例Give
ε0thereexistsapositive
umberNsuchthataaεforall
≥
N译文使得a都成立译文对给定的ε0存在一个正数N使得aε对所有的
≥N都成立
例2Si
cehxisharmo
ico
a
eighborhoodofBarwehave译文内调和译文因为hx在某个邻域Bar内调和故在某个邻域内调和
B
∫hxdσxha
例该等式成立的充分必要条件是a0且b3译文：译文：Thesufficie
ta
d
ecessaryco
ditio
fortheequalityisa0a
db3Theequalityisvalidwhe
a
do
lywhe
a0a
db3Theequalityisvalidifa
do
lyifa0a
db3Theequalityisvalidiffa0a
db3
例对任意数ε0存在一个数d译文译文Forevery
umberwhe
everx－x0d－0使得只要x－x0d就有fx－fx0使得只要－就有
ε
ε0thereexistsa
umberd0suchthatfx－fx0ε－
使得例对取定的ε0存在d0使得fx－fx0－立
ε对所有满足－x0d的x都成对所有满足x－
译文译文Give
ε0thereexistsd0suchthatfx－fx0εforallxwithx－x0d－例2Equatio
sareoftwoki
dside
titiesa
dequatio
sofco
ditio
thelatteriscalledequatio
sforshort等式分为两类，恒等式和条件等式，后者简称为方程等式分为两类，恒等式和条件等式，后者简称为方程例3A
arithmeticora
algebraicide
tityisa
equatio
译文代数和算数的恒等式都是等式译文代数和算数的恒等式都是等式例4Afterchecki
gwesee－3does
otsatisfytheorigi
alequatio
but3does译文：经检验得知不满足原方程但是却是方程的根译文：经检验得知－3不满足原方程但是3却是方程的根例5Thesquarerootofa
egative
umberisapureimagi
ary译文：负数的平方根是纯虚数
f例Thevaluesoftheco
structedfu
ctio
should
otexceedthemaximumpermissible译文：构造的函数的值不应超过所允许的最大数值例11Wecallatria
glea
obtusetria
glewhe
o
ea
gleisa
obtusea
gle译文：有一个角为钝角的三角形被称为钝角三角形例12Agreatdealofpracticalproblemsca
besolvedwiththediffere
tialequatio
s译文：使用微分方程人们可以解决大量的实际问题22Themostfamousqua
tityi
mathematicsistheratioofthecircumfere
ceofacircletoitsdiameterwhichisalsok
ow
asthe
umberpia
dde
otedbytheGreekletterp23“Li
earalgebra”isthestudyofli
earsetsofequatio
sa
dtheirtra
sformatio
propertiesLi
earalgebraallowsthea
alysisofrotatio
si
spaceleastsquaresfitti
gsolutio
ofcoupleddiffere
tialequatio
sdetermi
atio
ofacirclepassi
gthroughthreegive
poi
tsaswellasma
yotherproblemsi
mathematicsphysicsr