高等數學（上冊）（英文）

Chapter1Functionsandlimits1
1.1Mappingsandfunctions1
1.1.1Sets1
1.1.2Mappings4
1.1.3Functions5
Exercises1-119
1.2Limitsofsequences23
1.2.1Conceptoflimitsofsequences23
1.2.2Propertiesofconvergentsequences27
Exercises1-229
1.3Limitsoffunctions30
1.3.1Definitionsoflimitsoffunctions30
1.3.2Thepropertiesoffunctionallimits33
Exercises1-334
1.4Infinitesimalandinfinityquantity36
1.4.1Infinitesimalquantity36
1.4.2Infinityquantity36
Exercises1-438
1.5Rulesoflimitoperations38
Exercises1-543
1.6Principleoflimitexistence—twoimportantlimits44
Exercises1-649
1.7Comparingwithtwoinfinitesimals50
Exercises1-752
1.8Continuityoffunctionsanddiscontinuouspoints52
1.8.1Continuityoffunctions52
1.8.2Discontinuouspointsoffunctions54
Exercises1-856
1.9Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions57
1.9.1Continuityofthesum，difference，productandquotientofcontinuousfunctions57
1.9.2Continuityofinversefunctionsandcompositefunctions58
1.9.3Continuityofelementaryfunctions59
Exercises1-959
1.10Propertiesofcontinuousfunctionsonaclosedinterval60
1.10.1Boundednessandmaximum-minimumtheorem60
1.10.2Zeropointtheoremandintermediatevaluetheorem61
1.10.3Uniformcontinuity62
Exercises1-1063
Exercises163

Chapter2Derivativesanddifferential66
2.1Conceptofderivatives66
2.1.1Examples66
2.1.2Definitionofderivatives70
2.1.3Geometricinterpretationofderivative77
2.1.4Relationshipbetweenderivabilityandcontinuity78
Exercises2-179
2.2Fundamentalderivationrules81
2.2.1Derivationrulesforsum，difference，productandquotientoffunctions81
2.2.2Therulesofderivativeofinversefunctions83
2.2.3Therulesofderivativeofcompositefunctions（TheChainRule）85
2.2.4Basicderivationrulesandderivativeformulas89
Exercises2-291
2.3Higher-orderderivatives93
Exercises2-395
2.4Derivationofimplicitfunctionsandfunctionsdefinedbyparametricequations97
2.4.1Derivationofimplicitfunctions97
2.4.2Derivationofafunctiondefinedbyparametricequations101
2.4.3Relatedratesofchange103
Exercises2-4103
2.5TheDifferentialsoffunctions105
2.5.1Conceptofthedifferential105
2.5.2Geometricmeaningofthedifferential107
2.5.3Formulasandrulesondifferentials108
2.5.4Applicationofthedifferentialinapproximatecomputation109
Exercises2-5110
Exercises2111

Chapter3Meanvaluetheoremsindifferentialcalculusandapplicationsofderivatives113
3.1Meanvaluetheoremsindifferentialcalculus113
Exercises3-1120
3.2L’Hospital’srules121
Exercises3-2125
3.3Taylorformula126
Exercises3-3130
3.4Monotonicityoffunctionsandconvexityofcurves131
3.4.1Monotonicityoffunctions131
3.4.2Convexityofcurvesandinflectionpoints132
Exercises3-4136
3.5Extremevaluesoffunctions,maximumandminimum137
3.5.1Extremevaluesoffunctions137
3.5.2Maximumandminimumoffunction140
Exercises3-5143
3.6Differentiationofarcandcurvature145
3.6.1Differentiationofanarc145
3.6.2curvature146
Exercises3-6149
Exercises3149

Chapter4Indefiniteintegral151
4.1Conceptandpropertyofindefiniteintegral151
4.1.1Conceptofantiderivativeandindefiniteintegral151
4.1.2Tableoffundamentalindefiniteintegrals153
4.1.3Propertiesoftheindefiniteintegral155
Exercises4-1157
4.2Integrationbysubstitutions158
4.2.1Integrationbysubstitutionofthefirstkind158
4.2.2Integrationbysubstitutionofthesecondkind163
Exercises4-2167
4.3Integrationbyparts169
Exercises4-3173
4.4Integrationofrationalfunction173
4.4.1Integrationofrationalfunction173
4.4.2Integrationwhichcanbetransformedintotheintegrationofrationalfunction175
Exercises4-4177
Exercises4178

Chapter5Definiteintegrals180
5.1Conceptandpropertiesofdefiniteintegrals180
5.1.1Examplesofdefiniteintegralproblems180
5.1.2Thedefinitionofdefineintegral182
5.1.3Propertiesofdefiniteintegrals184
Exercises5-1186
5.2Fundamentalformulaofcalculus188
5.2.1Therelationshipbetweenthedisplacementandthevelocity188
5.2.2Afunctionofupperlimitofintegral188
5.2.3Newton-Leibnizformula189
Exercises5-2192
5.3Integrationbysubstitutionandpartsfordefiniteintegrals194
5.3.1Integrationbysubstitutionfordefiniteintegrals194
5.3.2Integrationbypartsfordefiniteintegral198
Exercises5-3199
5.4Improperintegrals201
5.4.1Improperintegralsonaninfiniteinterval201
5.4.2Improperintegralsofunboundedfunctions203
Exercises5-4205
5.5TestsforconvergenceofimproperintegralsΓ-function206
5.5.1Testforconvergenceofinfiniteintegral206
5.5.2Testforconvergenceofimproperintegralsofunboundedfunctions209
5.5.3Γ-function209
Exercises5-5211
Exercises5212

Chapter6Applicationsofdefiniteintegrals214
6.1Methodofelementsfordefiniteintegrals214
6.2Theapplicationsofthedefiniteintegralingeometry215
6.2.1Areasofplanefigures215
6.2.2Thevolumesofsolid219
6.2.3Lengthofplanecurves222
Exercises6-2224
6.3TheapplicationsofthedefiniteIntegralinphysics227
6.3.1Workdonebyvariableforce227
6.3.2Forcebyaliquid228
6.3.3Gravity229
Exercises6-3230
Exercises6230

Chapter7Differentialequations232
7.1Differentialequationsandtheirsolutions232
Exercises7-1236
7.2Separableequations237
Exercises7-2240
7.3Homogeneousequations241
7.3.1Homogeneousequations241
7.3.2Reductiontohomogeneousequation243
Exercises7-3245
7.4Afirst-orderlineardifferentialequations245
7.4.1Linearequations245
7.4.2Bernoulli’sequation248
Exercises7-4249
7.5Reduciblesecond-orderequations250
Exercises7-5254
7.6Second-orderlinearequations254
7.6.1Constructionofsolutionsofsecond-orderlinearequation254
7.6.2Themethodofvariationofparameters257
Exercises7-6259
7.7Homogeneouslineardifferentialequationwithconstantcoefficients259
Exercises7-7263
7.8Nonhomogeneouslineardifferentialequationwithconstantcoefficients264
Exercises7-8270
7.9Euler’sdifferentialequation270
Exercises7-9271
Exercises7271

Appendix273
References280

Englishisthemostimportantlanguageininternationalacademia.Inordertostrengthenacademicexchangewithwesterncountries,manyuniversitiesinChinapaymoreandmoreattentiontothebilingualteachinginclassroomsinrecentyears.Consideringtheimportanceofadvancedmathematicsandscarcityofbilingualmathematicstextbook,wehavewrittenthisbook.

Themainsubjectofthisbookiscalculus.Besides,italsoincludesdifferentialequation,spaceanalyticgeometry,vectoralgebraandinfiniteseries.Thisbookisdividedintotwovolumes.Thefirstvolumecontainscalculusoffunctionsofasinglevariableanddifferentialequation.Thesecondvolumecontainsspaceanalyticgeometryandvectoralgebra,calculusofmultivariatefunction,curveintegralandsurfaceintegral,infiniteseries.

Wehaveattemptedtogivethisbookthefollowingcharacteristics.

①ThemainusersofthisbookarethoseforeignstudentsstudiedinChina.Therearelotsofdifferencesintheirmathematicalabilitybecausetherearenostandardadmissiontestforenrollingforeignstudents.Throughtheteachingexperienceinthepast5years,inthisedition,webeginwithpreteststoassessthenecessarymathematicalability.

②ThecontentsofthisbookarebasedontheChinesetextbook“AdvancedMathematics(seventhedition)”whichiswrittenbydepartmentofmathematicsofTongjiUniversity.ThereadersmayreadthisbookandusetheChinesetextbook“AdvancedMathematics”asareference.ItmayhelpreaderstounderstandthemathematicalcontentsandtoimprovetheleveloftheirEnglish.

③Inordertotrainthemathematicalqualityandabilityofthestudents,weusesomemodernideas,languageandmethodsofmathematics.Wealsobringinsomemathematicalandlogicalsymbol.

④Wepaymoreattentiontotheapplicationofmathematicsinpracticalproblems.Wehaveaddedsomeadditionalexamplesandexercisesinphysics,chemistry,economiesandevendailylife.

⑤Consideringthedifferentteachingrequirementsindifferentuniversity,wemarksomedifficultsectionsandexercisesbythesymbol“*”.Teachersandstudentsmaychoosesuitablecontentsasrequired.

Inthisvolume,Chapter1toChapter3arewrittenbyBinPan,Chapter4iswrittenbyJinqiuLi,Chapter5iswrittenbyJingxianYu,Chapter6iswrittenbyHongNiu,Chapter7iswrittenbyLiChen.AllthechaptersarecheckedandrevisedbyBinPan.

Wehopethisbookcanbringreaderssomehelpinthestudyingandteachingofbilingualmathematics.Duetothelimitofourability,itisimpossibletoavoidsomeunclearexplanations.Wewouldappreciateanyconstructivecriticismsandcorrectionsfromreaders.

Authors
2019-5