTER 3. GEOMETRY

30

Figure 13.

181. =++ NUM

182. ~ >+ I~ >+ I~ >+ K

183. ~


39/334

CHAPTER 3. GEOMETRY

31

185. i~ `

+= O~ OOO I

+= ~O~ OOO I

+= ~O~ OOO K

186. i~ p

oO

~=

=

=

I

o ~ K

187.pQ

~

O

~

O

~

O

O

O

O

~o

~

====

=

=

=

188.( )( )( )

~O

= I

~ N

N

N

N ++= K

189.( )( )

O

=

I

( )

~

O

=

I

( )( )( )~

O~

=

K

190. ( )( )( )~~

O~ = I

( )( )( )~O

= I

( )( )( )~

O = K


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CHAPTER 3. GEOMETRY

32

191. == ~ I== ~ I

== ~ K

192.Q

~

O

OOOO~

+= I

Q

O

~

OOOO

+= I

Q

O~

OOO

O += K

Figure 15.

193.~

P

O^j = I

P

O_j = I

P

O`j = EcKNRFK

194.( )

( )OO~

~Q

+

= I

( )( )O

O

~

~Q

+

= I

( )( )O

O

~

~Q +

= K


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CHAPTER 3. GEOMETRY

33

195.O

O

O

~p ~ === I

O

O~

O~p === I

( )( )( )~p = Ee c~FIp = I

oQ

~p = I

= oOpO

I

O~

O~

O~p O

= K

3.5 Square

p ~ ~W ~a~~W o~ W oo~ W mW i^~W p

Figure 16.


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CHAPTER 3. GEOMETRY

34

196. O~ =

197.O

O~

O

o ==

198.O

~ =

199. ~Qi =

200. O~p =

3.6 Rectangle

p ~ ~W ~I a~~W o~ W omW i^~W p

Figure 17.

201. OO ~ +=


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CHAPTER 3. GEOMETRY

35

202.O

o =

203. ( )~Oi +=

204. ~p =

3.7 Parallelogram

p ~ ~~~W ~I a~~W ON I

` ~W I^ ~~W ^W mW i^~W p

Figure 18.

205. =+ NUM

206. ( )OOOOON ~O +=+


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CHAPTER 3. GEOMETRY

36

207. ==

208. ( )~Oi +=

209. == ~~p I

= O

Np ON K

3.8 Rhombus

p ~ W ~a~~W ON I

` ~W I^W eo~ W mW i^~W p

Figure 19.


45/334


46/334

CHAPTER 3. GEOMETRY

38

Figure 20.

216.O

~

+=

217. O

~p =

+=

3.10 Isosceles Trapezoid

_~ ~ ~W ~I iW

jW ^W a~~W o~ W o^~W p


47/334

CHAPTER 3. GEOMETRY

39

Figure 21.

218.O

~

+=

219. O~ +=

220. ( )OO ~Q

N =

221.( )( )~O~O

~o

O

++

+=

222. O

~p =

+=


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CHAPTER 3. GEOMETRY

40

3.11 Isosceles Trapezoid with

Inscribed Circle

_~ ~ ~W ~I iW jW ^W a~~W o~ W oo~ W mW i^~W p

Figure 22.

223. O~ =+

224. O

~ =

+=

225. OOO +=


49/334


50/334


51/334

CHAPTER 3. GEOMETRY

43

Figure 24.

234. =++ PSMO

235. ( )~Oi +=

236.O

p ON=

3.14 Cyclic Quadrilateralp ~ ~~~W ~I I I a~~W ON I

^ ~~W f~ ~W IIIo~ W omW ipW ^~W p


52/334

CHAPTER 3. GEOMETRY

44

Figure 25.

237. =+=+ NUM

238. m q

ON~ =+

239. ~i +++=

240.( )( )( )( )( )( )( )~

~~~

Q

No

+++

= I

O

i = K

241. = O

Np ON I

( )( )( )( )~p = I

O

i = K


53/334

CHAPTER 3. GEOMETRY

45

3.15 Tangential Quadrilateral

p ~ ~~~W ~I I I a~~W ON I

^ ~~W o~ W mW ipW ^~W p

Figure 26.

242. ~ +=+

243. ( ) ( )O~O~i +=+=+++=

244.( ) ( )

O

~~

OOOO

ON += I

Oi = K


54/334


55/334

CHAPTER 3. GEOMETRY

47

248. = O

Np ON

3.17 Regular Hexagon

pW ~f~ ~W p~ W o~ W o~ W omW ipW ^~W p

Figure 28.

249. = NOM

250.O

P~ ==


56/334

CHAPTER 3. GEOMETRY

48

251. ~o =

252. ~Si =

253.O

PP~p

O

== I

O

i = K

3.18 Regular Polygon

pW ~k W f~ ~W

p~ W o~ W o~ W omW ipW ^~W p


57/334

CHAPTER 3. GEOMETRY

49

Figure 29.

254.

= NUMO

O

255. = NUMO O

256.

O

~o

=

257.Q~o

~O

~

O

O ===

258. ~i =

259.

O

O

op

O = I

Q

~op

OO == I


58/334

CHAPTER 3. GEOMETRY

50

O

i = K

3.19 Circle

o~W oa~W `W ~p~ W I q~ W `~ ~W f ~W mW i^~W p

260.O

oO~

=

Figure 30.


59/334


60/334

CHAPTER 3. GEOMETRY

52

Figure 33.

264.O

=

Figure 34.

265. oOi ==

266. O

io

Q

op

OO

=

==


61/334

CHAPTER 3. GEOMETRY

53

3.20 Sector of a Circle

o~ ~ W o^ W `~ ~ E ~~FW `~ ~ E FW mW i^~W p

Figure 35.

267. o =

268.

=NUM

o

269. oOi +=

270.

===PSM

o

O

o

O

op

OO

62/334

CHAPTER 3. GEOMETRY

54

3.21 Segment of a Circle

o~ ~ W o^ W `W ~`~ ~ E ~~FW `~ ~ E FW e W mW i

^~W p

Figure 36.

271. OoOO~ =

272. OO ~oQO

No = I o K

494.N

~~O

~~

= I M< K

495.N

~~~ = I M> K

496. = N~~~ I M< K

497. ( ) ~~ =

498. ~~O

~

=

499.ON

N~~

+= I M> K

500.ON

N~~

+= I M< K

501.ON

~~+

=

502.N

~~~ = I M> K

503.N

~~~ += I M< K


114/334

CHAPTER 4. TRIGONOMETRY

106

4.20 Trigonometric Equations

t W

504. ~ = I ( ) ~~N +=

505. ~ = I O~~ +=

506. ~~ = I ~~~ +=

507. ~ = I ~~ +=

4.21 Relations to Hyperbolic Functions

f~~ W

508. ( ) =

509. ( ) ~~ =

510. ( ) =

511. ( ) =

512. ( ) =


115/334


116/334


117/334


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119/334


120/334

CHAPTER 5. MATRICES AND DETERMINANTS

112

+++

+++

+++

=+

OONN

OOOOOOONON

NNNONONNNN

~~~

~~~

~~~

_^

K

MMM

K

K

K

534. f ~ ~~I ~ ~^ = ~ ~I

[ ]

==

ON

OOOON

NNONN

~~~

~~~

~~~

~^

K

MMM

K

K

K

535. j~ q j~q ~ ~ ~

K

f

[ ]

==

ON

OOOON

NNONN

~~~

~~~

~~~

~^

K

MMM

K

K

I

[ ]

==

ON

OOOON

NNONN

_

K

MMM

K

K

I


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113

==

ON

OOOON

NNONN

`^_

K

MMM

K

K

I

=

=+++=

NOONN ~~~~ K

E IIOIN K= X IIOIN K= FK

q

[ ]

==

OPOOON

NPNONN

~~~

~~~~^ I [ ]

==

P

O

N

_ I

=

=

POPOOONON

PNPONONNN

P

O

N

OPOOON

NPNONN

~~~

~~~

~~~

~~~^_ K

536. q~ ~ j~f ~ ~ ~ ~ ~I

~ ~ ~ ~ ~Kf ^ ~ ~I ~ q^

^

K

537. q ~ ^ ~ f^^q = K

538. f ~ ^_ I

( )

qqq

^_^_ = K


122/334


114

539. ^ j~f ^ ~ ~ ~I ~I ^~ I

~ ~ ~ ` W

[ ]q`^~ = K

540. q~ ~ j~f ^ ~ ~ ~I ~I ^ I ~ ~~W

OONN ~~~^ +++= K K

541. f ~ j~f ^ ~ ~ ~ ~ ~ ~

^ I N^

^

^~^ N = K

542. f ~ ^_ I ( ) NNN ^_^_ = K

543. f ^ ~ ~ ~I u ~ ~

u^u = I ~ ~ ~~ ~

Mf^ = K

5.5 Systems of Linear Equations

s~~W I I I N I KIOo~ W KI~I~II~I~I~ NONNNPON


123/334


115

a~W aI a I a I a

j~W ^I _I u

544.

=+

=+

OOO

NNN

~

~I

a

a = I

a

a

= E`~ FI

NOON

OO

NN ~~~

~a == I

NOON

OO

NN

a == I

NOON

OO

NN

~~~

~a == K

545. f Ma I ~ ~ W

a

a = I

a

a

= K

f Ma = ~ Ma E Ma FI ~

Kf Maaa

=== I ~ ~

K

546.

=++

=++

=++

PPPP

OOOO

NNNN

~

~

~

I

a

a

= I a

a

= I aa

= E`~ FI


124/334


125/334


117

=

ON

OOOON

NNONN

~~~

~~~

~~~

^

K

MMM

K

K

I

=

O

N

uM

I

=

O

N

_M

K

549. p ~ p i~ b~ _^u N = I

N

^

^K


126/334


127/334


128/334

CHAPTER 6. VECTORS

120

Figure 75.

555. f ( ) ( )NNNN wIvIuwIvIurr

= I

Nuu = I Nvv = I Nww = K

6.2 Vector Addition

556. rrr

+=

Figure 76.


129/334

CHAPTER 6. VECTORS

121

Figure 77.

557. PON r

Krrrr

++++=

Figure 78.

558. `~ i~rrrr

+=+

559. ^~ i~( ) ( ) rrrrrr ++=++

560. ( )ONONON wwIvvIuu +++=+rr


130/334

CHAPTER 6. VECTORS

122

6.3 Vector Subtraction

561. rrr

= rrr

=+ K

Figure 79.

Figure 80.

562. ( )rrrr

+=

563. ( )MIMIMM ==rrr

564. MM =r

565. ( )ONONON wwIvvIuu =rr

I

6.4 Scaling Vectors

566. rr

=


131/334

CHAPTER 6. VECTORS

123

Figure 81.

567. rr

=

568. ( )wIvIu =r

569. = rr

570. ( ) rrr

+=+

571. ( ) ( ) ( )rrr

==

572. ( ) rrrr

+=+

6.5 Scalar Product

573. p~~ m s r

~r

= rrrr

I

~ r

~r

K


132/334

CHAPTER 6. VECTORS

124

Figure 82.

574. p~~ m `~ cf ( )NNN wIvIu =r I ( )OOO wIvIu =

r I

ONONON wwvvuu ++=rr

K

575. ^ _ q sf ( )NNN wIvIu =r

I ( )OOO wIvIu =r

I

O

O

O

O

O

O

O

N

O

N

O

N

ONONON

wvuwvu

wwvvuu

++++

++= K

576. `~ mrrrr

=

577. ^~ m( ) ( )

rrrr=

578. a m( )

rrrrrrr+=+

579. M =rr

r

Ir

~ ~ EO

= FK

580. M >rr

OM


133/334

CHAPTER 6. VECTORS

125

581. M


134/334

CHAPTER 6. VECTORS

126

Figure 83.

588.

OOO

NNN

wvu

wvu

rrr

rrr==

589.

==OO

NN

OO

NN

OO

NN

vu

vuI

wu

wuI

wv

wvrrr

590. == prrrr

EcKUPF

591. ^ _ q s EcKUPF

rr

rr

=

592. k~ m( )

rrrr=

593. ^~ m( ) ( )

rrrr=


135/334

CHAPTER 6. VECTORS

127

594. a m( )

rrrrrrr+=+

595. Mrrr

= r

~ r

~ ~~ E M= FK

596. Mrrrrrrr

===

597. rrr

= I rrr

= I rrr

=

6.7 Triple Product

598. p~~ q m[ ] ( ) ( ) ( )

rrrrrrrrrrrr===

599. [ ] [ ] [ ] [ ] [ ] [ ] rrrrrrrrrrrrrrrrrr =====

600. ( ) [ ]rrrrrr

=

601. p~~ q m `~ c

( )PPP

OOO

NNN

wvuwvu

wvu

=

rrr

I

( )NNN wIvIu =

rI ( )OOO wIvIu =r

I ( )PPP wIvIu =r

K

602. s m~~( )s

rrr=


136/334

CHAPTER 6. VECTORS

128

Figure 84.

603. s m~

( )S

Ns

rrr=

Figure 85.

604. f ( ) M =rrr

I r

I r

I ~ r

~ ~

I rrr

+= ~~ ~ K

605. f ( ) M rrr

I r

I r

I ~ r

~ ~

K


137/334

CHAPTER 6. VECTORS

129

606. s q m( ) ( ) ( )

rrrrrrrrr=


138/334

130

Ch a p t e r 7

Analy t i c 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

ONNO ^_ ===

Figure 86.

608. a ~ i p o~

++

=N

ONM I

`_

^`= I N K

Figure 87.


139/334

CHAPTER 7. ANALYTIC GEOMETRY

131

609. j ~ i p

O

ON

M

+

= I N= K

7.2 Two-Dimensional Coordinate System

m ~WM

IN

IO

IM

IN

IO

m~ ~W Io~ W m ~ W ~I I Ia~ W ^~W p

610. a~ _ q m( ) ( )ONO

O

NO ^_ +==

Figure 88.


140/334


132

611. a ~ i p o~

+

+

= N

ON

M I +

+

= N

ON

M I

`_

^`= I N K

Figure 89.


141/334


133

Figure 90.

612. j ~ i p

O

ONM

+= I

O

ONM

+= I N= K

613. ` Ef j~F ~ q~

P

PONM

++= I

P

PONM

++= I

( )NN I^ I ( )OO I_ I ~ ( )PP I` ~ ~ ^_` K


142/334


134

Figure 91.

614. f Ef ^ _F ~ q~

~

~

PONM ++

++= I ~

~

PONM ++

++= I

_`~ = I `^ = I ^_ = K

Figure 92.


143/334


135

615. ` Ef p m~_F ~ q~

N

N

N

O

N

N

N

PP

OO

NN

POP

OP

OOO

OO

NON

ON

M

+

++

= I

N

N

N

O

N

N

N

PP

OO

NN

OP

OPP

OO

OOO

ON

ONN

M

+

++

=

Figure 93.


144/334


136

616. l Ef ^F ~ q~

N

N

N

N

N

N

PP

OO

NN

OPONP

OONPO

ONPON

M

+

+

+

= I

N

N

N

N

N

N

PP

OO

NN

PONOP

ONPOO

NPOON

M

+

+

+

=

Figure 94.

617. ^~ ~ q~

( ) ( )NPNP

NONO

PP

OO

NN

O

N

N

N

N

O

Np

==


145/334


137

618. ^~ ~ n~~~

( ) ( )( ) ( )( )[ ++++= POPOONON O

N

p( )( ) ( )( )]NQNQQPQP ++++

Figure 95.

kW f ~ SNTI SNU EHF EF ~ ~ ~ ~~K

619. a~ _ q m m~ `~

( )NOONO

OO

N O^_ +==


146/334


138

Figure 96.

620. ` o~~ `~ m~ `~= I = K

Figure 97.

621. ` m~ `~ o~~ `~

OO += I

~ = K


147/334


139

7.3 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 k~ W

r

m W r

I ~r

I r

622. d~ b~ ~ p~ iM`_^ =++

623. k~ s ~ p~ iq ( )_I^

r ~ M`_^ =++ K

Figure 98.

624. b b~ ~ p~ i Ep-f cF += K


148/334


140

q ~ = ~ K

Figure 99.

625. d~ ~ i

NO

NO

~

==

Figure 100.


149/334


141

626. b~ ~ i d ~ m ~ d~( )MM += I

~I ( )MM Im ~ K

Figure 101.

627. b~ ~ i q~ m~ q q m

NO

N

NO

N

=

MNN

N

OO

NN = K


150/334


142

Figure 102.

628. f c

N

~

=+

Figure 103.


151/334


143

629. k~ cM =+

Figure 104.

630. m a c

v

u

NN =

I

( )vIu ~ ( )NNN Im K


152/334


144

Figure 105.

631. s~ i~=

632. e~ i =

633. s b~ ~ p~ i

~rrr

+= I

l ~Iu ~ ~~ I~r

~ ^ I

r

~ I ~~ I ~ ~~I

= lur

~ u K


153/334


145

Figure 106.

634. p~ i m~~ c

+=

+=

OO

NN

~

~

I( )I ~ ~ ~ I( )ON ~I~ ~ ~ ~ I( )ON I ~ ~ ~ ~~ I ~ ~~K


154/334


146

Figure 107.

635. a~ c ~ m q ~ iq ~ ( )I~m

M`_^ =++

OO _^

`_^~

+++= K

Figure 108.


155/334


147

636. m~~ iq NN += ~ OO += ~ ~~

ON = Kq M`_^ NNN =++ ~ M`_^ OOO =++ ~~~

O

N

O

N

_

_

^

^= K

Figure 109.

637. m~ iq

NN

+= ~OO

+= ~ ~

N

O

N = I ~I N ON = K

q M`_^ NNN =++ ~ M`_^ OOO =++ ~~

M__^^ ONON =+ K


156/334


148

Figure 110.

638. ^ _ q i

ON

NO

N

~

+

= I

OO

OO

ON

ON

ONON

_^_^

__^^

++

+= K


157/334


149

Figure 111.

639. f q if M`_^ NNN =++ ~ M`_^ OOO =++ -I ~ ~

NOON

NOONM

_^_^

_`_`

+

= INOON

NOONM

_^_^

`^`^

+

= K

7.4 Circle

o~W o

` W ( )I~m ~W I I N I N I o~ W ^I _I `I aI bI cI


158/334


150

640. b~ ~ ` ` ~ l Ep~~cF

OOO o =+

Figure 112.

641. b~ ~ ` ` ~ ^ m ( )I~

( ) ( ) OOO o~ =+

Figure 113.


159/334


151

642. q m c

M

N

N

N

N

PPOP

OP

OOOO

OO

NNON

ON

OO

=

+

+

+

+

Figure 114.643. m~~ c

=

=

o

o

I OM K

644. d~ c

Mcba^^ OO =++++ E^ I ^cQba OO >+ FKq ~ ~ ( )I~ I

^O

a~ = I

^O

b = K

q ~


160/334


161/334


162/334


154

651. d~ c

Mcba`_^ OO =+++++ I

M^`Q_O K

653. `

( )~bQi = I b ~ K

654. ^~ c~ `

( )( )~~RKNi += I( )OO ~Oi += K

655. ~p =

7.6 Hyperbola

q~ ~W ~`~ ~W cW ( )MIcN I ( )MIcOa~ W ObW ^W I o~ W ^I _I `I aI bI cI I


163/334


155

656. b~ ~ e~ Ep~~ cF

N

~

O

O

O

O

=

Figure 117.

657. ~O ON = I N I O ~ ~ ~ ( )Im

~ K


164/334


156

Figure 118.

658. b~ ^

~

=

659. OOO ~ +=

660. b

N~

>=

661. b~ a

~

~

O

==

165/334


157

662. m~~ b~ o _~ ~ e~

=

=

~

I OM K

663. d~ c

Mcba`_^ OO =+++++ I

M^`Q_O > K

664. d~ c ^ m~~ `~ ^

Mcba`^ OO =++++ I M^` < K

665. ^ c

Q

O

= I

= I

Q

O

= K

f ~ I ~ ~ ~ M= ~M = K


166/334


158

Figure 119.

7.7 Parabola

c~ ~~W cW c

sW ( )MM Ijo~ W ^I _I `I aI bI cI I ~I I

666. b~ ~ m~~~ Ep~~ cF

OO =


167/334


159

Figure 120.

b~

O

= I

`~

MIO

c I

`~ ( )MIMj K

667. d~ c

Mcba`_^ OO =+++++ I

M^`Q_O = K

668. O~ = I ~ON = K

b~


168/334


160

O

= I

`~

O

IMc I

`~ ( )MIMj K

Figure 121.

669. d~ cI ^ m~~ -~Mcba^O =+++ E^I b FI

~ O ++= I~O

N = K

b~

O

M = I

`~


169/334


161

+

O

Ic MM I

`~

~O

M = I

~Q

~Q~

O

MOMM

=++= K

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 psW s


170/334


162

670. a~ _ q m

( ) ( ) ( )ONOO

NO

O

NO ^_ ++==

Figure 123.

671. a ~ i p o~

+

+=

N

ONM I +

+=

N

ONM I +

+=

N

ONM I

`_^`= I N K


171/334


163

Figure 124.

Figure 125.


172/334


164

672. j ~ i p

O

ON

M

+

= I O

ON

M

+

= I O

ON

M

+

= I N= K

673. ^~ ~ q~q ~~ ~ ~ ( )NNNN IIm I

( )OOOO IIm I ~ ( )PPPP IIm O

PP

OO

NN

O

PP

OO

NN

O

PP

OO

NN

NN

N

NN

N

NN

N

O

N

p ++= K

674. s ~ q~q ~ ~ ( )NNNN IIm I

( )OOOO IIm I ( )PPPP IIm I ~ ( )QQQQ IIm

N

N

N

N

S

Ns

QQQ

PPP

OOO

NNN

= I

QPQPQP

QOQOQO

QNQNQN

S

Ns

= K

kW t EHF EF ~ ~ ~ K


173/334


165

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 aIN

^ IO

^ I ~I I IN

~ IO

~ I I I I k~ W r I N

r I Or

a W I I a~ ~W

675. d~ b~ ~ m~Ma`_^ =+++


174/334


166

676. k~ s ~ m~q ( )`I_I^

r ~ ~

Ma`_^ =+++ K

Figure 127.

677. m~~ `~ b~ ~ m~Ma`_^ =+++

f M^ = I ~ ~~ -~Kf M_ = I ~ ~~ -~Kf M` = I ~ ~~ -~Kf Ma = I ~ K

f M_^ == I ~ ~~ -~Kf M`_ == I ~ ~~ -~K

f M`^ == I ~ ~~ -~K


175/334


167

678. m a c( ) ( ) ( ) M`_^ MMM =++ I

( )MMM IIm ~I ~ - ( )`I_I^ ~ ~K

Figure 128.

679. f c

N

~ =++


176/334


168

Figure 129.

680. q m c

M

POPOPO

PNPNPN

PPP

=

I

M

N

N

NN

PPP

OOO

NNN = K


177/334


169

Figure 130.

681. k~ cM =++ I

~ ~ ~ I ~ I I ~ ~ ~ ~K


178/334


179/334


171

Figure 132.

683. a~ ^ _ q m~f ~ ~

Ma`_^ NNNN =+++ IMa`_^ OOOO =+++ I

~ ~

OO

OO

OO

ON

ON

ON

ONONON

ON

ON

`_^`_^

``__^^

++++

++=

= rr

rr

K


180/334


172

Figure 133.

684. m~~ m~q ~ Ma`_^ NNNN =+++ ~

Ma`_^ OOOO =+++ ~ ~~

O

N

O

N

O

N

`

`

_

_

^

^== K

685. m~ m~q ~ Ma`_^ NNNN =+++ ~

Ma`_^ OOOO =+++ ~ ~ M``__^^ ONONON =++ K

686. b~ ~ m~ q ( )NNN IIm ~ m~~ q s ( )NNN II~ ~ ( )OOO II~ EcKNPOF


181/334


173

M

~

~

OOO

NNN

NNN

=

687. b~ ~ m~ q ( )NNNN IIm ~ ( )OOOO IIm I~ m~~ q s ( )II~

M

~

NONONO

NNN

=

Figure 134.

688. a~ c ~ m q ~ m~q ~ ( )NNNN IIm ~

Ma`_^ =+++


182/334


174

OOO

NNN

`_^

a`_^

++

+++= K

Figure 135.

689. f q m~f ~ Ma`_^ NNNN =+++ ~

Ma`_^ OOOO =+++ I ~

+=

+=

+=

~

N

N

N

I

~

NNN =

=

I


183/334


175

OO

NN

`_

`_~ = I

OO

NN

^`

^` = I

OO

NN

_^

_^ = I

OOO

OO

NN

OO

NN

N~

_a

_a

à

à

++

= I

OOO

OO

NN

OO

NN

N~

à

à~

â

â

++

= I

OOO

OO

NN

OO

NN

N~

â

â

_a

_a~

++

= K

7.10 Straight Line in Space

m ~W I I I N I N I N I

a W I I o~ W Î _I Ì aI ~I I I N~ I O~ I I

a ~ W r

I Nr

I Or

k~ ~ ~W r

^ W

690. m a c b~ ~ i

~

NNN =

=

I

( )NNNN IIm I ~ ( )II~ K


184/334


176

Figure 136.

691. q m c

NO

N

NO

N

NO

N

=

=

Figure 137.


185/334


177

692. m~~ c

+=+=

+=

N

N

N

I

( )NNNN IIm ~ I I I ~

I ~~ ~ ~ K

Figure 138.

693. ^ _ q p~ i

OO

OO

OO

ON

ON

ON

ONONON

ON

ON

~~

~~

++++

++=

= rrrr


186/334


187/334


179

O

O

O

O

O

O

~

=

=

M

~

~

OOO

NNN

NONONO

=

K

697. m~~ i ~ m~

q ~

~

NNN =

=

~ ~

Ma`_^ =+++ ~ ~~ M =

rrI

M`_^~ =++ K

Figure 140.


188/334


180

698. m~ i ~ m~

q ~

~

NNN

=

=

~ ~Ma`_^ =+++ ~ ~

rr

I

`_

~

^== K

Figure 141.

7.11 Quadric Surfaces

m ~ ~ ~W I I o~ W ^I _I `I ~I I I PON II I

189/334


181

699. d~ n~~ b~

MaoOnOmOeOdOcO`_^ OOO =+++++++++

700. `~~ n~ p~

`~ o~EF o~EbF q p~N P Q M< p~ o~ bO P Q M> p~ f~~ bP P Q M> a e N pQ P Q M< a e O pR P P a o~ n~ S P P p~ f~~ n~ T O Q M< p~ b m~~U O Q M> a e m~~V O P p~ o~ b

NM O P p~ f~~ b NN O P a e NO O O a o~ f m~

NP O O p~ f~~ f m~NQ N P m~~ NR N O o~ m~~ m~NS N O f~~ m~~ m~NT N N ` m~

e

=

`cd

c_e

de^

I

=

aonm

o`cdnc_e

mne^

b I ( )b= I

PON II ~ ~I

M

`cd

c_e

de^

=

K


190/334


182

701. o~ b E`~ NF

N

~

O

O

O

O

O

O

=++

Figure 142.

702. f~~ b E`~ OF

N

~

O

O

O

O

O

O

=++

703. e N p E`~ PF

N

~

O

O

O

O

O

O

=+


191/334


183

Figure 143.

704. e O p E`~ QF

N

~

O

O

O

O

O

O

=+

Figure 144.


192/334


184

705. o~ n~ ` E`~ RF

M

~

O

O

O

O

O

O

=+

Figure 145.

706. f~~ n~ ` E`~ SF

M

~

O

O

O

O

O

O

=++

707. b m~~ E`~ TF

M

~

O

O

O

O

=+


193/334


185

Figure 146.

708. e m~~ E`~ UF

M

~

O

O

O

O

=

Figure 147.


194/334


186

709. o~ b ` E`~ VF

N

~

O

O

O

O

=+

Figure 148.

710. f~~ b ` E`~ NMF

N

~

O

O

O

O

=+

711. e ` E`~ NNFN

~

O

O

O

O

=


195/334


187

Figure 149.

712. o~ f m~ E`~ NOF

M

~

O

O

O

O

=

713.f~~ f m~ E`~ NPF

M

~

O

O

O

O

=+

714. m~~ ` E`~ NQF

M~

O

O

=


196/334


188

Figure 150.

715. o~ m~~ m~ E`~ NRF

N~

O

O

=

716. f~~ m~~ m~ E`~ NSF

N

~

O

O

=

717. ` m~ E`~ NTF

MO =


197/334


189

7.12 Sphere

o~ ~ W om ~W I I I N I N I N I

` ~ W ( )II~o~ W ^I aI bI cI j

718. b~ ~ p ` ~ l Ep~~cF

OOOO o =++

Figure 151.719. b~ ~ ` ` ~ ^ m ( )II~

( ) ( ) ( ) OOOO o~ =++

720. a~ c

( )( ) ( )( ) ( )( ) M ONONON =++ I


198/334


190

( )NNNN IIm I ( )OOOO IIm ~ ~ ~K

721. c m c

M

N

N

N

N

N

QQQ

O

Q

O

Q

O

Q

PPPOP

OP

OP

OOOOO

OO

OO

NNNON

ON

ON

OOO

=

++

++

++

++

++

722. d~ c

Mjcba^^^ OOO =++++++ E^ FKq ~ ~ ( )II~ I

Ô

a~ = I

Ô

b = I

Ô

c = K

q ~

Ô

j^Qcbao

OOOO ++= K


199/334

191

Ch a p t e r 8

Diff erent ial 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. b c( ) ( ) =

724. l c( ) ( ) =

725. m c( ) ( )q =+

726. f c

( ) = ~ I ( )= ( ) N= K


200/334

CHAPTER 8. DIFFERENTIAL CALCULUS

192

Figure 152.

727. ` c( ) = I ( ) = I ( )( ) = ~ K

728. i~ c~ += I o I = ~~ I

-K


201/334


193

Figure 153.

729. n~~ cO = I o K

Figure 154.


202/334


194

730. ~ O ++= I o K

Figure 155.

731. ` cP = I o K


203/334


195

Figure 156.

732. ~ OP +++= I o K


204/334


196

Figure 157.

733. m c = I k K


205/334


197

Figure 158.

Figure 159.


206/334


198

734. p~ o c

= I [ ) IM K

Figure 160.

735. b~ c~ = I M~ > I N~ I = ~ = I KSTNUOUNUOUQKO =

Figure 161.


207/334


199

736. i~ c ~= I ( ) IM I M~ > I N~ I

= ~ = I M> K

Figure 162.

737. e p c

= IO

= I o K


208/334


200

Figure 163.

738. e ` c = I

O

+= I o K

Figure 164.


209/334


210/334


202

Figure 166.

741. e p~ c

= I

O

N +

=== I o K

Figure 167.


211/334


203

742. e `~ c

= I

O

N

=== I o I M K

Figure 168.

743. f e p c~ = I o K


212/334


204

Figure 169.

744. f e ` c~ = I [ ) IN K

Figure 170.

745. f e q~ c~~ = I ( )NIN K


213/334


205

Figure 171.

746. f e `~ c~ = I ( ) ( ) INNI K


214/334


206

Figure 172.

747. f e p~ c~ = I ( ]NIM K


215/334


207

Figure 173.

748. f e `~ c

~ = I o I M K

Figure 174.


216/334


208

8.2 Limits of Functions

cW ( ) I ( )^W o~ ~W ~I

749. ( ) ( )[ ] ( ) ( )~~~

+=+

750. ( ) ( )[ ] ( ) ( )~~~

=

751. ( ) ( )[ ] ( ) ( )~~~

=

752.( )

( )

( )

( )

~

~

~

= I ( ) M

~

K

753. ( )[ ] ( )~~

=

754. ( )( ) ( )~~

=

755.( ) ( )~~

= I ( ) ~ ~

=K

756. N

M

=

757. N~

M

=

758. N

N

M=


217/334


209

759. N~

N

M=

760.( )

NN

M

=+

761.

NN

=

+

762.

N =

+

763. N~ M

=

8.3 Definition and Properties of the Derivative

cW I I I I f ~~W o~ ~W ^W

764. ( )( ) ( )

MM=

=

+=


218/334


210

Figure 175.

765. = ~

766.( )

+=

+

767.( )

=

768.( )

=

769. m o( )

+=


219/334


211

770. n o

O

=

771. `~ o( )( ) = I ( ) = I

= K

772. a~ f c

N

= I

( ) ( ) K

773. o~ o

O

N

=

774. i~ a~( ) = I ( ) = I

( ) ( )[ ]

= K

8.4 Table of Derivatives

f ~~W o~ ~W `I ~I I k~~ W


220/334


212

775. ( ) M`

=

776. ( ) N

=

777. ( ) ~~

=+

778. ( ) ~~ O +=++

779. ( ) N

=

780. ( )N

+

=

781.O

N

N

=

782. ( )O

N

=

783. ( ) N

N

=

784. ( )N

=

785. ( )~

N

~ = I M~ > I N~ K


221/334


213

786. ( ) ~~~

= I M~ > I N~ K

787. ( )

=

788. ( )

=

789. ( ) =

790. ( )

N~

OO

==

791. ( )

N

OO

==

792. ( ) ~

=

793. ( )

=

794. ( )ON

N~

=

795. ( )ON

N~

=

796. ( ) ONN~~

+=


222/334


214

797. ( )ON

N~

+=

798. ( )N

N~

O

=

799. ( )N

N~

O

=

800. ( )

=

801. ( )

=

802. ( ) N~OO ==

803. ( )

N

OO

==

804. ( ) ~

=

805. ( )

=

806. ( )N

N~

O +

=

807. ( )N

N~

O

=

223/334


215

808. ( )ON

N~~

= I N < K

809. ( )N

N~

O

= I N > K

810. ( )

N +=

8.5 Higher Order Derivatives

cW I I I f ~~W k~~ W

811. p ~

( )O

O

=

=

==

812. e-l ~

( ) ( ) ( )( ) === N

813. ( )( ) ( ) ( ) +=+

814. ( )( ) ( ) ( ) =

815. i c~

( ) O ++=


224/334


216

( ) PP +++=

( )( ) ( ) ( ) ( ) ( ) ( )ON ON N ++++= K

816. ( )( )( )

>

>

=

817. ( )( ) > =

818. ( )( )( ) ( )

~

>NN

N

~

=

819. ( )( )( ) ( )

N >NN

=

820. ( )( )

~~~

=

821. ( )( ) =

822. ( )( ) ~~~ =

823. ( )( )

+= O

824. ( )( )

+=

O


225/334


217

8.6 Applications of Derivative

cW I I m ~ W sW ^~W f ~~W qW k~~ W

825. s ~ ^~( ) = ~ ~ ~

~ ~ ~ I( ) == ~~ I

( ) === ~~ ~~

K

826. q~ i( )( )MMM =

226/334


218

Figure 176.

827. k~ i

( )( )M

M

M

N

= Ec NTSF

828. f~ ~ a~ cK

f ( ) M M > I EF ~ ~ M K Ec NTTI N< IO < FI

f ( ) M M


227/334


219

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~Kf EF ~ E ( ) M > F ~ ~( ]NI~ ~ EF ~ E ( ) M

228/334


220

832. f EF ~ E ( ) M F ~ ~ [ )IO I EF ~ ~ ~ ~ O KEcKNTTFK

833. p a~ q i~ b~Kf ( ) M N = ~ ( ) M N I EF ~ ~ ~

~ O K EcKNTTF

834. `~KEF ~ ~ ~ M ~ ( ) ~ ~ M EcKNTTI P < FKEF ~ ~ ~ M ~ ( ) ~ ~ M K EcKNTTI P< FK

835. p a~ q `~Kf ( ) M M > I EF ~ ~ ~ M Kf ( ) M M


229/334


221

8.7 Differential

cW I I f ~~W a~ ~ W ( ) I ( )o~ ~W `a~ ( ) = W a~ W p~ ~ W p~ ~ W

838. =

839. ( ) ( ) ( ) +=+

Figure 178.


230/334


231/334


223

847. p l m~~ a~

O

O

=

=

I

O

O

=

=

I

O

=

=

I

O

=

=

K

f ~ ~ I

OO

=

K

848. `~ o

f ( ) ( )( )II = E ~ ~~ FI ( )( )

I

=

I ( )( )

I

=

K

f ( ) ( ) ( )( )I = I ( )

+

= K

f ( ) ( )( )III = I

+

=

I

+

=

K

849. p~ `~

+

232/334


224

850. i~ j~~ ~ j~( )I ~ ~ ~ ~ ~ ( )MM I ( ) ( )MM II

~ ( )I ( )MM I K

( )I ~ ~ ~ ~ ( )MM I ( ) ( )MM II ~ ( )I ( )MM I K

851. p~~ m

M

=

=

K

i~ ~~ ~ ~ ~ ~ ~~ K

852. p~ m^ ~~ ~ ~ ~ ~ ~

853. p a~ q p~~ m

i ( )MM I ~ ~~ E M

=

=

FK

( ) ( )

( ) ( )MMMM

MMMM

II

IIa = K

f Ma > I ( ) MI MM > I ( )MM I ~ ~ ~Kf Ma > I ( ) MI MM < I ( )MM I ~ ~ ~~Kf Ma < I ( )MM I ~ ~ Kf Ma = I ~K

854. q~ m~q ~ ~ ~ ~ ( )I =~ ( )

MMM

II

( )( ) ( )( )MMMMMMM II += K


233/334


225

855. k~ p~q ~ ~ ~ ( )I = ~

( )MMM II

( ) ( ) N

I

I

M

MM

M

MM

M

=

=

K

8.9 Differential Operators

r ~ ~ ~W r

I r

I r

p~~ E~~ FW ( )II I ( )ON III Kd~ ~ ~~ W ~ I

a~ ~W

s E FW ( )oInImcr

a ~ W cr

I cr

` ~ W cr

I cr

i~~~ ~W O

856. d~ ~ p~~ c

==

I

I

~ I

==ON

II

I

~ K K

857. a~ a~

+

+

=

I


234/334


226

( ) IIr

I N OOO =++ K

858. a ~ s c

o

n

mcc

+

+

==rr

859. ` ~ s c

onm

cc

==

rrr

rr

m

n

o

m

n

o rrr

+

+

=

860. i~~~ l~

O

O

O

O

O

OO

+

+

=

861. ( ) ( ) Mcc =rr

862. ( ) ( ) M~ =

863. ( ) ( ) ~ O==

864. ( ) ( ) ( ) cccc~c OOrrrrr

==


235/334

227

Ch a p t e r 9

Integral 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. ( ) ( ) `c += ( ) ( )c = K

866. ( )( ) ( ) =

867. ( ) ( ) =

868. ( ) ( )[ ] ( ) ( ) +=+

869. ( ) ( )[ ] ( ) ( ) =

870. ( ) ( ) `~c~

N~ +=


236/334

CHAPTER 9. INTEGRAL CALCULUS

228

871. ( ) ( ) `~c~

N~ ++=+

872. ( ) ( ) ( ) `O

N O +=

873.( )( )

( ) `

+=

874. j p( ) ( )( ) ( ) = ( )= K

875. f~ m~

= I ( ) I ( ) ~ ~ K

9.2 Integrals of Rational Functions

876. `~~ +=

877. `O

O

+=

878. `P

PO +=

879. `N

N

++=

+

I N K


237/334


229

880. ( )( )

( )`

N~

~~

N +

++

=++

I N K

881. `

+=

882. `~~

N

~

++=

+

883. `

~~

~

O+++=

++

884.( )( )

`~

~

N

~

+

++

=

++ I ~ K

885. ( ) `~~~N

~

O +++=+

886. ( ) ( ) `~~~~O~O

N

N

~

OOP

O

+

++++=

+

887.( )

`

~

~

N

~

+

+=

+

888.( )

`

~

~

~

N

~

OO

++

+=+

889.( )

`~

~~

N

~

OO

+

+++=

+


238/334


230

890.( )

`~

~~~O~

N

~

O

PO

O

+

+++=

+

891.( ) ( )

`

~

~

N

~~

N

~

OO

++

++

=+

892. `N

N

O

N

N

O

++

=

893. `N

N

O

N

N

O

++

=

894. `~

~

~O

N

~

OO

++

=

895. `~~

~ON

~

OO+

+=

896. `~N

NO

+=+

897. `~

~~

N

~

NOO +=+

898. ( ) `~O

N

~

OOOO

++=+

899. `~

~~

~

N

~

O

+

=

+ I M~ > K


239/334


231

900. `

~

O

N

~

OO

++=+

901.( )

`~

~O

N

~

O

O

O+

+=

+

902. `~

~

~O

N

~

OOO

++

=

903. `~Q~O

~Q~O

~Q

N

~

O

O

OO+

++

+

=

++ I

M~QO > K

904. `

~Q

~O~~

~Q

O

~

OOO

+

+

=++ I

M~QO

240/334


232

908.( )

( ) `~~NR

O~PO~ O

P

O++

=+

909.( )

`~~

~~

~

N

~

+

++

+

=

++ I

M~ > K

910.

( )

`

~

~~~

~

N

~

+

+

=

++ I

M~ K

912. ( )( ) ++=++

~N

~

( )( )

`~

~~~

~

~+

++

I E M~ FK

913. ( ) ( ) `~NMR NR~NO~UO~P

P

OOO

O +++=+

914.( )

`~NR

P~Q~UO

~

P

OOOO

+++

=+

915. `~~

~~

~

N

~

+++

+

=+ I M~ > K

241/334


233

916. `~

~~~

~

O

~

+

+

=

+ I M~ < K

917. ( )( ) ( ) `~

~~~

~+

++

+++=+

918. ( )( ) ( ) `~

~~~

~+

+

++=+

919. `~NN

N O ++=+

920.( )( )

`~

~~O

~~

+

=

921. ++=+ OO ~QO~

`~Q

O~

U

~QOP

O

++

+

922. ( ) `~~O~O~

N

~

OO

+++++=

++

I

M~ > K

923. `~Q~Q

~O~

~

N

~

OO

++

=++

I M~ < K

924. `~

O

~~

O

~ OO

OOOOO +++++=+


242/334


234

925. ( ) `~P

N~ O

POOOO ++=+

926. ( ) ++=+ OOOOOOO ~~OU

~

`~U

~ OOQ

+++

927. `~~

~ OO

OO

O

OO

+++++

=+

928. `~~

OOOO

+++=+

929. `

~~

~~

~

OO

OOOO

+

++

++=+

930. `~

~

OOOO

++=+

931. `~O

~~

O

~

OOO

OO

OO

O

++++=+

932. `~~

~

N

~

OOOO

+++

=+

933. `~O

~~

O

~ OO

OOOOO ++=

934. ( ) `~P

N~ O

POOOO +=


243/334


235

935. `~

~~~~ OO

OO

++=

936. `~~

~ OO

OO

O

OO

+++

=

937. `~~

OOOO

++=

938. `~~

OOOO

+=

939. `~O

~~

O

~

OOO

OO

OO

O

+++=

940. `~~

~N

~

OO

+=

941.( )

`~

~

~

N

~~

OO

++

=+

942.( )

`~~

~N

~~

OO+

+=

943. `~

~

~

O

OO

OOO+

=

944. ( ) `~~

~

OOOO

POO +=


244/334


236

945. ( ) ( ) += OOOOOP

OO ~~ROU

~

`~U

~P OOQ +++

946. `~

~

O

~~

O

~

OOOOO ++=

947. ( ) `~PN

~O

POOOO

+=

948. ( ) `~

~

U

~~~O

U

~

QOOOOOOO ++=

949. `

~~

~~

~OO

OOOO

+

+

+=

950. `~

~

~

~ OO

O

OO

+

=

951. `~N

O

+=

952. `~

~

OO

+=

953. `~~

OOOO

+=

954. `~

~

O

~~

O

~

OOOOO

O

++=

245/334


237

955.

( )

`~

~

O

N

~~

OO

++

=

+

956.( )

`~

~

O

N

~~

OO

++

=

957.( ) ( )

`~

~~

~

N

~

O

OOOO+

++

=

+ I ~ > K

958.( )

I`~~~

~

N

~

OOOOOOOOO

+++

+

=

+

~ < K

959. `~

~

~

O

OO

OOO+

=

960. ( ) ( ) `~

~

U

~P~O~R

U

~

QOOOOO

POO ++=

961.

( )

`~~

~

OOOO

POO

+

=

9.4 Integrals of Trigonometric Functions

962. ` +=

963. ` +=


246/334


238

964. ÒQ

N

O

O +=

965. ÒQ

N

O

O ++=

966. `Q

PP

NO

N`

P

N PP +=+=

967. `QPP

NON`

PN PP ++=+=

968. Ò

~

+==

969. `QO

~

+

+==

970. `

OO

+==

971. `~

OO

+==

972. Ò

~

O

N

O

O

P

P++==

973. `QO

~

O

N

O

O

P

P+

++==

974. ÒQ

N +=


247/334


239

975. `P

N PO +=

976. `P

N PO +=

977. `QPO

N

U

OO +=

978. `~ +=

979. ``

N

O

+=+=

980. `QO

~

O+

+=

981. `~~O +=

982. ` +=

983. ``

N

O +=+=

984. `O

~

O++=

985. `O +=

986. `~

+=


248/334


240

987. `QO

~

N

O

+

++=

988. Ò

~

N

O

++=

989. `~

OO

+=

990.( )( )

( )( )

Ò

O

+

+++

= IOO K

991.( )( )

( )( )

Ò

O

+

++

= IOO

K

992.( )( )

( )( )

Ò

O

+

+++

= IOO K

993. `~ +=

994. ` +=

995. `N

N +

+=

+

996. `N

N

++=

+


249/334


241

997. `N~~ O ++=

998. `N~~ O +=

999. ( ) `NO

N~~~~ O ++=

1000.

( )`N

O

N~~ O +++=

9.5 Integrals of Hyperbolic Functions

1001. ` +=1002. ` +=

1003. `~ +=

1004. ` +=1005. `~O +=

1006. ` O +=

1007. `~ +=


250/334


242

1008. ` +=

9.6 Integrals of Exponential and LogarithmicFunctions

1009. ` +=

1010. `

~

~~

+=

1011. `~

~~ +=

1012. ( ) `N~~

O

~~ +=

1013. ` +=

1014. `

+=

1015.( )

`N

N

N

O

N +

+

+= +

1016. `~

~ ~

OO

~ ++

=


251/334


243

1017. `~

~ ~

OO

~ +++

=

9.7 Reduction Formulas

1018. =

N N

1019.( ) +=

N

N

N

N

I N K

1020.

=

N

N ON

1021.( )

=

N

O

N

ON

I N K

1022.

+=

N

N ON

1023. ( )

+= N ON ON I N K

1024.

NN

+=

+

+

+

N O

1025.

NN

+=

+


252/334


244

+

N O

1026. += ~~NN

~ ON I N K

1027. += NN

ON I N K

1028. +

=

NO

N~ O

O I N K

1029.

+=

N

N ON

1030.

( )

+

=

N

O

N

ON

I N K

1031.

+=

N

N ON

1032.( )

+

=

N

O

N

ON

I N K

1033.

NN

+=

+

+

+

N O

1034.

NN

+=

+


253/334


245

+

+

N O

1035. = ~~NN

~ ON I N K

1036. = NN

ON I N K

1037. +

=

NO

N~ O

O I N K

1038.

+

= N

O

N

O

O I N K

1039.

+

++= N

N

NN

1040.( )

+

=

N

N

N

N

I N K

1041. = N

1042. = N

1043. = N

1044. += N

1045. = N


254/334


246

1046. +

+

=+

+

N

N

N

N

O

NN

NN

1047. +

++

=+

+

N

N

N

N

O

NN

NN

1048. +++=+

+

N

N

N~

N

~

O

NN

NN

1049. +=+ ~

~

~

~

1050.( ) ( )( )( ) NOOO ~~QN

~O

~

++

=

++

( )

( )( ) ( ) ++

NOO ~

~QN

~POO

I N K

1051.( ) ( ) ( ) ( ) ( )

I~

~NO

PO

~~NO

~

NOOONOOOOO +

+

+=

+N K

1052. ( ) ( ) ( ) NOOOOO ~~NO

~

=

( ) ( )

NOOO ~

~NO

POI N K


255/334


247

9.8 Definite Integral

a ~ ~ W ( )

~

I ( )

~

I

o~ W ( )=

N

p~ ~W ^~W ( )c I ( )di ~W ~I I I

1053. ( ) ( )=

=

N

M~

~

I

N = I N K

Figure 179.


256/334


248

1054. ~N

~

=

1055. ( ) ( ) =

~

~

1056. ( ) ( )[ ] ( ) ( ) +=+

~

~

~

1057. ( ) ( )[ ] ( ) ( ) =

~

~

~

1058. ( ) M~

~

=

1059. ( ) ( ) =~

~

1060. ( ) ( ) ( ) +=

~

~

~


257/334


249

1063. c~~ q `~

( ) ( ) ( ) ( )~ccc

~

~ == ( ) ( )c = K

1064. j pf ( )= I

( ) ( )( ) ( ) =

~

I

( )~ N= I ( ) N= K

1065. f~ m~

( ) =

~

~

~

1066. q~~ o

( ) ( ) ( ) ( )

++

=

=

N

NM

~

OO

~


258/334


250

Figure 180.

1067.p

o( ) ( ) ( ) ( ) ( )[ ++++

= PONM

~

QOQP

~

( ) ( ) ( )]NQ QO ++++ K I

~~

+= I IIOINIM K= K


259/334


251

Figure 181.

1068.^~ r ~ `

( ) ( ) ( )~ccp

~

== I

( ) ( )c = K


260/334


252

Figure 182.

1069.^~ _ q `

( ) ( )[ ] ( ) ( ) ( ) ( )~d~cdcp

~

+== I

( ) ( )c = I ( ) ( )d = K


261/334


253

Figure 183.

9.9 Improper Integral

1070. q ~ ( )

~

~ ~ ~

~ I ( ) ~

~ [ ]I~ K

1071. f ( ) ~ [ )I~ I

( ) ( )

=

~

~

K


262/334


254

Figure 184.

1072. f ( ) ~ ( ]I I

( ) ( ) =

K

Figure 185.


263/334


255

k W q ~ NMTNI NMTO ~ ~ ~ X ~ ~

K

1073. ( ) ( ) ( )

+=

Figure 186.

f ~ I ~

~ I ~ ( )

~

X K

1074. `~ qi ( ) ~ ( ) ~ [ )I~ K p ~ ( ) ( )M ~ [ )I~ K


264/334


256

f ( )

~

I ( )

~

~

I

f ( )

~

I ( )

~

~ K

1075. ^ `

f ( )

~

I ~ ( )

~

~-

K

1076. a f~i ( ) ~ ~[ )I~ ~ = K q

( ) ( )

+=

~M

~

Figure 187.


265/334


257

1077. i ( ) ~ ~ ~ ~ [ ]I~ ( )I~ K q

( ) ( ) ( )+

+

++=

M

~M

~

K

Figure 188.

9.10 Double Integral

c ~~W ( )I I ( )I I

a ~W ( )o

I I ( )o

I I

o~ W ( )= =

N

N I

p~ ~W I o ~W oI pm~ ~W I


266/334


258

^~W ^p~ ~~W p

s ~ W sj~ ~ ~~W aW ( )Ic W j I j

j ~W f I f I Mf

`~ ~ ~W n`~ W ( )I

`~ ~W I ^~ ~ W

1078. a a f~q ~ ~ ~ [ ] [ ]II~

( )[ ] [ ] ( ) = = =

N

N M~M~

II~

I^I

I

I ~

( ) NN II I ~ N = I N = K

Figure 189.


267/334


259

q ~ ~ ~ o

( ) ( )[ ] [ ]

=

II~o

ÎÎ I

~ [ ] [ ]II~ ~ oI( ) ( )II = ( )I o ~ ( ) MI = K

Figure 190.

1079. ( ) ( )[ ] ( ) ( ) +=+ooo

ÎÎÎI

1080. ( ) ( )[ ] ( ) ( )

=

ooo

ÎÎÎI

1081. ( ) ( ) =oo

ÎÎ I

~ ~K

1082. f ( ) ( )II oI ( ) ( ) oo

ÎÎ K

1083. f ( ) MI o ~ op I


268/334


260

( ) ( ) op

ÎÎ K

Figure 191.

1084. f ( ) MI o ~ o ~ p ~ -~

I ( ) ( ) ( ) += popo ÎÎÎ Ke po o ~ pK

Figure 192.


269/334


261

1085. f~ f~ ~ c q

( ) ( )( )

( )

=

~

oI^I

~ fI( ) ( ) ( ){ }I~Io = K

Figure 193.

( ) ( )( )

( )

=

o

I^I

~ ffI( ) ( ) ( ){ }IIo = K


270/334


262

Figure 194.

1086. a f~ o~~ o

f o ~~ [ ] [ ]II~ I

( ) ( ) ( )

=

=

~

~

o

III K

f ~ ~ ~ ( )I ~ - ~ ( ) ( ) ~

( ) ( ) ( ) ( ) ( )

==

~oo

I K

1087. `~ s~~

( ) ( ) ( )[ ]( )( )

I

IIIII

po

= I

( )( )

M

I

I

=

~~ ~-

~ ( ) ( )II I ~ p ~ o


271/334


263

~ ( )I= I ( )I = - oK

1088. m~ `~= I = K

Figure 195.

1089. a f~ m~ `~

q a~ m~ `~

( )( )

=

=

I

I K

i o ~ W( ) ( ) M I I O Kq

( ) ( )( )

( )

=

o

II K


272/334


264

Figure 196.

f o ~ ~ ~M I I O I

( ) ( )

=

~o

II K

Figure 197.


273/334


265

1090. ^~ ~ o

( )

( )

=

~

^ E ~ f FK

Figure 198.

( )

( )

=

^ E ~ ff FK

Figure 199.


274/334


266

1091. s ~ p

( )

=

o

Îs K

Figure 200.

f o ~ f ~= I = I ( ) = I( ) = I

( ) ( )( )

( )

==

~

o

IÎs K

f o ~ ff = I = I ( )= I( )= I

( ) ( )( )

( )

==

o

IÎs K


275/334


267

f ( ) ( )II ~ oI ( )IN = ~ ( )IO = o

( ) ( )[ ] =o

^IIs K

1092. ^~ ~ s m~ `~f p ~ -~ = I = I

( )= I ( )= I

( )

( )

==

p

^^ I

( ) =p

Is K

Figure 201.

1093. p~ ^~

Np

o

OO

+

+=


276/334


268

1094. j~ ~ i~~

( )

=

o

Î I

~~ ~ o ~ ~~ EIF ( )I K

1095. jq ~~ ~ -~ -~

( ) = o Îj K

q ~~ ~ -~

( ) =o

Îj K

q ~ ~ -~

( ) =o

O Îf K

q ~ ~ -~

( ) =o

O Îf K

q ~ ~ ( ) ( ) +=

o

OOM Îf K

1096. ` j~

( )

( )

( )

===

o

o

o

Î

Î

Î

N

j I


277/334


269

( )

( )

( )

===

o

o

o

Î

Î

Î

N

j K

1097. `~ ~ m~

( )=o

În I

~ ~ ~ o ~

~ ~ ~ EIF ( )I K

1098. ^~ ~ c

( )=o

Îp

NI

=o

^p K

9.11 Triple Integral

c ~~W ( )II I ( )II I

q ~W ( )

d

sII I ( )

d

sII I

o~ W ( )= = =

N

N

N II

p~ ~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


278/334


270

j~ ~ W aW ( )II

`~ ~W I I c W j I j I j

j ~W f I f I f I f I f I f I Mf

1099. a q f~q ~ ~ ~~ [ ] [ ] [ ]III~

( )[ ] [ ] [ ]

( ) IIIsII

N

N

N

M~M~M~

III~

= = =

=

II ~~

( ) ( )NNN III I ~ N = I

N

= IN

= K

1100. ( ) ( )[ ] ( ) ( ) +=+ddd

sIIsIIsIIII

1101. ( ) ( )[ ] ( ) ( ) =ddd

sIIsIIsIIII

1102. ( ) ( ) =dd

sIIsII I

~ ~K

1103. f ( ) MII ~ d ~ q ~ ~ ~I

( ) ( ) ( ) += qdqd

sIIsIIsII K

e qd d ~ qK


279/334


271

1104. b~~ q f~ o~ f~f d ( )II ~

( ) ( ) ( )IIIoI ON I

( ) ( )( )

( )

IIIIo

I

Id

O

N

=

I

o d -~K

f d ( )II ~

( ) ( ) ( ) ( )IIII~ ONON I

( ) ( )( )

( )

( )

( )

IIII

~

I

Id

O

N

O

N

=

1105. q f~ m~~f d ~ ~~ [ ] [ ] [ ]III~ I

( ) ( ) IIII

~

d

= K

f ~ ~ ~ ( )II ~ ~ ( ) ( ) ( ) ~

( ) ( ) ( ) ( )

=

~d

II K

1106. `~ s~~

( ) =d

II

( ) ( ) ( )[ ]( )( )

III

IIIIIIIIII

p

=


280/334


272

( )( )

M

IIII

= ~~

~~ ( ) ( )IIII I ~ p -~ d ~ ( )II= I

( )II =

( )II = dK

1107. q f~ `~ `~q ~ ~ ~

( )( )

II

II =

= K

i d ~ W( ) ( ) ( )IIIoI ON I

o d -~K q

( ) ( ) =pd

IIII

( )

( )

( )

( )

=

II

Io

I

I

O

N

K

e p ~ d ~ ~K

1108. q f~ p~ `~q a~ p~ `~

( )( )

=

=

II

II O

( ) =d

II


281/334


273

( ) =p

O II I

p ~ d ~ -

~K q ~ ~ M O I ~ ~ M K

Figure 202.

1109. s ~ p

=d

s

1110. s `~ `~

( )

sIIp

=

1111. s p~ `~

( )

=

s

IIp

O


282/334


274

1112. j~ ~ p

( )

=

d

sII I

~ d ~ ~~ ( )II ( )II K

1113. ` j~ ~ p

j

= I

j = I

j

= I

( ) =

d

sIIj I

( ) =d

sIIj I

( ) =d

sIIj

~ ~ ~ ~ M= IM = I M = I I ( )II K

1114. j f~ ~ -~ E M = FI -~E M= FI ~ -~ E M = F

( ) =d

O sIIf I

( ) = dO

sIIf I

( ) =d

O sIIf K

1115. j f~ ~ -~I -~I ~ -~

( ) ( ) +=+=d

OO sIIfff I

( ) ( ) +=+=d

OO sIIfff I


283/334


275

( ) ( ) +=+=d

OO sIIfff K

1116. m~ j f~

( ) ( ) ++=++=d

OOOM sIIffff

9.12 Line Integral

p~~ W ( )IIc I ( )Ic I ( )p~~ ~W ( )II`W `I N` I O`

i ~W ~I I I m~~W I m~ ~W I s W ( )oInImcr

m W ( )r

r W r

I r

I r

I r

^~ W pi ~ W ij~ ~ W aW

( )II I

( )I

`~ ~W I I

c W j I j I j

j ~W f I f I f

s ~ W stW t

j~ W _r

`W fb W j~ W


284/334


276

1117. i f~ ~ p~~ ci ~ ` ( )

rr= I

pM I ~ ~ ~~ c `Kq

( )( ) ( ) ==``

p

M

cIIccr

I

~ ~K

1118.

+=

ONON ````

ccc

Figure 203.

1119. f ` ~~ ( )rr

= I I

( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )

++= IIcIIc OOO

`

K

1120. f ` ~ -~ ~( ) = I ~ I

( ) ( )( ) ( )( ) +=

~

O

`

NIcIc K

1121. i f~ p~~ c m~ `~


285/334


277

( ) ( )

+=

IcIc

O

O

`

I

` ~ FE K

1122. i f~ s ci ~ ` ( )

rr= I

pM K q

( )== II

rr

~ K

Figure 204.

i ~ ( )oInImcr

`K

q ~ cr

~ `

( ) ++=++p

M`

onmonm K


286/334


278

1123. m i f~ s c

( ) ( )

=

``

ccrrrr

I

-` ~K

( ) ( ) ( ) ( ) +== ONON `````

ccccrrrrrrrr

I

` N` ~ O` K

1124. f ` ~~ ( ) ( ) ( ) ( )II =r

I I

=++`

onm

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )

++=

IIo

IIn

IIm

1125. f ` -~ ~ ~ ( ) = I

( )( ) ( )( )

+=+

~`

InImnm K

1126. d q

+= `onm

m

n I

( ) ( )InImcrrr

+= ~ -

~~ ~

m

In

~

~ oI ~ I `K


287/334


279

1127. ^~ ~ o o _ ` `

== `o ON

p

1128. m~ f i f~

q ~ ~ onmcrrrr

++= ~ ~ I ~ mI nI ~ o ~ ~ ~ aI ~ ~~ ( )II = E~ ~~ ~F a ~

~c =r

I m

= I n

= I o

=

K

q

( ) ( ) ( )^_onmc``

=++= rrr

K

1129. q ~ `~ c

^ ~c =r ~ ~ ~

K q ~ ~ onmcrrrr

++= ~ ~

M

onm

cr

rrr

r=

= K

f ~ ~ -~ ~

( ) ( )^_nm`

=+ I

~ ~~

nm

=

K


288/334


280

1130. i ~ `

( )

+

+

===

i

OOO

`

r

I

` ~ ~ - ( )

rI K

f ` -~I

( )

+

===

i

OO

`

r

K

f ` ~ ~ ( ) = -~ ( )~ I

+=

~

O

Ni K

1131. i ~ ` m~ `~

+

=

i O

O

I

` ~ ( )= I ~ ~K

1132. j~ ~ t

( )=`

II I

( )II ~ K

f ` ~ ~~ ( ) ( ) ( ) ( )II =r

I ~ ~

~


289/334


281

( ) ( ) ( )( )

+

+

=

II

OOO

K

f ` ~ -~I ~

( )=`

I I

( ) ( )( )

+

=

I

OO

E ~~ FK

1133. ` j~ ~ t

j

= I

j = I

j

= I

( ) = ` IIj I

( ) =`

IIj I

( ) =`

IIj K

1134. j f~q ~ ~ -~I -~I ~ -~~ ~

( ) ( ) +=`

OO IIf I

( ) ( ) +=`

OO IIf I

( ) ( )+=

`

OO

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