1、数学常用短语与从句(Phrasesorclausesfrequentlyusedinmathematics)常见的涉及运算的短语或句子1.Differentiatebothsidesoftheequationandweget.Integratebothsidesoftheequationandweget.2.Differentiatingbothsidesoftheequation,weget.Integratingbothsidesoftheequation,weget.3.Add(A)to(B)andwehave(其中(A)、(B) 为某些表达式，如不等式、等式、方程等，下同)4Subtr
2、act(B)from(A)andwehave5.Multiplyingeachtermoftheequationby,weobtain6.Dividingtheequationthroughby.,wehave.7.(A)and(B)togethergive(A)and(B)togetheryield(A)and(B)togetherimply8.Comparing(A)with(B),itiseasytoseethat9.Substituting(A)into(B),weobtain10.Eliminating(theparameter)tfrom(A)and(B),wehave11.Byi
3、ntroducinganewvariable,wecanthenrewrite(A)asfollowsByintroducinganewvariable,wecanthenrewrite(A)inthefollowingform12.Byasimplecalculation,weobtainfrom(A)定理证明过程中常见的短语和句子1.下面的句型可用来表达“根据什么即可得到什么 ”的意思Accordingtodefinition,itfollowsAccordingtohypothesis,itfollowsAccordingtoassumptions,itfollowsAccordingt
4、otheorem(N),itfollowsAccordingtolemma(A),itfollowsAccordingtocorollary(B),itfollowsAccordingtotheremark,itfollowsAccordingtothefactthat,itfollows(可以把上面的“accordingto”换成“by”)Since,itfollows2.如果一个论断可以通过一些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可以自行推算，因而只需直接写出论断来，这时可用下面句型：(1)ItiseasytoseethatItiseasytoshowthatItis
5、easytoprovethatItiseasytoverifythatItiseasytocheckthat(2)ItcaneasilybeseenthatItcaneasilybeshownthatItcaneasilybeprovedthatItcaneasilybeverifiedthatItcaneasilybecheckedthat3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：(1)ItisclearthatItisobviousthatItisevidentthatItiswell-knownthat(2)Clearly,Obviously,E
6、vidently,4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：LetusfirstdefinethefunctionLetusintroduceanewfunctionLetusconsiderthefunctionLetusfirstinvestigatethefunctionLetSetDefinePutConsider5.在一个定理中，有几个结论需要证明，其中有些结论比较明显，可不用证明，仅需证明余下结论即可，这时可用下面句型：Since(A)and(B)areobvious,weneedonlyprove(C).Since(A)and(B)aretrivial,wene
7、edonlyprove(C).Since(A)and(B)aretrivial,itsufficestoprove(C)6.为了证明一个定理，有时我们并不是直接去证明，而是证明一个新的论断，一旦新的论断得到证明，已给定理不难由此而得证，这时可用下面句型：以下各句用于新的论断被证明之前：ThetheoremwillbeprovedifwecanshowTheresultwillbeprovedifwecanshowThetheoremwillbeprovedbyshowingthatIfwecanprovethenthetheoremfollowsimmediately.以下各句用于新的论断被证
8、明之后：Thetheoremisnowadirectconsequenceofwhatwehaveproved.Thetheoremfollowsimmediatelyfromwhatwehaveproved.Thetheoremisnowevidentfromwhatwehaveproved.Itisevidenttoseethatthetheoremholds.7.在证明过程中，有时要用到一些早已学过的知识或技巧，这时可用下面句子，以提醒读者：RecallthatNoticethatNotethatObservethatInordertoprovethetheorem,weneedthek
9、nowledgeofInordertoobtainthefollowingequation,weneed8.如果需要证明的定理的假设条件是一般条件，但是，只要定理在特殊条件下成立，就不难推出定理在一般条件下也成立，这时仅需要在特殊情况下去证明定理就够了，为此可用下面句型：Withoutlossofgenerality,wemayconsiderWithoutlossofgenerality,wemayassumeWithoutlossofgenerality,wemayprovethetheoreminthecaseItsufficestoprovethetheoreminthecaseWen
10、eedonlyconsiderthecaseForsimplicity,wemaytake9.如果待证的论断可用以前用过的相似的方法或步骤进行证明，则可用下面句型：Thistheoremcanbeprovedinthesamewayasshownbefore.Thisstatementcanbeprovedinasimilarwayasshownbefore.Thistheoremcanbeprovedbythesamemethodasemployedinthelastsection.Thistheoremcanbecompletedbythemethodanalogoustothatused
11、above.UsingthesameargumentasintheproofoftheoremN,wecaneasilycarryouttheproofofthistheorem.WenowproceedasintheproofoftheoremN.WeshalladoptthesameprocedureasintheproofoftheoremN.10.如果我们用的是反证法，则其开头及结尾可用下面句型：Ifthestatement(orassertion,conclusion)werefalse(ornottrue,notright)thenIftheassertionwouldnothol
12、d,thenThisiscontrarytoThiscontradictsthefactthatThisleadstoacontradiction.11.表示定理已证毕或者把前面所证的总结为一结论Wehavethusprovedthetheorem.Thiscompletestheproof.Theproofofthetheoremisnowcompleted.Itisnowobviousthatthetheoremholds.ThuswehavederivedthatConsequently,weinferthatThusweconcludethatThusweareledtotheconc
13、lusionthatThuswearriveattheconclusionthatThuswecansummarizewhatwehaveprovedasthefollowingtheorem.12.其它Thereexist(s)suchthatWeclaiminfactWearenowinapositiontoIfotherwiseProvidedthat第九章微分方程Chapter12DifferentialEquation解微分方程 solveadirrerentialequation常微分方程 ordinarydifferentialequation偏微分方程 partialdiffe
14、rentialequation,PDE微分方程的阶 orderofadifferentialequation微分方程的解 solutionofadifferentialequation微分方程的通解 generalsolutionofadifferentialequation初始条件 initialcondition微分方程的特解 particularsolutionofadifferentialequation初值问题 initialvalueproblem微分方程的积分曲线 integralcurveofadifferentialequation可分离变量的微分方程 variablesep
15、arabledifferentialequation隐式解 implicitsolution隐式通解 inplicitgeneralsolution衰变系数 decaycoefficient衰变 decay齐次方程 homogeneousequation一阶线性方程 lineardifferentialequationoffirstorder非齐次 non-homogeneous齐次线性方程 homogeneouslinearequation非齐次线性方程 non-homogeneouslinearequation常数变易法 methodofvariationofconstant暂态电流 tr
16、ansientstatacurrent稳态电流 steadystatecurrent伯努利方程 Bernoulliequation全微分方程 totaldifferentialequation积分因子 integratingfactor高阶微分方程 differentialequationofhigherorder悬链线 catenary高阶线性微分方程 lineradifferentialequationofhigherorder自由振动的微分方程 differentialequationoffreevibration强迫振动的微分方程 differentialequationofforce
17、doscillation串联电路的振荡方程 oscillationequationofseriescircuit二阶线性微分方程 secondorderlineradifferentialequation线性相关 linearlydependence线性无关 linearlyindependce二阶常系数齐次线性微分方程secondorderhomogeneourlineardifferentialequationwithconstantcoefficient二阶变系数齐次线性微分方程secondorderhomogeneouslineardifferentialequationwithvar
18、iablecoefficient特征方程 characteristicequation无阻尼自由振动的微分方程 differentialequationoffreevibrationwithzerodamping固有频率 naturalfrequency简谐振动 simpleharmonicoscillation,simpleharmonicvibration微分算子 differentialoperator待定系数法 methodofundeterminedcoefficient共振现象 resonancephenomenon欧拉方程 Eulerequation幂级数解法 powerseriessolution数值解法 numerialsolution勒让德方程 Legendreequation微分方程组 systemofdifferentialequations常系数线性微分方程组 systemoflineradifferentialequationswithconstantcoefficient
