大一年级数学分析双语

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,数学分析,语码转换,1,实数及其性质,real number and its properties,有理数,rational numbers,无理数,irrational numbers,定义,definition,命题,proposition,加,plus,减,minus,乘,multiplied by,；,times,除,over; is to; divided by,2,确界原理,principles of supremum and,infimum,数集,set of the numbers,区间,interval,开区间,open interval,闭区间,closed interval,半开区间,semi-open interval,半闭区间,semi-closed interval,有限区间,finite interval,3,邻域,neighborhood,去心邻域,deleted neighborhood,和,sum,差,difference,积,product,商,quotient,数轴,number axis; number line,封闭性,closeness,稠密性,the density,无限区间,infinite interval,4,有界集,bounded set,无界集,unbounded set,下确界,infimum (or greatest lower bound,),函数的定义,definition of function,定义域,domain of definition,值域,region 5ri:dVEn,5,数列极限,the limits of sequences,发散序列,divergent sequence,无穷小序列,infinitesimal sequence,收敛序列,convergent sequence,6,唯一性定理,uniqueness theorem,有界性定理,boundedness theorem,保号性,inheriting order properties,保不等式性,inheriting inequality,子列,subsequence,7,严格递增,strictly increasing,单调递增序列,monotone increasing sequence,单调递减序列,monotone decreasing sequence,严格递减,strictly decreasing,必要条件,necessary condition,充分条件,sufficient condition,夹逼定理,squeeze principle,8,函数极限,the limits of functions,无穷极限,infinite limits,单侧极限,one-sided limit,右（左）极限,right (left) limit ; right(left),hand limit,函数极限的性质,properties of limits of functions,9,局部有界性,local boundedness,海涅定理,Heine theorem,无穷大量,infinities,无穷小量的阶,the order of infinitesimal,高（低）阶无穷小量,infinitesimal of higher (lower) order,10,同阶无穷小量,infinitesimal of the same order,等价无穷小量,equivalent infinitesimals,垂直渐近线,vertical asymptote,斜渐近线,oblique asymptote,水平渐近线,horizontal asymptote,11,f,在点,x,连续,f is continuous at a point x,函数的增量,increment of function,自变量的增量,increment of independent,variable,12,右（左）连续,right(left) continuous,有界性定理,boundedness theorem,间断点及其分类,discontinuity point,and classification,可去间断点,removable discontinuity,跳跃间断点,jump discontinuity,第一类间断点,discontinuity of the first kind,第二类间断点,discontinuity of the second kind,13,局部有界性,local boundedness,连续函数的局部性质,local properties of continuous functions,连续函数的复合性质,composition property for continuous functions,闭区间上连续函数的性质,properties of continuous functions over closed interval,14,最大值和最小值定理,maximum and minimum value theorem,介值性定理,intermediate value theorem,零点定理,zero-point theorem,一致连续性定理,uniform continuity theorem,反函数的连续性,continuity of inverse function,初等函数的连续性,continuity of elementary function,15,导数与微分,Derivatives and Differentials,有限增量公式,finite increment formula,左导数,left derivative,右导数,right derivative,单侧导数,one-sided derivative,导函数,derivative function,可导函数,derivable function,16,导数的几何意义,geometric meaning of derivative,费马定理,Fermat theorem,导函数的介值定理,intermediate value theorem of derivative function,导数的四则运算,the algebra of derivatives,17,和的导数,derivative of a sum,差的导数,derivative of a difference,积的导数,derivative of a product,商的导数,derivative of a quotient rule,反函数的导数,derivatives of inverse functions,复合函数的导数,derivatives of composite functions,18,圆的参数方程,parametric equation of circle,椭圆的参数方程,parametric equation of ellipse,摆线，旋轮线的参数方程,parametric equation of cycloid,星形线的参数方程,parametric equation of asteroid,19,对数求导法,logarithmic derivative,基本初等导数公式,derivative formulas of basic elementary function,二阶导数,second derivative,三阶导数,third derivative,四阶导数,fourth derivative,n,阶导数,nth derivative,20,微分的概念,concept of differential,线性主部,linear principle part,自变量的微分,differential of the independent variable,加速度,acceleration,微分的运算法则,operational rules of differentials,物理解释,physical interpretation,21,和的极限等于极限的和,.,The limit of a sum is equal to the sum of the limits.,幂的极限等于极限的幂,.,The limit of a power is equal to the power of the limits.,收敛数列的和收敛于极限的和,.,The sum of convergent sequences convergences to the sum of the limits.,22,达布定理,Darboux theorem,高阶微分,higher differential,微分形式的不变性,preservation of the form of differential,微分的几何意义,geometric meaning of a differential,23,微分中值定理及其应用,Mean Value Theorems of Differentials and Its Applications,罗尔中值定理,Rolle mean value theorem,拉格朗日中值定理,Lagrange mean value theorem,24,柯西中值定理,Cauchy mean value theorem,洛必达法则,Lhospitals Rule,0/0,型不定式极限,limits of indeterminate forms of the type 0/0,/,型不定式,indeterminate forms of the type,/,25,泰勒公式的余项,Remainder of Taylor formula,带有佩亚诺型余项的泰勒公式,Taylor formula with Peano Remainder,带有拉格朗日型余项的麦克劳林公式,Maclaurin formula with Lagrange Remainder,带有拉格朗日型余项的泰勒公式,Taylor formula with Lagrange Remainder,26,函数的极值,extreme values of function,极值判别,test of extreme value,凸函数,convex function,凹函数,concave function,极值的第一充分条件,the first sufficient condition of extreme values,极值的第二充分条件,the second sufficient condition of extreme valu,es,27,函数的凸点与拐点,convexity of a function and inflection,严格凸函数,strictly convex function,严格凹函数,strictly concave function,28,换元积分法,integration by substitution,第一换元公式,the formula of the first kind of substitution,第二换元公式,the formula of the second kind of substitution,分部积分法,integration by parts,29,有理函数的不定积分,indefinite integrals of rational functions,真分式,proper fraction,假分式,improper fraction,部分分式分解,decomposition into partial fractions,待定系数法,method of the undetermined coefficients,30,被积函数,integrand function,积分符号,integral sign,被积表达式,element of integration,积分变量,integral variable,积分常数,integral constant,31,定积分,definite integrals,曲边梯形,curvilinear trapezoid,分割,dividing; partition,近似求和,summing approximately,取极限,taking the limit,32,黎曼和,Riemann sums,黎曼积分,Riemann integrals,黎曼可积,integrable in the sense of,Riemann,积分区间,interval of integration,上限,upper limit,下限,lower limit,33,可积的必要条件,a necessary condition for the integrability,可积的充要条件,necessary and sufficient conditions for the integrability,达布小和,Darboux lower sum,达布大和,Darboux upper sum,可积函数类,class of integrable functions,34,定积分的线性性质,linearity of the definite integral,关于积分区间的可加性,additive with respect to integral intervals (additivity over integrals),积分不等式性,monotonicity of the integral(comparison property),积分中值定理,mean value theorems of integrals,积分第一中值定理,first mean value theorem of integrals,积分第一中值定理的推广,generalized first mean value theorem of integrals,35,变限积分,integral with variant upper limit,原函数的存在性,the existence of the primitive,原函数的存在定理,existence theorem of the primitive,定积分换元积分法,the substitution method for definite integrals,定积分分部积分法,integration by parts for definite integrals,泰勒公式的积分型余项,integral form of remainder of Taylor formula,36,定积分的应用,Applications of Definite Integrals,平面图形的面积,areas of plane figures,由平行截面面积求体积,volume of a,solid with known parallel sections,37,平面曲线的弧长与曲率,arclength of plane curves and curvature,光滑曲线,smoothed curve,弧微分,differential of arc length,曲率圆,circle of curvature,曲率半径,radius of curvature,曲率中心,center of curvature,38,旋转曲面的面积,area of a surface of revolution,定积分在物理中的某些应用,applications of definite integral in physics,功与平均功率,work and average power,定积分的近似计算,approximate computation of definite integral,梯形法,trapezoidal method,抛物线法,parabola method,引力,gravitation,39,反常积分的概念,notion of improper integrals,无穷限反常积分,improper integrals with infinite limits,无界函数的反常积分,improper integrals with an unbounded integrand,收敛,convergent,40,无穷积分的性质与收敛判别,Properties,of in finite integral and tests of,convergence,比较判别法,comparison test,柯西判别法,Cauchys test,狄利克雷判别法,Dirichlets test,阿贝尔判别法,Abels test,绝对收敛,absolutely convergent,条件收敛,conditional convergent,41,关于实数集完备性的基本定理,fundamental theorems of completeness in the set of real numbers,区间套定理,the nested interval theorem,柯西收敛准则,Cauchy,s general principle of convergence,聚点定理,accumulation theorem,42,有限覆盖定理,finite covering theorem,实数完备性基本定理的等价性,equivalence of the completeness of real numbers,43,级数的收敛性,convergence of series,部分和序列,sequence of partial sums,几何级数,geometric series,调和级数,harmonic series,级数收敛的柯西准则,Cauchy,convergence criterion for series,收敛的必要条件,a necessary condition,for convergence,44,正项级数,positive term series,比较原则,comparison test,比式判别法,(,达朗贝尔判别法,) DAlemberts ratio test,根式判别法,root test,正项级数收敛的一般判别原则,general criterion of convergence for positive terms series,45,比式判别法的极限形式,limit form of ratio test,柯西判别法（根式判别法）,Cauchys root test,根式判别法的极限形式,limit form of root test,高斯判别法,Gausss test,积分判别法,integral test,拉贝判别法,Raabes test,46,一致收敛性,uniform convergence,收敛域,convergence domain; region of convergence,一致收敛于,f converge uniformly to f,函数项级数及其一致收敛性,series of functions and uniform convergence,一致有界,uniform boundedness,47,一致收敛判别法,test of uniform convergence,魏尔斯特拉斯判别法,Weierstrass,s test;,test of uniform convergence ; the,Weierstrass uniform convergence,Criterion;Weierstrass M-test for uniform convergence,狄利克莱判别法,Dirichlet,s test,48,幂级数,power series,收敛区间,interval of convergence,收敛半径,radius of convergence,幂级数的运算,operations of power series,49,泰勒级数,Taylor-series,初等函数的幂级数展开式,the power series expansion of elementary functions,复变量的指数函数,exponential function of a complex variable,欧拉公式,Euler,s Form,ula,50,三角傅里叶级数,trigonometric Fourier series,正交函数系,orthogonal function stem,简谐振动,simple harmonic vibration,角频率,angular frequency,收敛定理,convergence theorem,分段光滑,piecewise smooth,傅里叶系数,Fourier coefficient,51,收敛定理,convergence theorem,奇函数与偶函数的傅里叶级数,Fourier series of even and odd functions,振幅,amplitude,以 为周期的函数的傅里叶级数,Fourier series for functions of period,52,