(高等数学英文课件)8.2 Subsequences Bounded Sequences
高数双语课件section1_5.pptx
kind [第一类间断点] of the function; all other discontinuous points are called discontinuity of the second kind[第二类间断点].
y
y
O
x
First kind
x O
Second Kind
11
The Classification of Discontinuous Points
Finish.
7
The Continuity of Function
2x 1, 1 x 0
Example
Prove
f
(
x
)
x
2
3,
0 x1
does not continuous at
x0 .
Proof Since f (0) 3 and
xlim0 f ( x) xlim0( x2 3) 3 f (0)
x) sin( x0)
2cos
x0
x 2
sin
x 2
then
lim
x0
y
2
lim cos
x0
x0
x 2
sin
x 2
0.
Hence sin x is continuous at x x0. Since, x0 is arbitrary point
in the interval (,), we have sin x C(,) .
(
x0
)
x x0
lim
x x0
f (x) f (x)
f ( x0)
.
f ( x0)
4
The Continuity of Function
微积分教学资料——chapter8.1,8.2-文档资料
For example:
(1){1}: 1, 1 , 1 , 1 , n 23 n
(2){ (1)n1 1}: 1, 1 , 1 ,(1)n1 1 ,
n
23
n
(3){n (1)n }: 0, 3 , 2 , 5 n (1)n ,
n
234
n
(4){8}: 8,8,8,88
1 lim 0 n n 2
Theorem:
Example:
Find lim n3 n n n3 2n2 1
Solution:
1
lim
n3 n
1
lim
n2
1
n n3 2n2 1
n
1
2
1
n n3
The Sandwich Theorem for Sequences 'sænwidʒ]
n2
We should choose
N
1
2.Showing that this N works.
given 0, Let N 1
If
then
1 n2
0
n N,
Therefore , by the definition of a limit,
2
2
example
Find a formula for the general term an of the sequence assuming that the pattern of the first few terms continues.
1, 2 , 4 , 8 , 3 9 27
subsequence的短语
subsequence的短语单词：subsequence1.1 词性：名词1.2 释义：随后；接着；后续；子序列（在数学或计算机科学等领域中，指序列中的一部分，其元素按原序列顺序排列）1.3 英文解释：A following part or sequence; in mathematics or computer science, a subset of a sequence where the elements are in the same order as in the original sequence.1.4 相关词汇：同义词有 sequel，succession；派生词有 subsequent（形容词，随后的；后来的）。
---2 起源与背景2.1 词源：源于拉丁语“sub-”（在……之下；低于；次于）和“sequi”（跟随），组合起来表示在某个序列之后的部分或跟随而来的情况。
2.2 趣闻：在一些古老的家族族谱记载中，会详细描述家族成员的出生顺序以及后续（subsequence）的重大事件，这些记录不仅是家族传承的一部分，也为研究当时的社会文化和人口发展提供了有价值的线索。
例如，通过分析不同家族的 subsequence 信息，可以了解到特定时期的婚姻习俗、寿命情况等。
---3 常用搭配与短语3.1 短语：- in subsequence：在随后；接下来。
例句：We will discuss the details in subsequence. 翻译：我们将在接下来讨论细节。
- subsequence of events：事件的后续发展。
例句：The subsequence of events led to a complete change in the situation. 翻译：事件的后续发展导致了局势的彻底改变。
- a subsequence of numbers：数字子序列。
(高等数学英文课件)1.3 Limits Involving Infinity
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1.3
Limits Involving Infinity
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1.3.1 Finite Limits as x … 1.3.2 Rules for Limits 1.3.3 Horizontal Asymptotes 1.3.4 Infinite Limits 1.3.5 Vertical Asymptotes 1.3.6 Oblique Asymptotes
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1.3.4
Infinite Limits
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x
x1
0
上式中分子多项式的次数应为零,
故有1－a=0,a－b=0,由此得 a=b=1
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1.3.3
Horizontal Asymptotes
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limsinx, limsinx.
x
x
11
lim , lim .
x x x0
x0
_________________________________________________
Longest common subsequence
If I is an IS of π with length equal to the size of a cover C of π, then I is a LIS of π and C is the smallest cover of size c.
5
Proof
If I is an increasing sequence, it cannot contain more than one element from a decreasing sequence. This means that no increasing subsequence can have size more than the size of any cover C, that is, if a maximum of one element from each can participate in any increasing sequence. Thus, an IS derived from this decomposition can have a maximum length of |C |=c. Conversely, C must be the smallest. If not, let c’ be the length of a cover C’ such that |C’|=< c i.e., if we derive IS from C, it must contain more than one element from one of the decreasing sequence of C’, which is not possible. Hence C has to be of smallest size.
高等数学(微积分学)专业术语名词、概念、定理等英汉对照.
目录第一部分英汉微积分词汇Part 1 English-Chinese Calculus Vocabulary第一章函数与极限Chapter 1 function and Limit (1)第二章导数与微分Chapter 2 Derivative and Differential (2)第三章微分中值定理Chapter 3 Mean Value theorem of differentials and theApplicati on of Derivatives (3)第四章不定积分Chapter 4 Indefinite Intergrals (3)第五章定积分Chapter 5 Definite Integral (3)第六章定积分的应用Chapter 6 Application of the Definite Integrals (4)第七章空间解析几何与向量代数Chapter 7 Space Ana lytic Geomertry and Vector Algebra (4)第八章多元函数微分法及其应用Chapter 8 Differentiation of functions Several variablesand Its Application (5)第九章重积分Multiple Integrals (6)第十章曲线积分与曲面积分Chapter 10 Line(Curve ) Integrals and Sur face Integrals……………………6 第十一章无穷级数Chapter 11 Infinite Series……………………………………………………6 第十二章微分方程Chapter 12 Differential Equation (7)第二部分定理定义公式的英文表达 Part 2 English Expression for Theorem, Definition and Formula第一章函数与极限Chapter 1 Function and L imit (19)1.1 映射与函数(Mapping and Function ) (19)1.2 数列的极限(Limit of the Sequence of Number) (20)1.3 函数的极限(Limit of Function) (21)1.4 无穷小与无穷大(Infinitesimal and Inifinity) (23)1.5 极限运算法则(Operation Rule of L imit) (24)1.6 极限存在准则两个重要的极限(Rule for theExistence of Limits Two Important Limits) (25)1.7 无穷小的比较(The Comparison of infinitesimal) (26)1.8 函数的连续性与间断点(Continuity of FunctionAnd Discontinuity Points) (28)1.9 连续函数的运酸与初等函数的连续性(OperationOf Continuous Functions and Continuity ofElementary Functions) (28)1.10 闭区间上联系汗水的性质(Properties ofContinuous Functions on a Closed Interval) (30)第二章导数与数分Chapter2 Derivative and Differential (31)2.1 导数的概念(The Concept of Derivative) (31)2.2 函数的求导法则(Rules for Finding Derivatives) (33)2.3 高阶导数(Higher-order Derivatives) (34)2.4 隐函数及由参数方程所确定的函数的导数相关变化率(Derivatives of Implicit Functions and Functions Determined by Parametric Equation and Correlative Change Rate) (34)2.5 函数的微分(Differential of a Function) (35)第三章微分中值定理与导数的应用Chapter 3 Mean Value Theorem of Differentials and theApplication of Derivatives (36)3.1 微分中值定理(The Mean Value Theorem) (36)3.2 洛必达法则(L’Hopital’s Rule) (38)3.3 泰勒公式(Taylor’s Formula) (41)3.4 函数的单调性和曲线的凹凸性(Monotonicityof Functions and Concavity of Curves) (43)3.5 函数的极值与最大最小值(Extrema, Maximaand Minima of Functions) (46)3.6 函数图形的描绘(Graphing Functions) (49)3.7 曲率(Curvature) (50)3.8 方程的近似解(Solving Equation Numerically) (53)第四章不定积分Chapter 4 Indefinite Integrals (54)4.1 不定积分的概念与性质(The Concept andProperties of Indefinite Integrals) (54)4.2 换元积分法(Substitution Rule for Indefinite Integrals) (56)4.3 分部积分法(Integration by Parts) (57)4.4 有理函数的积分(Integration of Rational Functions) (58)第五章定积分Chapter 5 Definite Integrals (61)5.1 定积分的概念和性质(Concept of Definite Integraland its Properties) (61)5.2 微积分基本定理(Fundamental Theorem of Calculus) (67)5.3 定积分的换元法和分部积分法(Integration by Substitution andDefinite Integrals by Parts) (69)5.4 反常积分(Improper Integrals) (70)第六章定积分的应用Chapter 6 Applications of the Definite Integrals (75)6.1 定积分的元素法(The Element Method of Definite Integra (75)6.2 定积分在几何学上的应用(Applications of the DefiniteIntegrals to Geometry) (76)6.3 定积分在物理学上的应用(Applications of the DefiniteIntegrals to Physics) (79)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geometry and Vector Algebar (80)7.1 向量及其线性运算(Vector and Its Linear Operation) (80)7.2 数量积向量积(Dot Produc t and Cross Product) (86)7.3 曲面及其方程(Surface and Its Equation) (89)7.4 空间曲线及其方程(The Curve in Three-space and Its Equation (91)7.5 平面及其方程（Plane in Space and Its Equation) (93)7.6 空间直线及其方程（Lines in and Their Equations) (95)第八章多元函数微分法及其应用Chapter 8 Differentiation of Functions of SeveralVariables and Its Application (99)8.1 多元函数的基本概念（The Basic Concepts of Functionsof Several Variables) (99)8.2 偏导数(Partial Derivative) (102)8.3 全微分(Total Differential) (103)8.4 链式法则（The Chain Rule) (104)8.5 隐函数的求导公式(Derivative Formula for Implicit Functions). (104)8.6 多元函数微分学的几何应用(Geometric Applications of Differentiationof Ffunctions of Severalvariables) (106)8.7方向导数与梯度(Directional Derivatives and Gradients) (107)8.8多元函数的极值(Extreme Value of Functions of Several Variables) (108)第九章重积分Chapter 9 Multiple Integrals (111)9.1二重积分的概念与性质(The Concept of Double Integralsand Its Properities) (111)9.2二重积分的计算法(Evaluation of double Integrals) (114)9.3三重积分(Triple Integrals) (115)9.4重积分的应用(Applications of Multiple Itegrals) (120)第十章曲线积分与曲面积分Chapte 10 Line Integrals and Surface Integrals………………………………121 10.1 对弧长的曲线积分(line Intergrals with Respect to Arc Length) ………121 10.2 对坐标的曲线积分(Line Integrals with respect toCoordinate Variables) ……………………………………………………123 10.3 格林公式及其应用(Green's Formula and Its Applications) ………………124 10.4 对面积的曲面积分(Surface Integrals with Respect to Aarea) ……………126 10.5 对坐标的曲面积分(Surface Integrals with Respect toCoordinate Variables) ………………………………………………………128 10.6 高斯公式通量与散度(Gauss's Formula Flux and Divirgence) …… 130 10.7 斯托克斯公式环流量与旋度(Stokes's Formula Circulationand Rotation) (131)第十一章无穷级数Chapter 11 Infinite Series (133)11.1 常数项级数的概念与性质(The concept and Properties ofThe Constant series) ………………………………………………………133 11.2 常数项级数的审敛法(Test for Convergence of the Constant Series) ……137 11.3 幂级数(powe r Series). ……………………………………………………143 11.4 函数展开成幂级数(Represent the Function as Power Series) ……………148 11.5 函数的幂级数展开式的应用(the Appliacation of the Power Seriesrepresentation of a Function) (148)11.6 函数项级数的一致收敛性及一致收敛级数的基本性质(The Unanimous Convergence of the Ser ies of Functions and Its properties) (149)11.7 傅立叶级数（Fourier Series）.............................................152 11.8 一般周期函数的傅立叶级数(Fourier Series of Periodic Functions) (153)第十二章微分方程Chapter 12 Differential Equation……………………………………………155 12.1 微分方程的基本概念（The Concept of DifferentialEqu ation) ……155 12.2 可分离变量的微分方程(Separable Differential Equation) ………156 12.3 齐次方程(Homogeneous Equation) ………………………………156 12.4 一次线性微分方程(Linear Differential Equation of theFirst Order) (157)12.5 全微分方程(Total Differential Equation) …………………………158 12.6 可降阶的高阶微分方程(Higher-order DifferentialEquation Turned to Lower-order DifferentialEquation) (159)12.7 高阶线性微分方程(Linear Differential Equation of Higher Order) …159 12.8 常系数齐次线性微分方程(Homogeneous LinearDifferential Equation with Constant Coefficient) (163)12.9 常系数非齐次线性微分方程(Non HomogeneousDifferential Equation with Constant Coefficient) (164)12.10 欧拉方程(Euler Equation) …………………………………………164 12.11 微分方程的幂级数解法(Power Series Solutionto Differential Equation) (164)第三部分常用数学符号的英文表达Part 3 English Expression of the Mathematical Symbol in Common Use第一部分英汉微积分词汇Part1 English-Chinese Calculus Vocabulary映射 mappingX到Y的映射 mapping of X ontoY 满射 surjection 单射 injection一一映射 one-to-one mapping 双射 bijection 算子 operator变化 transformation 函数 function逆映射 inverse mapping复合映射 composite mapping 自变量 independent variable 因变量 dependent variable 定义域 domain函数值 value of function 函数关系 function relation 值域 range自然定义域 natural domain 单值函数 single valued function 多值函数 multiple valued function 单值分支 one-valued branch 函数图形 graph of a function 绝对值函数 absolute value 符号函数 sigh function 整数部分 integral part 阶梯曲线 step curve 第一章函数与极限Chapter1 Function and Limit 集合 set元素 element 子集 subset 空集 empty set 并集 union交集 intersection 差集 difference of set 基本集 basic set补集 complement set 直积 direct product笛卡儿积 Cartesian product 开区间 open interval 闭区间 closed interval 半开区间half open interval 有限区间 finite interval区间的长度 length of an interval 无限区间 infinite interval 领域 neighborhood领域的中心 centre of a neighborhood 领域的半径 radius of a neighborhood 左领域left neighborhood 右领域 right neighborhood当且仅当 if and only if(iff) 分段函数 piecewise function 上界 upper bound 下界lower bound 有界 boundedness 无界 unbounded函数的单调性 monotonicity of a function 单调增加的 increasing 单调减少的decreasing单调函数 monotone function函数的奇偶性 parity(odevity) of a function对称 symmetry 偶函数 even function 奇函数 odd function函数的周期性 periodicity of a function 周期 period反函数 inverse function 直接函数 direct function 复合函数 composite function 中间变量 intermediate variable 函数的运算 operation of function基本初等函数 basic elementary function 初等函数 elementary function 幂函数 power function指数函数 exponential function 对数函数 logarithmic function 三角函数 trigonometric function反三角函数 inverse trigonometric function 常数函数 constant function 双曲函数hyperbolic function 双曲正弦 hyperbolic sine 双曲余弦 hyperbolic cosine 双曲正切hyperbolic tangent反双曲正弦 inverse hyperbolic sine 反双曲余弦 inverse hyperbolic cosine 反双曲正切 inverse hyperbolic tangent 极限 limit数列 sequence of number 收敛 convergence 收敛于 a converge to a 发散 divergent极限的唯一性 uniqueness of limits收敛数列的有界性 boundedness of aconvergent sequence子列 subsequence函数的极限 limits of functions函数f(x)当x趋于x0时的极限 limit of functions f(x) as x approaches x0 左极限 left limit 右极限 right limit单侧极限 one-sided limits水平渐近线 horizontal asymptote 无穷小 infinitesimal 无穷大 infinity铅直渐近线 vertical asymptote 夹逼准则 squeeze rule单调数列 monotonic sequence高阶无穷小 infinitesimal of higher order 低阶无穷小 infinitesimal of lower order 同阶无穷小 infinitesimal of the same order 等阶无穷小 equivalent infinitesimal 函数的连续性 continuity of a function 增量 increment函数f(x)在x0连续 the function f(x) is continuous at x0左连续 left continuous 右连续 right continuous区间上的连续函数 continuous function 函数f(x)在该区间上连续 function f(x) is continuous on an interval 不连续点 discontinuity point第一类间断点 discontinuity point of the first kind第二类间断点 discontinuity point of the second kind初等函数的连续性 continuity of the elementary functions定义区间 defined interval最大值 global maximum value (absolute maximum)最小值 global minimum value (absolute minimum)零点定理 the zero point theorem介值定理 intermediate value theorem 第二章导数与微分Chapter2 Derivative and Differential 速度 velocity匀速运动 uniform motion 平均速度 average velocity瞬时速度 instantaneous velocity 圆的切线 tangent line of a circle 切线 tangent line切线的斜率 slope of the tangent line 位置函数 position function 导数 derivative 可导derivable函数的变化率问题 problem of the change rate of a function导函数 derived function 左导数 left-hand derivative 右导数 right-hand derivative 单侧导数 one-sided derivativesf(x)在闭区间【a,b】上可导 f(x)isderivable on the closed interval [a,b] 切线方程 tangent equation 角速度 angular velocity 成本函数 cost function 边际成本 marginal cost 链式法则 chain rule隐函数 implicit function 显函数 explicit function 二阶函数 second derivative 三阶导数 third derivative 高阶导数 nth derivative莱布尼茨公式 Leibniz formula 对数求导法 log- derivative 参数方程 parametric equation 相关变化率 correlative change rata 微分 differential 可微的 differentiable 函数的微分 differential of function自变量的微分 differential of independent variable微商 differential quotient间接测量误差 indirect measurement error 绝对误差 absolute error相对误差 relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives 罗马定理Rolle’s theorem 费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点 stationary point 稳定点 stable point 临界点 critical point辅助函数 auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule0/0型不定式 indeterminate form of type 0/0不定式 indeterminate form泰勒中值定理Taylor’s mean value theorem泰勒公式 Taylor formula 余项 remainder term拉格朗日余项 Lagrange remainder term 麦克劳林公式Maclaurin’s formula 佩亚诺公式 Peano remainder term 凹凸性 concavity凹向上的 concave upward, cancave up 凹向下的，向上凸的concave downward’ concave down拐点 inflection point函数的极值 extremum of function 极大值 local(relative) maximum 最大值global(absolute) mximum 极小值 local(relative) minimum 最小值 global(absolute) minimum 目标函数 objective function 曲率 curvature弧微分 arc differential平均曲率 average curvature 曲率园 circle of curvature 曲率中心 center of curvature 曲率半径 radius of curvature渐屈线 evolute 渐伸线 involute根的隔离 isolation of root 隔离区间 isolation interval 切线法 tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数 primitive function(antiderivative) 积分号 sign of integration 被积函数integrand积分变量 integral variable 积分曲线 integral curve 积分表 table of integrals换元积分法 integration by substitution 分部积分法 integration by parts分部积分公式 formula of integration by parts有理函数 rational function 真分式 proper fraction 假分式 improper fraction第五章定积分Chapter5 Definite Integrals 曲边梯形 trapezoid with 曲边 curve edge窄矩形 narrow rectangle曲边梯形的面积 area of trapezoid with curved edge积分下限 lower limit of integral 积分上限 upper limit of integral 积分区间 integral interval 分割 partition积分和 integral sum 可积 integrable矩形法 rectangle method积分中值定理 mean value theorem of integrals函数在区间上的平均值 average value of a function on an integvals牛顿－莱布尼茨公式 Newton-Leibniz formula微积分基本公式 fundamental formula of calculus换元公式 formula for integration by substitution递推公式 recurrence formula 反常积分 improper integral反常积分发散 the improper integral is divergent反常积分收敛 the improper integral is convergent无穷限的反常积分 improper integral on an infinite interval无界函数的反常积分 improper integral of unbounded functions绝对收敛 absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法 the element method 面积元素 element of area平面图形的面积 area of a luane figure 直角坐标又称“笛卡儿坐标 (Cartesian coordinates)”极坐标 polar coordinates 抛物线 parabola 椭圆 ellipse旋转体的面积 volume of a solid of rotation旋转椭球体 ellipsoid of revolution, ellipsoid of rotation曲线的弧长 arc length of acurve 可求长的 rectifiable 光滑 smooth 功 work水压力 water pressure 引力 gravitation 变力 variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量 vector自由向量 free vector 单位向量 unit vector 零向量 zero vector 相等 equal 平行parallel向量的线性运算 linear poeration of vector 三角法则 triangle rule平行四边形法则 parallelogram rule 交换律 commutative law 结合律 associative law 负向量 negative vector 差 difference分配律 distributive law空间直角坐标系 space rectangular coordinates坐标面 coordinate plane 卦限 octant向量的模 modulus of vector向量a与b的夹角 angle between vector a and b方向余弦 direction cosine 方向角 direction angle向量在轴上的投影 projection of a vector onto an axis数量积，外积，叉积 scalar product,dot product,inner product曲面方程 equation for a surface 球面 sphere旋转曲面 surface of revolution 母线 generating line 轴 axis圆锥面 cone 顶点 vertex旋转单叶双曲面 revolution hyperboloids of one sheet旋转双叶双曲面 revolution hyperboloids of two sheets柱面 cylindrical surface ,cylinder 圆柱面 cylindrical surface 准线 directrix抛物柱面 parabolic cylinder 二次曲面 quadric surface 椭圆锥面 dlliptic cone 椭球面ellipsoid单叶双曲面 hyperboloid of one sheet 双叶双曲面 hyperboloid of two sheets 旋转椭球面 ellipsoid of revolution 椭圆抛物面 elliptic paraboloid旋转抛物面 paraboloid of revolution 双曲抛物面 hyperbolic paraboloid 马鞍面 saddle surface椭圆柱面 elliptic cylinder 双曲柱面 hyperbolic cylinder 抛物柱面 parabolic cylinder 空间曲线 space curve空间曲线的一般方程 general form equations of a space curve空间曲线的参数方程 parametric equations of a space curve 螺转线 spiral 螺矩 pitch 投影柱面 projecting cylinder 投影 projection平面的点法式方程 pointnorm form eqyation of a plane法向量 normal vector平面的一般方程 general form equation of a plane两平面的夹角 angle between two planes 点到平面的距离 distance from a point to a plane空间直线的一般方程 general equation of a line in space方向向量 direction vector直线的点向式方程 pointdirection form equations of a line方向数 direction number直线的参数方程 parametric equations of a line两直线的夹角 angle between two lines 垂直 perpendicular直线与平面的夹角 angle between a line and a planes平面束 pencil of planes平面束的方程 equation of a pencil of planes行列式 determinant系数行列式 coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application 一元函数 function of one variable 多元函数 function of several variables 内点 interior point 外点 exterior point 边界点 frontier point,boundary point 聚点 point of accumulation 开集 openset 闭集 closed set 连通集 connected set 开区域 open region 闭区域 closed region有界集 bounded set 无界集 unbounded setn维空间 n-dimentional space 二重极限 double limit 多元函数的连续性 continuity of function of seveal 连续函数 continuous function 不连续点 discontinuity point 一致连续 uniformly continuous 偏导数 partial derivative 对自变量x的偏导数 partial derivative with respect to independent variable x 高阶偏导数 partial derivative of higher order 二阶偏导数 second order partial derivative 混合偏导数 hybrid partial derivative 全微分 total differential 偏增量 oartial increment 偏微分 partial differential 全增量 total increment 可微分 differentiable 必要条件 necessary condition充分条件 sufficient condition 叠加原理 superpostition principle 全导数 total derivative中间变量 intermediate variable 隐函数存在定理 theorem of the existence of implicit function 曲线的切向量 tangent vector of a curve 法平面 normal plane 向量方程vector equation 向量值函数 vector-valued function 切平面 tangent plane 法线 normal line 方向导数 directional derivative梯度 gradient数量场 scalar field 梯度场 gradient field 向量场 vector field 势场 potential field 引力场 gravitational field 引力势 gravitational potential 曲面在一点的切平面 tangent plane to asurface at a point 曲线在一点的法线 normal line to asurface at a point 无条件极值 unconditional extreme values 条件极值 conditional extreme values 拉格朗日乘数法 Lagrange multiplier method 拉格朗日乘子 Lagrange multiplier 经验公式 empirical formula 最小二乘法 method of least squares 均方误差mean square error 第九章重积分 Chapter9 Multiple Integrals 二重积分 double integral 可加性 additivity累次积分 iterated integral 体积元素 volume element 三重积分 triple integral 直角坐标系中的体积元素 volumeelement in rectangular coordinate system 柱面坐标 cylindrical coordinates 柱面坐标系中的体积元素 volumeelement in cylindrical coordinate system 球面坐标 spherical coordinates 球面坐标系中的体积元素 volumeelement in spherical coordinate system 反常二重积分 improper double integral 曲面的面积 area of a surface 质心 centre of mass 静矩 static moment 密度 density 形心centroid 转动惯量 moment of inertia 参变量 parametric variable 第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分 line integrals with respect to arc hength第一类曲线积分 line integrals of the first type对坐标的曲线积分 line integrals with respect to x,y,and z第二类曲线积分 line integrals of the second type有向曲线弧 directed arc单连通区域 simple connected region 复连通区域 complex connected region 格林公式Green formula第一类曲面积分 surface integrals of the first type对面的曲面积分 surface integrals with respect to area有向曲面 directed surface对坐标的曲面积分 surface integrals with respect to coordinate elements第二类曲面积分 surface integrals of the second type有向曲面元 element of directed surface 高斯公式 gauss formula拉普拉斯算子 Laplace operator 格林第一公式Green’s first formula 通量 flux散度 divergence斯托克斯公式 Stokes formula 环流量 circulation 旋度 rotation,curl第十一章无穷级数Chapter11 Infinite Series 一般项 general term 部分和 partial sum 余项 remainder term 等比级数 geometric series 几何级数 geometric series 公比 common ratio调和级数 harmonic series柯西收敛准则 Cauchy convergence criteria, Cauchy criteria for convergence 正项级数series of positive terms 达朗贝尔判别法D’Alembert test 柯西判别法 Cauchy test交错级数 alternating series 绝对收敛 absolutely convergent 条件收敛 conditionally convergent 柯西乘积 Cauchy product 函数项级数 series of functions 发散点 point of divergence 收敛点 point of convergence 收敛域 convergence domain 和函数 sum function 幂级数 power series幂级数的系数 coeffcients of power series 阿贝尔定理 Abel Theorem收敛半径 radius of convergence 收敛区间 interval of convergence 泰勒级数 Taylor series麦克劳林级数 Maclaurin series 二项展开式 binomial expansion 近似计算approximate calculation舍入误差 round-off error,rounding error 欧拉公式Euler’s formula魏尔斯特拉丝判别法 Weierstrass test 三角级数 trigonometric series 振幅 amplitude 角频率 angular frequency 初相 initial phase 矩形波 square wave谐波分析 harmonic analysis 直流分量 direct component 基波 fundamental wave 二次谐波 second harmonic三角函数系 trigonometric function system 傅立叶系数 Fourier coefficient 傅立叶级数 Forrier series 周期延拓 periodic prolongation 正弦级数 sine series 余弦级数cosine series 奇延拓 odd prolongation 偶延拓 even prolongation傅立叶级数的复数形式 complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程 solve a dirrerential equation 常微分方程 ordinary differential equation偏微分方程 partial differential equation,PDE微分方程的阶 order of a differential equation微分方程的解 solution of a differential equation微分方程的通解 general solution of a differential equation初始条件 initial condition微分方程的特解 particular solution of a differential equation初值问题 initial value problem微分方程的积分曲线 integral curve of a differential equation可分离变量的微分方程 variable separable differential equation隐式解 implicit solution隐式通解 inplicit general solution 衰变系数 decay coefficient 衰变 decay齐次方程 homogeneous equation一阶线性方程 linear differential equation of first order非齐次 non-homogeneous齐次线性方程 homogeneous linear equation非齐次线性方程 non-homogeneous linear equation常数变易法 method of variation of constant暂态电流 transient stata current 稳态电流 steady state current 伯努利方程 Bernoulli equation全微分方程 total differential equation 积分因子 integrating factor高阶微分方程 differential equation of higher order悬链线 catenary高阶线性微分方程 linera differentialequation of higher order自由振动的微分方程 differential equation of free vibration强迫振动的微分方程 differential equation of forced oscillation串联电路的振荡方程 oscillation equation of series circuit二阶线性微分方程 second order linera differential equation线性相关 linearly dependence 线性无关 linearly independce二阶常系数齐次线性微分方程 second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程 second order homogeneous linear differential equation with variable coefficient 特征方程 characteristic equation无阻尼自由振动的微分方程 differential equation of free vibration with zero damping 固有频率 natural frequency简谐振动 simple harmonic oscillation,simple harmonic vibration微分算子 differential operator待定系数法 method of undetermined coefficient共振现象 resonance phenomenon 欧拉方程 Euler equation幂级数解法 power series solution 数值解法 numerial solution 勒让德方程 Legendre equation微分方程组 system of differential equations常系数线性微分方程组 system of linera differential equations with constant coefficient第二部分定理定义公式的英文表达Part2 English Expression for Theorem, Definition and Formula第一章函数与极限Chapter 1 Function and Limit1.1 映射与函数 (Mapping and Function)一、集合 (Set)二、映射 (Mapping)映射概念 (The Concept of Mapping) 设X, Y是两个非空集合 , 如果存在一个法则f，使得对X中每个元素x，按法则f，在Y中有唯一确定的元素y与之对应 ,则称f为从X到 Y的映射 , 记作f:X→Y。
高等数学中常见的英文
V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simp son’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。
离散数学英文版PPT
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Prerequisite and Description
• Prerequisite: MATH 170 Calculus I and CSCI 185 Programming II • Description: An introduction to discrete structures with applications to computing problems. Topics include logic, sets, functions, relations, proof techniques and algorithmic analysis. Graph theory and trees may be studied as well
Course Objectives
1. Relate practical examples to the appropriate set, function, or relation model, and interpret the associated operations and terminology in context 2. Apply proof techniques, including logic, to problems 3. Differentiate between dependent and independent events 4. Apply the binomial theorem to independent events and Bayes’ theorem to dependent events, and solve problems such as Hashing 5. Relate ideas of mathematical induction(归纳) to recursion and apply it to problems in computer science setting 6. Apply the basic counting principles, permutations and combinations to problems in computer science setting 7. Implementing algorithms in C or C++ programs
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n
Solution. We must find the integer N such that for all n
nNCC
It is obvious that we can obtain N as
N 1
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Example 3. Show that 1 n1 diverges.
Solution.
1n
lim n n
0
1n
n
1 n
, lim 1 0. n n
1n
lim
0
n n
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Example 6. Applying the Sandwich Theorem
cosnsinn
lim
n
3n2 n
Solution.
0
cos n sin n 3n2 n
Solution. Suppose the sequence converges to some number L.
nN1n1L
For 1 , we have
2
1n1 L 1
2
L 1 1 1 L1 1 1 L 3
2
2
22
2
L 1 1 1 L11 3 L1
2
2
22
2
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8.1
Limits of Sequence of Numbers
(数列的极限)
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Example 1. Applying the definition to show that
n
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Example 8. Applying the L’Hospital Rule
Solution.
ln n lim n n
lim ln x lim 1 0
x x
x x
lim ln n 0 n n
lnn 1n 1 lim lim lim0 n n n 1 n n
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Example 9. Applying the L’Hospital Rule
limn41n4
n
Solution.
e n4 e 1n4 lnn41n4
1 lnn4
n4
e limn41n4
lim1lnn4
en n4
lnn4
lim n n4
n
lim 1
e n n4
limlnn1lnn1 1
1 n
en
n
lim2n2 n21
en
e2
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Exercises
P617 14, 18, 24, 26, 29, 30, 32, 35, 43, 56.
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lim 1 1 n n 1 1 n
lim 1 1 n
n
lim 1 1 n
1 1
1
n
lim
n
4 4n6 n6 3
lim 4 4n6 n n6 3
4 4n6
lim
n
n6 3
4
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Example 5. Applying the Sandwich Theorem
Convergence 8.6 Power Series 8.7 Taylor and Maclaurin Series 8.8 Applications of Power Series
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Introduction
1 1 1 1 1
2 4 8 16
1 1 1 1 1 1
lim 1 0 n n
Solution.
We must find the integer N such that
for all n
n N 1 0
n
1us that we can obtain N as
N1
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Example 2. Applying the definition to show that
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Example 4. Applying the limit rules
lim
n
1 n
Solution.
lim n 1 n n 1
4 4n6
lim
n
n6 3
lim
n
1 n
lim 1 1
n
n
lim1lim1 10
n
n n
0
lim n 1 n n 1
1 3n2 n
,lim 1 0. n 3n2 n
lni mco3snn2sinnn0
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Example 7. Applying the Theorem 3
1
lim 2 n
n
Solution.
1
lim 1
li m 2 n 2 n n 2 0 1
e0
1
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Example 10. Applying the L’Hospital Rule
Solution.
lim
n
n n
1 1
n
n 1n n 1
eln
n1n n1
n ln n 1
e n1
lni mnn11n
e e limnlnn1 n n1
lim ln n1 n n1
Chapter 8 Infinite Series
8.1 Limits of Sequences of Numbers 8.2 Subsequences, Bounded Sequences, and… 8.3 Infinite Series 8.4 Series of Nonnegative Terms 8.5 Alternating Series, Absolute and Conditional