Geometric Series - Sum to infinity
A-level数学词汇(P1-P4)
Pure mathematics 1 1 Algebraic expression 代数表达式Index laws 指数定律Indices (index的复数形式) 指数Notation 注释Simplify 化简Power 指数Base 底Exponent 指数Expression 表达式Term 项Numerator 分子Expand 展开Possible 可能Fraction 分数Bracket 括号Product 乘积Multiply 乘Collecting like terms 合并同类项Linear 一次的Diagram 图形Rectangle 长方形Square 正方形Length 长度Width 宽Side length 边长Area 面积Shade 阴影Cuboid 长方体Dimension 维Show that 证明V olume 体积Given that 已知Constant 常数Value 值Factorize 因式分解Factor 因子Opposite 相反的Completely 完全地Common factor 公因式Quadratic 二次的Form 形式Real number 实数Positive 正的Negative 负的Include 包含Surd 无理数Add 加Sum 和Take out 提取Difference 差Difference of two squares 平方差Cancel 取消，相互抵消Similarly 同样的Rational 有理的Rational number 有理数Integer 整数Square root 平方根Evaluate 求…的值Substitute 代替Calculator 计算器Square number 平方数Irrational number 无理数Decimal 小数的Expansion 展开式Never-ending 无限的Never repeat 不循环的Exact 准确的Answer 答案Manipulate 操作Denominator 分母Rationalizing denominator 分母有理化Rearrange 调整Prime 质数Work out 计算Hence 然后Fully 完全地State 陈述Solve 解决Equation 方程2 Quadratics 二次方程式Quadratic equation 二次方程Solution 解Real solution 实根Set 设置Root 根Distinct 不同的Repeated root 重根Case 情况Straightforward 简单直接的Symbol 符号Plus 加，正Minus 减，负Factorization 因式分解Shape 形状Section 部分Formula 公式Reading off 读取Coefficient 系数Necessary 必要的Significant figures 有效数字Choose 选择Suitable 适当的Method 方法Trapezium 梯形Height 高Discard 丢弃Completing the square 完全平方（配方）Frequently 经常的Useful 有用的Process 过程Original 最初的Determine 决定Otherwise 另外Function 函数Mathematical 数学上的Relationship 关系Map 映射Set 集合Input 输入Output 输出Single 单一的Notation 符号Represent 代表Domain 定义域Range 值域，范围Member 成员Define 定义Minimum 最小的Occur 发生Explain 解释Consider 考虑Graph 图像Curve 曲线Parabola 抛物线Sketch 画图Identify 确定Key 关键的Feature 特征Overall 整体的Cross 交叉，横过Axis 轴Coordinate 坐标Turning point 转折点（顶点）Maximum 最大的Since 因为Symmetrical 对称的Symmetry 对称性Line of symmetry 对称轴Half-way 位于中途的Explore 探测Technology 技术Plot 绘制Scale 刻度However 但是，不管怎样Smooth 平滑的Relevant 相关的Intercept 截距Label 标记Axes (axis的复数)坐标轴Discriminant 判别式Sign 符号Check 核实Inequality 不等式Calculate 计算Match 匹配Prove 证明Algebra 代数学Diver 跳水运动员Launch 发射Springboard 跳板Meter 米Pool 水池Second 秒Model 模型High 高的Hit 撞击Reach 达到Non-zero 非零3 Equations and inequalities 方程和不等式Simultaneous 联立的Linear simultaneous equations 一次方程组Elimination 消元法Substitution 置换Quadratic simultaneous equations 二次方程组Up to 直到，多达Make sure 确保Correctly 正确地Simplest 最简的Graphically 以图表形式As 因为Satisfy 满足Intersection 相交Simultaneously 同时地Intersect 相交Once 一次Twice 两次Result 结果，导致Produce 产生Graph paper 坐标纸Accurately 准确地Verify 验证Linear inequalities 一次不等式Set notation 集合符号Number line 数轴Overlap 重叠Separately 单独地Illustrate 图解，阐明Quadratic inequalities 二次不等式Corresponding 相应的Critical 临界的Require 要求Describe 描述Interpret 解释Region 区域，范围Coordinate grid 坐标网Dotted line 虚线Solid line 实线Vertex 顶点Vertices (vertex的复数)顶点Within 在内部，之内4 Graphs and transformations 图像和转换Cubic 三次的Cubic function 三次函数Several 几个Depend on 取决于Touch 接触Coordinate axes 坐标轴Indicate 表明，显示Reciprocal 倒数的Reciprocal function 反比例函数Such as 例如Asymptote 渐近线Approach 接近Reach 到达Quadrant 象限Point of intersection 交点Steeper 更陡峭的Eventually 最后，终于Reason 理由，原因Appropriate 恰当的Number 数量Translate 平移Transform 改变Alter 改动Subtract 减Outside 在外面Vertically 竖直地Translation 平移Vector 矢量Horizontally 水平地Direction 方向In terms of 用…来表示Slide 滑动Stretch 伸缩Scale factor 比例系数Double 两倍Halve 减半，对分Inside 在里面Triple 三倍的Reflection 反射(镜面对称) Alternatively 二选一Parallel 平行Lie on 坐落在Pass through 穿过Apply 应用Unfamiliar 陌生的，不熟悉的Specific 特殊的Origin 原点Position 位置Image 像Suggest 提议Mark 标记5 Straight line graphs 直线图像Gradient 斜率Straight line 直线Join 连接Distance 距离Formula 公式Collinear 共线的Intercept 截距Define 定义Either 两者中的任一个Condition 条件Triangle 三角形General equation 一般式Parallel 平行Perpendicular 垂直Whether 是否Quadrilateral 四边形Trapezium 梯形Right angle 直角Congruent 全等的Neither 两者都不Hypotenuse 直角三角形斜边Line segment 线段Scalene 不等边的Respectively 分别地Go through 通过6 Trigonometric ratios 三角比Cosine rule 余弦定理Miss 缺失Version 版本Exchange 交换Standard 标准Prove 证明Opposite 对边Adjacent 邻边Pythagoras’ theorem 勾股定理Letter 字母Round 四舍五入Final 最终的Coastguard 海岸警卫队Station 驻地Bearing 方位Away from 远离Appropriate 适当的Mark 标记Airport 机场Due north 正北Due east 正东Due west 正西Due south 正南Sail 航行Helicopter 直升飞机Tee 球座Flag 旗Particular 特定的Hole 孔，洞Golf course 高尔夫球场Yard 码（1码=3英尺）Tee shot 发球台Land 着陆Largest 最大的Farmer 农场Field 场地Fence 栅栏Cargo 货物Plane 平面Kilometer 千米Sine rule 正弦定理Refer to 涉及Data 数据Remain 剩余Located on 坐落于Zookeeper 动物管理员Enclosure 围场Llama 骆驼Diagonal 对角线Surveyor 检验员Measure 测量Elevation 高程，仰角Apart 相距Assumption 假设Mathematical 数学的Model 模型Obtuse 钝角Acute 锐角Isosceles 等腰的Circle 圆Radius 半径Centre 圆心Least 最小的Instead 代替Crane 吊车Anchored 固定Wreck 破坏Suspend 悬挂Cable 缆绳Rotate 旋转Level 对准Proof 证明Triangular plot 三角图Involve 涉及Trigonometry 三角函数Encounter 遇到Decide 决定Mast 桅杆In order that 为了Interfere 干扰Efficient 有效的Hiker 徒步旅行者Radar 雷达Perimeter 周长Tangent 正切Periodic 周期性的Repeat 重复的Certain 确定的Interval 间距Period 周期Undefined 无意义的Knowledge 知识Periodicity 周期性Verify 证明Variation 变化Rock pool 潮汐潭Midday 中午During 在…期间Non-exact 非精准的Significant figure 有效数字Windmill 风车Sail 帆Tower 塔Deduce 推导Dune 沙丘Realistic 现实的7 Radians 弧度Radian 弧度So far 到目前为止Probably 大概，可能Degree 度Revolution 循环Around 围绕Circle 圆Subtend 朝着Arc 圆弧Circumference 周长Convert 转换Without 没有Multiple 倍数Arc length 弧长Sector 扇形Radius 半径Contain 包含Perimeter 周长Border 边界Pond 池塘Consist 由…组成Edge 边缘Minor arc 劣弧Major arc 优弧Chord 弦Diameter 直径Template 模板Brooch 胸针Ferris wheel 摩天轮Pod 蚕茧，豆荚Estimate 估计Speed 速率Patio 露台Lawn 草坪Design 设计Earring 耳环Nearest 最近点（精确到）Segment 弓形Radii (radius的复数形式) A plot of …的一块Erect 建造Along 沿着Subtract 减Tangent 切线Ratio 比例Bound 关，围入Decimal place 小数Midpoint 中点Semicircular 半圆Drawer 抽屉Handle 把手Difference 差Badge 徽章Equilateral 等边的Railway 铁路Track 轨迹Prism 三棱镜Attempt 尝试Mistake 错误8 Differentiation 微分Gradient 斜率Constantly 不断地Although 然而Comment on 对…评论Copy 抄写，复制Complete 完成Table 表格Hypothesis 假设Derivative 导数Principle 原理Detail 细节Account 解释Originate 起源Formalize 确定，形成Approach 方式，方法Limit 极限Tend to 趋向Gradient function 斜率函数Evaluate 求…的值Fixed value 定值Limiting value 定值Definition 定义One-at-a-time 一次一个Turning point 转折点（顶点）Slope 斜率Disappear 消失Polynomial 多项式Normal 切线First order derivative 一阶导数Second order derivative 二阶导数Rate of change 变化率Respect to 关于Displacement 位移Acceleration 加速度Local 局部的9 Integration 积分Reverse 相反的Differ 不同Integrate 求积分Integral 积分Indefinite 不确定的Indefinite integral 不定积分Elongated 拉长的，伸长的Arrow 箭Fire 射击Castle 城堡Drop off 下降Cliff 悬崖Cyclist 骑行者Pure mathematics 2 1Algebraic methods 代数方法Division 除法Dividing polynomial 多项式除法Finite 有限的Whole number 整数Long division 长除法Quotient 商Remainder 余数Factor theorem 因式定理Remainder theorem 余数定理Logical 逻辑的Structured 有组织的Argument 论据Statement 命题Conjecture 猜想Previously 预先Establish 建立Deduction 推导Desired 想要的Conclusion 结论Odd number 奇数Demonstration 示范，演示Even number 偶数Identical 完全相等的Identity 恒等式Parallelogram 平行四边形Rhombus 菱形Congruent 全等的Exhaustion 穷举法Consecutive 连续的Square number 平方数Break into 拆分Is suited to 适合于Disprove 反驳Counter-example 反例Sufficient 充分的Prime number 质数Divisible 可整除的Either … or…二者择一的Cube number 立方数Hold 有效Claim 宣称Opposite edge 对边Hexagon 六边形Regular hexagon 正六边形Side length 边长Reason 原因2Coordinate geometry in the (x,y) plane 解析几何Bisector 二等分线Perpendicular bisector 中垂线Averaging 求平均值Endpoint 端点Circumcentre 外心Equidistant 等距的Fixed point 定点Vector 向量Property 性质Unique 独一无二的Circumcircle 外接圆3Exponentials and logarithms 指数和对数Exponential 指数的Decrease 减小Increase 增加Smooth 光滑的，平滑的Increasing function 增函数Decreasing function 减函数Justify 证明Logarithms 对数Specific 特定的Button 按钮Typically 典型的Natural logarithms 自然对数Instance 实例Multiplication law 乘法定律Division law 除法定律Power law 指数定律Recognize 识别Attention 注意Condition 条件Complicated 复杂的Whenever 无论何时Convenient 方便的Suppose 假设Notice 注意Particular 特别的4The binomial expansion 二项式展开Binomial 二项式Pascal’s triangle 杨辉三角（帕斯卡三角形）Immediately 直接地Pattern 图案Adjacent 相邻的Investment 投资Interest rate 利率Annum 年，岁Approximation 近似值Ignore 忽略Factorial notation 阶乘Combination 组合Superscript 上标Subscript 下标Probability 可能性Toss 投Likelihood 可能性Ascending powers 升幂Individual 个别的Estimation 估值Engineering 工程学Science 科学Percentage error 百分误差Microchip 微型集成电路片Faulty 有缺点的Chip 芯片Restrict 限制Achieve 达到School fair 学校园游会Prize 奖赏Digit 数字Display 显示5Sequences and series 数列和级数Arithmetic sequence 等差数列Arithmetic progression 等差数列Common difference 公差Arithmetic series 等差级数（等差数列前n 项求和）Exceed 超过Inclusive 包含的Stick 棒子Pentagon 五角形Geometric sequences 等比数列Geometric progression 等比数列Common ratio 公比Converge 收敛Alternating sequence 交错数列Million 百万Geometric series 等比级数（等比数列前n项求和）Sum to infinity 无限项求和Divergent 发散的Convergent 收敛的Recurring 循环的Sigma notation 求和符号Capital 首都，大写字母Signify 表示Recurrence relations 递推关系Previous term 前一项First term 初项Generate 生成，产生Periodic sequence 周期数列Period 周期Salary 薪水Profit 利润Predict 预言Annual 年度的Business 商业Financial 金融的Advisor 顾问Fold 折叠Thickness 厚度Unrealistic 不切实际的Investor 投资人Account 账户Thereafter 以后Deposit 存款，定金Wage 工资Rise 上升Gear 齿轮Successive 连续的Intermediate 中间的Valuable 有价值的Commission 佣金Insurance 保险Policy 政策Prospector 勘探者Drill 钻孔Subsequent 随后的Available 可获得的Payment 报酬Virus 病毒Infect 传染Diagnose 诊断Overfish 过度捕捞Chess 象棋Chessboard 棋盘Sponsored 赞助的Polygon 多边形Appointment 约会，任命6Trigonometric identities and equations 三角恒等式和方程Unit circle 单位圆Anticlockwise 逆时针Quadrant 象限Equivalent 相等的Equilateral triangle 等边三角形Isosceles right-angled triangle 等腰直角三角形Identity 恒等式Reflex 优角(大于180度，在第三、四象限)Principal value 主值Inverse trigonometric function 反三角函数Justification 理由7Differentiation 微分Strictly 严格地Interval 区间Stationary point 驻点Local maximum 局部最大Greatest value 最大值Local minimum 局部最小Least value 最小值Point of inflection 拐点，反曲点Immediate 最接近的Vicinity 邻近，附近Second derivative 二次求导Rate of change 改变的快慢Convex 凸Concave 凹Establish 建立，证实Liter 升Instant 瞬间Tank 水槽Cuboid 长方体的Sheet 薄片Metal 金属Sphere 球体Displacement 位移Cylinder 圆柱体Perimeter 周长Semicircular 半圆的Semicircle 半圆Frame 框架Split 分离，分开Motion 运动Damped 阻尼Spring 弹簧Bent 弯的Biscuit 饼干Tin 罐头Close-fitting 紧贴的Lid 盖子Thin 薄的，瘦的Wastage 损耗Obtain 获得Percentage 百分比Store 储存Capacity 容量Container 容器Calculus 微积分学8Integration 积分Definite integral 定积分Indefinite integral 不定积分Whereas 反之，然而Upper limit 上限Lower limit 下限Square bracket 中括号Magnitude 大小Negligible 可忽略的Straddle 跨坐Unless 除非Complicated 复杂的Trapezium 梯形Trapezium rule 梯形法则Beneath 在…下面Strip 条，带Boundary 边界Adjacent 相邻的Improve 改善Accuracy 精确度Approximation 近似值Underestimate 低估Overestimate 高估Compare 比较Pure mathematics 3 Common multiple 公倍数Improper fraction 假分数Partial fractions 部分分数Degree 次数Modulus function 模函数Absolute value 绝对值Argument 辐角Set notation 集合符号Piecewise-defined function 分段函数Composite function 复合函数Inverse function 反函数Secant 正割Cosecant 余割Cotangent 余切Interval 区间Symmetry 对称性Symmetrical 对称的Chord 弦Inverse trigonometric function 反三角函数Addition formulae 加法公式Compound-angle formulae 复合角公式Double-angle formulae 二倍角公式Round 四舍五入Exponential function 指数函数Natural logarithms 自然对数Trend 趋势Outlier 极值Chain rule 链式法则Product rule 乘法法则Quotient rule 除法法则Continuous 连续的Fixed point iteration 定点迭代Successive 连续的Converge 收敛Staircase diagram 梯形图Cobweb diagram 网状图Diverge 发散Pure mathematics 4 Contradiction 反驳Assert 主张Falsehood 虚假Negation 反论Prime number 质数Split 分解Separate 独立的Parametric equation 参数方程Variable 变量Parameter 参数Revolution 循环Plot 绘图Valid 有效的As long as 只要Condition 条件Accurate 精确的Ascending 上升的Approximation 近似值Implicit differentiation 隐函数微分Explicitly 明确的Implicit 隐含的Rate of change 变化率Hemisphere 半球Cylindrical 圆柱形的Conical 圆锥形的Concave 凹Convex 凸Integrand 被积函数Integration by substitution 换元积分法Integration by part 分部积分法Polynomial 多项式Separating the variables 分离变量General solution 通解Boundary condition 边界条件Directed line segment 有向线段Parallelogram 平行四边形Unit vector 单位向量Column vector 列向量Position vector 位置矢量Scalene 不等边的21Clockwise 顺时针Anticlockwise 逆时针Coplanar 共面的Parallelepiped 平行六面体Trisect 三等分Hexagon 六边形Regular hexagon 正六边形Direction vector 方向向量Anchor 固定Dot product 点乘22。
数学专业英语词汇
数学专业英语词汇相信自己比依赖别人重要。
用尽心机不如静心做事数学 mathematics, maths(BrE), math(AmE) 公理 axiom 定理 theorem 计算calculation 算 operation 证明 prove 假设 hypothesis, hypotheses(pl.) 命题 proposition 算术 arithmetic 加 plus(prep.), add(v.), addition(n.) 被加数 augend, summand 加数 addend 和 sum 减 minus(prep.), subtract(v.), subtraction(n.) 被减数 minuend 减数 subtrahend 差 remainder 乘times(prep.), multiply(v.), multiplication(n.) 被乘数 multiplicand, faciend 乘数 multiplicator 积 product 除 divided by(prep.), divide(v.), division(n.) 被除数 dividend 除数 divisor 商 quotient 等于 equals, is equal to, is equivalent to 大于 is greater than 小于 is lesser than 大于等于 is equal or greater than 小于等于 is equal or lesser than 运算符operator 平均数mean 算术平均数arithmatic mean 几何平均数geometric mean n个数之积的n次方根倒数(reciprocal) x的倒数为1/x 有理数 rational number 无理数 irrational number 实数 real number 虚数 imaginary number 数字 digit 数 number 自然数 natural number 整数 integer 小数 decimal 小数点 decimal point 分数 fraction 分子 numerator 分母 denominator 比ratio 正 positive 负 negative 零 null, zero, nought, nil 十进制 decimal system 二进制 binary system 十六进制 hexadecimal system 权 weight, significance 进位 carry 截尾 truncation 四舍五入 round 下舍入 round down 上舍入 round up 有效数字 significant digit 无效数字 insignificant digit 代数 algebra 公式 formula, formulae(pl.) 单项式 monomial 多项式polynomial, multinomial 系数 coefficient 未知数 unknown, x-factor, y-factor, z-factor 等式，方程式 equation 一次方程 simple equation 二次方程quadratic equation 三次方程 cubic equation 四次方程 quartic equation 不等式 inequation 阶乘 factorial 对数 logarithm 指数，幂 exponent 乘方power 二次方，平方 square 三次方，立方 cube 四次方 the power of four, the fourth power n次方 the power of n, the nth power 开方 evolution, extraction 二次方根，平方根 square root 三次方根，立方根 cube root 四次方根 the root of four, the fourth root n次方根 the root of n, the nth root sqrt(2)=1.414 sqrt(3)=1.732sqrt(5)=2.236 常量 constant 变量 variable 坐标系 coordinates 坐标轴x-axis, y-axis, z-axis 横坐标 x-coordinate 纵坐标 y-coordinate 原点origin 象限quadrant 截距(有正负之分)intercede (方程的)解solution 几何geometry 点 point 线 line 面 plane 体 solid 线段 segment 射线 radial 平行 parallel 相交 intersect 角 angle 角度 degree 弧度 radian锐角 acute angle 直角 right angle 钝角 obtuse angle 平角 straight angle 周角perigon 底 base 边 side 高 height 三角形 triangle 锐角三角形 acute triangle 直角三角形 right triangle 直角边 leg 边 hypotenuse 勾股定理Pythagorean theorem 钝角三角形 obtuse triangle 不等边三角形 scalene triangle 等腰三角形 isosceles triangle 等边三角形 equilateral triangle 四边形 quadrilateral 平行四边形 parallelogram 矩形 rectangle 长 length 宽 width 周长 perimeter 面积 area 相似 similar 全等 congruent 三角trigonometry 正弦 sine 余弦 cosine 正切 tangent 余切 cotangent 正割secant 余割 cosecant 反正弦 arc sine 反余弦 arc cosine 反正切 arc tangent 反余切 arc cotangent 反正割 arc secant 反余割 arc cosecant集合aggregate 元素 element 空集 void 子集 subset 交集 intersection 并集union 补集 complement 映射 mapping 函数 function 定义域 domain, field of definition 值域 range 单调性 monotonicity 奇偶性 parity 周期性periodicity 图象 image 数列，级数 series 微积分 calculus 微分differential 导数 derivative 极限 limit 无穷大 infinite(a.) infinity(n.) 无穷小 infinitesimal 积分 integral 定积分 definite integral 不定积分indefinite integral 复数 complex number 矩阵 matrix 行列式 determinant 圆 circle 圆心 centre(BrE), center(AmE) 半径 radius 直径 diameter 圆周率 pi 弧 arc 半圆 semicircle 扇形 sector 环 ring 椭圆 ellipse 圆周 circumference 轨迹 locus, loca(pl.) 平行六面体parallelepiped 立方体 cube 七面体 heptahedron 八面体 octahedron 九面体enneahedron 十面体 decahedron 十一面体 hendecahedron 十二面体dodecahedron 二十面体 icosahedron 多面体 polyhedron 旋转 rotation 轴axis 球 sphere 半球 hemisphere 底面 undersurface 表面积 surface area 体积 volume 空间 space 双曲线 hyperbola 抛物线 parabola 四面体 tetrahedron 五面体 pentahedron 六面体 hexahedron菱形 rhomb, rhombus, rhombi(pl.), diamond 正方形 square 梯形 trapezoid 直角梯形 right trapezoid 等腰梯形isosceles trapezoid 五边形 pentagon 六边形 hexagon 七边形 heptagon 八边形 octagon 九边形 enneagon 十边形 decagon 十一边形 hendecagon 十二边形dodecagon 多边形 polygon 正多边形 equilateral polygon 相位 phase 周期period 振幅 amplitude 内心 incentre(BrE), incenter(AmE) 外心excentre(BrE), excenter(AmE) 旁心 escentre(BrE), escenter(AmE) 垂心orthocentre(BrE), orthocenter(AmE) 重心 barycentre(BrE), barycenter(AmE) 内切圆 inscribed circle 外切圆 circumcircle 统计 statistics 平均数average 加权平均数 weighted average 方差 variance 标准差 root-mean-square deviation, standard deviation 比例 propotion 百分比 percent 百分点 percentage 百分位数 percentile 排列 permutation 组合 combination 概率，或然率 probability 分布 distribution 正态分布 normal distribution 非正态分布 abnormal distribution 图表 graph 条形统计图 bar graph 柱形统计图 histogram 折线统计图 broken line graph 曲线统计图 curve diagram 扇形统计图 pie diagram abscissa 横坐标 absolute value 绝对值 acute angle 锐角 adjacent angle 邻角 addition 加 algebra 代数 altitude 高 angle bisector 角平分线 arc 弧 area 面积 arithmetic mean 算术平均值(总和除以总数) arithmetic progression 等差数列(等差级数) arm 直角三角形的股 at 总计(乘法) average 平均值 base 底 be contained in 位于...上 bisect 平分center 圆心 chord 弦 circle 圆形 circumference 圆周长 circumscribe 外切，外接 clockwise 顺时针方向 closest approximation 最相近似的combination 组合 common divisor 公约数，公因子 common factor 公因子complementary angles 余角(二角和为90度) composite number 合数(可被除1及本身以外其它的数整除) concentric circle 同心圆 cone 圆锥(体积,1/3*pi*r*r*h) congruent 全等的 consecutive integer 连续的整数coordinate 坐标的 cost 成本 counterclockwise 逆时针方向 cube 1.立方数 2.立方体(体积,a*a*a 表面积,6*a*a) cylinder 圆柱体 decagon 十边形 decimal 小数 decimal point 小数点 decreased 减少 decrease to 减少到 decrease by 减少了 degree 角度 define 1.定义 2.化简 denominator 分母 denote 代表，表示 depreciation 折旧 distance 距离 distinct 不同的 dividend 1. 被除数 2.红利 divided evenly 被除数 divisible 可整除的 division 1.除 2.部分divisor 除数 down payment 预付款，定金 equation 方程 equilateral triangle 等边三角形 even number 偶数 expression 表达 exterior angle 外角face (立体图形的)某一面 factor 因子 fraction 1.分数 2.比例 geometric mean 几何平均值(N个数的乘积再开N次方) geometric progression 等比数列(等比级数) have left 剩余 height 高 hexagon 六边形 hypotenuse 斜边improper fraction 假分数 increase 增加 increase by 增加了 increase to 增加到 inscribe 内切，内接 intercept 截距 integer 整数 interest rate 利率in terms of... 用...表达 interior angle 内角 intersect 相交 irrational无理数 isosceles triangle 等腰三角形 least common multiple 最小公倍数least possible value 最小可能的值 leg 直角三角形的股 length 长 listprice 标价 margin 利润 mark up 涨价 mark down 降价 maximum 最大值 median, medium 中数(把数字按大小排列，若为奇数项，则中间那项就为中数，若为偶数项，则中间两项的算术平均值为中数。
数学专业英文对照表-review
1．Glossary 术语2．algebra, theorem 代数，定理3．perpendicular bisector 垂直平分线4．Radius 半径5．Parallelogram 平行四边形6．isosceles triangle 等腰三角形7．rational \ imaginary numbers 有理数\虚数8．Associative \ commutative \ distributive property 结合律\交换律\分配率9．Ordinal numbers 序数10．cardinal numbers 基数11．complementary angle 余角12．concentric circles 同心圆13．angle bisector 角平分线14．Binomial 二项式15．Triangle 三角形16．Factor \ fraction 因数，因子\分数17．Polygon \polyhedron 多边形\多面体18．prime number 素数19．Odd and Even Numbers 奇数和偶数20．Finite and Infinite Sets 有限集和无限集21．Additive Identity Property 加法单位元22．Commutative Law of Addition 加法交换律23．Associative Law of Addition 加法结合律24．Parentheses 括号25．Addition Sum 加法和26．Subtraction Difference 减法差27．Multiplication Product 乘法积28．Division Quotient Remainder 除法商余数29．The Greatest Common Divisor 最大公约数30．The Least Common Multiple 最小公倍数31．expansion 展开，展开式32．shuffle 洗纸牌33．arithmetic 算术34．arithmetic operation 算术运算35．arithmetic operator 算术运算符36．fraction 分数，分式37．algebra 代数38．algebraic 代数的39．algebraic expression 代数式40．coefficient 系数41．exponent 指数42．term 项43．middle term 中间项44．like terms 同类项45．variable 变量46．perfect square 完全平方47．structure 构造，结构48．reflexive Property 反射性49．symmetric Property 对称性50．transitive Property 传递性51．commutative law 交换律52．associative law 结合律53．distributive laws 分配律54．add(v) /plus(prep) 加55．addition/additive加法56．summand 被加数57．addend 加数58．sum 和59．subtract(v) /minus(prep)减60．subtraction减法61．minuend 被减数62．subtrahend 减数63．difference 差64．multiply乘65．multiplication/multiplicative乘法66．multiplicand被乘数67．multiplier乘数68．product积69．divide 除70．division 除法71．dividend 被除数72．divisor 除数73．quotient 商74．positive number 正数75．negative number 负数76．opposite numbers 相反数77．reciprocal 倒数78．factor out 提取公因式79．opposite sign 符号相反80．monomial 单项的，单项式81．binomial 二项的，二项式82．trinomial三项的，三项式83．polynomial 多项的，多项式84．dimension 尺度, 维(数), 85．rectangle 长方形，矩形86．area 面积87．Factorization 因式分解88．irreducible polynomial 不可约多项式89．fundamental theorem of algebra 代数基本定理90．expansion 展式91．leading coefficient 首项系数92．trinomial 三项式93．binomial 二项式94．Rewrite a trinomial to a product of two binomials95．将三项式改写为两个二项式的乘积96．algebraic 代数的，关于代数的97．exponent 指数，幂98．involution and Evolution 乘方与开方99．square , a square with side a, a-square二次方，平方100．cube三次方，立方101．a (raised) to the power of n, a to the nth (power) a 的n次方102．square root二次方根，平方根103．cube root三次方根，立方根104．the nth root of a the radical sign a的n次方根105．被开方数radicand106．2^-10 2 to the minus tenth107．perfect square trinomials 完全平方式108．sum/difference of two squares 平方和/差公式109．sum/difference of two cubes 立方和/差公式110．rational numbers 有理数111．Let a and b represent ……such that……112．Substitute the value for …into the equation to get corresponding value for…把……的值代入方程，计算出相应的……的值113．curved lines 曲线114．ellipse 椭圆115．partition 分割，分开116．Arithmetic 算术，计算117．Four arithmetic operation 四则运算118．cancellation （相）约，（相）消119．fraction 分数120．divide a by b a除以b121．hundreds digit 百位数122．tens digit 十位数123．units digit 个位数124．reverse the digits 颠倒数字顺序125．Number Patterns 数型，数据模式126．Pattern Recognition 模式识别127．Sequence 数列128．predecessor term前面的项129．triangular number 三角形数130．figurate number 形数131．subsequent 随后的132．legitimate 合理的133．general term 通项134．Pascal triangle 帕斯卡三角形135．horizontal sum 横向之和136．Arithmetic Progression （A.P.）等差数列137．the first term 首项138．nth term 通项139．common difference 公差140．Sn sum of the first n terms 前n项和141．G.P. 等比数列142．H.P. 调和数列143．Arithmetic mean 算术平均数144．Geometric mean几何平均数145．Harmonic mean-调和平均数146．Trimmed Mean切尾均值147．root mean square平方平均148．An arithmetic progression is a sequence in which each term (except the first term) is obtained from the previous term by adding a constant known as the common difference。
Geometric Series
6 ( 31 ) 1 186
5
Geometric Sequences and Series Summing terms of a G.P. e.g. 2 Find the sum of the first 20 terms of the geometric series, 2 6 18 54 . . leaving your answer in index forห้องสมุดไป่ตู้ 3 6 Solution: a 2, r 3 12 n 20
a , ar , ar 2 , ar 3 , . . .
The nth term of an G.P. is
u n ar
n 1
Geometric Sequences and Series Exercises 1. Use the formula for the nth term to find the term indicated of the following geometric sequences
Subtracting the expressions gives
S 5 rS 5 a ar ar 2 ar 3 ar 4 ar ar 2 ar 3 ar 4 ar 5
Move the lower row 1 place to the right
Geometric Sequences and Series Summing terms of a G.P.
Multiply by r:
S 5 a ar ar ar ar
2 3 2 3 4
4 5
rS 5 ar ar ar ar ar
英国A-level数学教材内容汇总
A-Leve l数学(Mathe matl cs)由四亍郃分姐成.换业数学・C ore Mathe matlcs h 力学数学t M ech an les Mathemati csx 轨计數H (Stali^tl cs Mathematitsy 决第數学Decision M ath&mati c& □选择学耳数学(Mathematics)^生，際了孩心数学心时Mathemahcs：^必修的基础数学之尔学生还需^据自己将来的犬学报读若业选择茸中T磯学『力Mechanics Mathematics},统计数学:Statistics f.fathemstics}.决董数学Decision Mathematics]・50将采读工程删]字主.可追力学数学谢xhanlcs);读社会科学觀金融经桥类的.可选:比计數字(Slatistlcs)：僂计算机嗽件类的.町选: 决策数韋Decision Maltieinalics^Core Mathematicsl (AS/A2) ------ 核心数学11. Algebra and fun ctio ns --- 代数和函数2. Quadratic functions ----- 二次函数3. Equati ons and in equalities --- 等式和不等式4. Sketchi ng curves ----- 画图(草图)5. Coordinate geometry in the (x, y) plane--------- 平面坐标系中的坐标几何6. Seque nces and series——数歹U7. Differe ntiation ------ 微分8. In tegrati on --- 积分Core Mathematics2 (AS/A2) ----- 核心数学21. Algebra and fun ctio ns --- 代数和函数2. The sine and cos ine rule ---- 正弦和余弦定理3. Expo nen tials and logarithm ----- 指数和对数4. Coordinate geometry in the (x, y) plane--------- 平面坐标系中的坐标几何5. The bi no mial expa nsion --- 二项展开式6. Radia n measure and its applicati on --- 弧度制及其应用7. Geometric seque nces and series ---- 等比数歹U8. Graphs of trig ono metric functions ----- 三角函数的图形9. Differe ntiation ------ 微分10. Trigonometric identities and simple equations ------ 三角恒等式和简单的三角等式11. I ntegration ---- 积分Core Mathematics3 (AS/A2) ----- 核心数学31. Algebra fractio ns ------ 分式代数2. Functions ------ 函数3. The expo nen tial and log fun ctio ns --- 指数函数和对数函数4. Numerical method ----- 数值法5. Tran sform ing graph of functions ---- 函数的图形变换6. Trigon ometry ------- 三角7. Further trig ono metric and their applicati ons ---- 高级三角恒等式及其应用8. Differe ntiation ------ 微分Core Mathematics4 (AS/A2) ----- 核心数学41. Partial fractio ns ---- 部分分式2. Coordinate geometry in the (x, y) plane--------- 平面坐标系中的坐标几何3. The bi no mial expa nsion --- 二项展开式4. Differe ntiation ------ 微分5. Vectors ----- 向量6. In tegrati on --- 积分A-Level :核心数学 Core Maths ，力学数学，统计数学，决策数学 1 2 3 4 5 6 7oio14 14 1520 24 252b3S丽48b4Core Mathematics1 (AS/A2 ) ----- 核心数学 1 8. In tegrati on ------ 积分 每章内容：SketclSketching < \4.1 4.2 4.3 4.44.5Quadratic functions 2A2.2 2.3 2.4 2.5 2.6 1 Algebra and futictlons1.1 1.2 L3 1用 IS L6 L7 L8 Summary of key poinisPlotting the s^phs of Solvingquadratic eqi 】“ 沪 巧 Completingthe: Solving quddratiuSolving quadrate t ions by “Sketching Z> JSummary of kfy 尸為Equations 匚 M Solving sinSolving simultaneob.Using substISolving linGi 『 in 何亦It 華& Solving quadratic^^^ii^sj.jtions by elimination 屈tion* by substitutinn f equation Is linear and the other is quadraticSimplifying an expression by collecting like termsThe laws Qf indicesExpanding an expression Factorhing an expressionFactorising a quadr^k expressionThv las\s of indices for dll rational exponents The use and nianipulation of it rdsRationalising the iknonnridtor of a fraction ivhen 才二dw: XCxsE by ractor i sa ti映>；肯『 c equtijLArby comgTfctjng th 」square .' 'J u iht^m uh Quadratic formulae ； ^ncc^is liiKar rq 3.13.2 33 3J 3.5Suiiunary of 匕叮心疋试he 护ph$ of cubic functions Interpr^lW^yaphs nf cubk fuiKtioiuSketch inutile reciprocal function JK ■ttivinicr sect ion points of o[ functions to solv< equations of the triinsfbr mat ions f(x + ⑷ dnd 冃工-川 舉effect of the transforiiiations fiux) and'Fftrfotming transformations on the sketches of curves詁ry of key pointsAlgebra and fun ctio ns ----- 代数和函数 Quadratic functions ----- 二次函数Equati ons and in equalities --- 等式和不等式 Sketchi ng curves --- 画图(草图)Coordinate geometry in the (x , y ) plane -------- 平面坐标系中的坐标几何 Seque nces and series ——数列Differe ntiation ------ 微分5 Coordinate geonwtry in the (x9 y) plane 6S5.1 The equation of a straight line in the form y = nix + c or ax + 如 + c = 0 655.2 The gradient of a straight line 605.3 The equation of a straight line of the form y - y严ifi(x - 心) 7&5.4 The formula for finding the equation of a straight line5.5 The conditions for two straight lines to be parallel or perpendicular ' 75Summary of key points6 Sequences and series6.1 Introduction to sequences6.2 The nth term of a sequence 0 836.3 Sequences generated by a recurrence relationship i \ // ()856.4 Arithmetic sequences 二886.5 Arithmetic series \C/^ \ °°6.6 The sum to n of an arithmetic series 936.7 Using X notation 」97Summary of key points ' 〃丿) 101 7 Differentiation (//. 1027」The derivative of f(x) as the thiCpn^kto tft^ graph y = f(x) 102105109113114115116117121122122124125126128130Core Mathematics2 (AS/A2 ) 核心数学 21. Algebra and fun ctio ns ---- 代数和函数2. The sine and cos ine rule ---- 正弦和余弦定理3. Expo nen tials and logarithm ---- 指数和对数4. Coordinate geometry in the (x , y ) plane -------- 平面坐标系中的坐标几何5. The bi no mial expa nsion --- 二项展开式6. Radia n measure and its applicati on --- 弧度制及其应用7. Geometric seque nces and series --- 等比数歹 U8. Graphs of trig ono metric functions ---- 三角函数的图形9. Differe ntiation ------ 微分 10. Trigonometric identities and simple equations ------ 三角恒等式和简单的三角等式 11. I ntegration ---- 积分 每章内容： Aigcbrj dEid luiKtions 1J1.2 13 1.4 Simplifying algebraic fractions by division Dividing apolynomial by (x i p)Factorising a polynomial using the Factor TheoremUsing the Remainder Theorem Summary of k (?y pointsThe sint ： and cosine ruleUsing the sine rule to find missing sides Using the sine nde find unF^wn angles The rule andfinding two w* Using the cosine ruEc ia Fin# Using the cosine rule tc a Using the sine tl «Calculaikng the area 2A 2.2 23 2.4 2.5 2.6 2.7 f or a nih^F Eo切 Mck ssing an^ic^ L . ■ #4RI le 3 nr< !'『 庶耳竝遁 Theo re mot^Jy^ngle us)闵jExponctuiah an<r^ogaMh * 3J王2 玉33.43.5 3.6Summary of key pointsCk Coordinate in the (x, y\ plant4.1 The 4.2 The ciiibi Suniinjjy of key polrt 115 10 131718 18 21 23 24 27 30 32 36 37 37 39 4() 41 43 45 4ti 49 49 57 60 68 70 70 72 73 75 79tnsTh<bfunctk 严 Writing ns as a Calculating *丄耳 to Laws of JogarithmS Solvi ng equations 汐 a' - b Changing the mt ni ot A line M 峥曲亡two points on a line 4*3 The equatitJiiif a circle Summiiry of fr r/ points iriomTal expansion s triangle X Combinntions and factorialUsing (：) m the binomial expansion5-4 Expanding (d + bxY r using the binomial expansion Summary of key point*11o Kaaian measure ana its applications Using radians to measure angles The length ofthe arc of a circle The area of a sector of a circle The area of a segment of a circle Geometric sequences Geometric progressions and the nth term Usinggeometric sequences to solve problems The sum of a geometric seriesThe sum to infinity of a geometric series Graphs of trigonometric functionsSine, cosine andtangent(unctionsThe values of trigonomef/ functions in the Exact values and surds f Graphs of sine 0f cos J J 、 Simple transformants oDifferentiation9.1 Increasing s ・9.2 Stationarymaximun 、, minipjum and points of inflexion 9.3Using f^rninjf points to Summar 1 “ 亠 inisTrigonom^/ Jidentitie】0.161 6.2 6.3 6.4Summary of key pointsGeometric sequences and series 7.1 7.2 7.3 7.4 7.5Summary of key points8.1 8.2 8.3 8.4 &5Summary of key poE ；' le equations titles ometrical equations e formsin(nd + a), cos(n0 + a) and tan(n0 + a) = k ig?nometrical equationsSimple trigoSolving simj SolvingeqySolving qudIntegratio11.1 11.2 11.3 Are n.4 94 94 9598 100 103 109 110 110 114 117 118 121 127 141 141 146 149 151 156157 157 159 161 164 169 17710129 129 131135 1406 93 ms10.210.310.4Summary ote integrationa curvea curve that gives negative values n a straight line and a curve rapezium Rule of key points11Core Mathematics3 (AS/A2 ) 核心数学 31 2 3 4 5 6 7 8101Ki.S1 12每章内容:7.5 i'hc racloi tbrinuiai'Alxvbrdit Iriiciions1.1 1.2 1.3I'rigonomctry64 6.2AjipJying a corn^ixiatj Sketching trar^8 Differentiation8.1 B.2 8.3 K 4 8.58.68.71281301311322 E r unctioi-i^2,1 2.2 2-3 2.4 2.5Differentiating ti&ing the chain rule Differ ent tatlng using the product rult Differs nt latL ng using the quotient rukr I if fere nt iat j ng the exponential function Finding the differential of the logarithmic function. _Differentiating 5in x(C~Di fferenti ati ng cos xDifferent is tin^ t^n xDifferenliatkng further trigonometrLcaJ functions[differentiating functians formed by combining frigon 九丁贰#乎卜 cxprtncniiaL logfkritl-imLc and polynomial fLinctior^ ;Simplify algebraic fractions by LUI 1{.C IL UI ^ 口 Multiply dix jdici^frjLiujii->Adding and subtracting algvbrd k frautionsI nx alxvbriiit fr*ittic.jri^ jind tiir rcn )i»iii<lvr Ltit.-c.i^int <ln y 4-cn U>iTi 耳$ [JCX JJ€»TLOkl t k J I fLIIlC'Floriiir^^of <> gr^jj^/ica11 y 「_ ___2 =」cth^Js^lcrlind approximate root 萤 of 陶仟彳Tran 露Fermi 订呂 graf^/of fui^ctiini^5-1 Sketch! tig graphs ot 1^hhockx!^^ 4^u net ion y 一 lf(x)l 5u2 Sketching g^r^phs y = f(lxl) (A p olvin^a mcxluliis mictions to sketch erv«?« 什fih 订花JJ CFIM H 11 台 Mlielling lhe co-ordinates ofgiven H, cosecant 仇 and cotaingEfU 丹 f ^tant 也 cosecant 优 and cotangent 甘 xpressicmsj proving iclentiti^ iind solving equations, usingMapping diingrarns and 耳of opaeiiitions ( JFunc-tions <irid functioii notatk>tiRange, mapping diagrams,, graphs and definitionsJUsing composite functions #*f 丿Finding and using inverse-The exponential and log f u net ion s°3/1 Introdticing exponent ial ・ rtions of lhe I'omCj^ . h 心 3・立 Graphs of exponential旷卞：」^ 前m 扌匸卩 存占；逆"二tlxfn 琴， 严U^irig 护 ^Eidinwu b©■—主亠二亠亠」一■■Numerical method? ” 4.1 Finding approx if 4.2 U^ing ilerati algebraic iiicthi l ^irs'lw^nnd approximate rt>ots The fijnr/Q?6?/I TieSimplifying £ sec 他 cowO?R and cot Hidcnlitles l 十 lan 2^ = $2H and 1 + cot-^ = cosec 2IJs.iriglmerse trigcinometricai Uinclions and their graphs7 ..Further tngimonietrk identities and theif applies HonsSt/LMrig addition trigoiionietrical lormulac二Using double an^lc trigoiiDmctrical farmulae7?T Solving equdtiom and proving Idcntltiics using doubk iirigle foniiuLie ^^7 4 Usin^ the fonii a cos b sin B lin striving trigonotnetrical piobiennAlgebra fractions ------ 分式代数 Functions ----- 函数Transforming graph of functions -------函数的图形变换 Trigonometry ----- 三角Further trigonometric and their applications ------ 高级三角恒等式及其应用 Differentiation ------ 微分The exponential and log functions Numerical method ------ 数值法指数函数和对数函数Core Mathematics3 (AS/A2 ) 核心数学4Core Mathematics3 (AS/A2 ) 核心数学 51 2 3 4 556 vector37074 11B4 2110ft6J1111126139 SI6264 fi2 «2 AIXJUL L l||\ ULHJK2A3.2 33 Exam style paperFormulae you need to know List of symbols and notation AnswersIndexof two vectors n of a straightAdding and subtracting algehraic fractionsPartial fractions with two linear factors in the denominatorPartial fractions with ttnee or mor^ linear factors in th<? denominator Partial tract ions with repeated linear factors in the denominator Improper fractions into partial fractions1 Partial fractions L:y 1 ■jitrgrating £t^ndard Junctions Integrating using the reverse chain rule Using trigonometric identities in integrationUsing partial fractions to Integrate expressionsUsing standard patterns to integrdle expre^iorr liitvgraUon by subtjtiti.ition Integration by parts Numericalintegration Integration to find ateas and volumes 1Using integration to solvedifferential equations Difkrtntiai rquatjom in context2 Cootdinate geometry in the (x, y) 2」Parametric equations used toParametric equations used to dtiine the uxirdin^tes ot a Using paranictrkequ 訓 UKndinate 驴oimtr* Converting paramet^. jitions into cartesian 世qiut 档才 Finding the itrea ^iidche airve given by pannr 严旷 ^quations3 Fhe binomial ex3,i UMII^ VtXUMl IU UtSUilW J-^JJJLS I ；In 2 or 3 dimensions 二,二二 55Cartesian toniponeidi Gf a \yytor in 2dimensionsCartesian components ol in 3 dimensio%7^； Extending 2 /悸幺？冲results io ]he seal;| The vect*[nUT^clrnjfetraighi line vector 戸理逖石kFx linesJo between two straight Using partial fracti>#w$ Kjtw tiiv ■binamiai expanjy^f \、 Different la Uon4.1 Differentki(I nti ；ons givenpararnetricaifrf/4 2 Diffenyitiating^uationwhich arc implicitO43 Diffett»y^a!ing the function a 1 4.4｛垃 tSftitiibn and rates of change4.5 唏蛙少他rtrntjai equations 5 VecS^ ?<^54,Ve?tor d^fmitipns 4nd vector ^^iiAgrams r 、§,2 Vector arithmetic and the unit vectorThe binomial expulsion a - positive integral index Using the binomidexpand + l^x)"\ j ' 6. In tegrati on ------ 积分 每章内容：The bi no mial expa nsion --- 二项展开式 Differe ntiation ------微分 Vectors ----- 向量Partial fractio ns ---- 部分分式 Coordin ate geometry in the ( x , y ) pla ne 平面坐标系中的坐标几何。
调和级数求和近似公式
调和级数求和近似公式Finding the sum of a harmonic series can be a challenging task, particularly as the series diverges to infinity. Fortunately, there is an approximate formula that can provide a reasonable estimate of the sum. This formula involves taking the natural logarithm of the number of terms in the series and adding a constant known as Euler's constant, denoted by γ.寻找调和级数的和可能是一项具有挑战性的任务，特别是当级数发散到无穷大时。
幸运的是，有一个近似的公式可以提供对和的合理估计。
这个公式涉及取系列中的项数的自然对数，然后加上一个称为欧拉常数的常数，用γ表示。
When dealing with a harmonic series, it is crucial to first understand the nature of the series and how it diverges. A harmonic series is the sum of a sequence of reciprocal numbers, such as 1/1, 1/2, 1/3, and so on. As more terms are added to the series, it diverges to infinity, meaning that the sum grows without bound. This divergence presents a challenge when trying to calculate the exact sum of the series, which is why an approximate formula is often used instead.在处理调和级数时，首要的是要先了解级数的性质以及它的发散方式。
高等数学专业名词中英文对照(全面)
微积分英文词汇,高数名词中英文对照，高等数学术语英语翻译一览关键词：微积分英文，高等数学英文翻译,高数英语词汇来源:上海外教网| 发布日期：2008—05-16 17:12V、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 Simpson'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 ：二项级数微积分词汇第一章函数与极限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 a convergent sequence子列subsequence函数的极限limits of functions函数当x趋于x0时的极限limit of functions 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作者：新少年特工2007-10-8 18：37 回复此发言-—--——-—---——————-—----—--———----—---——-—-——-————--————-—-—————-——————————--———-2 高等数学-翻译等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数在x0连续the function is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数在该区间上连续function 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 derivatives在闭区间【a,b】上可导is derivable 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 a surface at a point曲线在一点的法线normal line to a surface 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volume element in rectangular coordinate system 柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system 球面坐标spherical coordinates球面坐标系中的体积元素volume element 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 differential equation 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 coefficientV、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。
geometric progression公式
一、引言几何级数（geometric progression）是数学中非常重要的一种数列形式，它具有简单并且清晰的规律，应用范围广泛。
在数学、金融、经济学等领域都有着重要的意义。
本文将带您深入了解几何级数的公式及其相关知识。
二、什么是几何级数几何级数是指一个等比数列的和。
等比数列是指数列中的每一项与它的前一项的比相等的数列。
其一般形式可以表示为a,ar,ar^2,ar^3,...,ar^n,...。
其中，a为首项，r为公比。
三、几何级数的公式1. 几何级数的前n项和公式设几何级数的首项为a，公比为r，前n项和为S_n。
则有以下公式：S_n = a(1-r^n)/(1-r)2. 几何级数的通项公式几何级数的通项公式可表示为：a_n = a * r^(n-1)四、几何级数的性质1. 收敛性当公比|r|小于1时，几何级数收敛，其和为S = a/(1-r)；当|r|大于等于1时，几何级数发散。
2. 无穷级数几何级数也可以是一个无穷级数，即n趋向无穷时的和，其公式为S = a/(1-r)。
当|r|小于1时，无穷几何级数的和存在且等于a/(1-r)；当|r|大于等于1时，无穷几何级数发散。
3. 利用几何级数求和只要公比不等于1，几何级数的和可以用简单的公式表示。
这一性质使得几何级数在数学和应用中具有重要的意义。
五、几何级数的应用1. 财务领域在投资和贷款中，几何级数的概念被广泛应用。
复利计算中的每期利息就是一个几何级数。
2. 自然科学几何级数也在自然科学中有着广泛的应用。
生物的繁殖、细胞的分裂等过程都可以用几何级数来进行建模和分析。
3. 工程技术在工程技术中，几何级数被用来描述增长率、递减率等问题，例如人口增长、物种灭绝等。
六、结语几何级数在数学和应用领域都具有重要的意义。
它的公式简洁清晰，性质明确，应用广泛。
通过深入理解几何级数的公式及其性质，我们可以更好地应用它来解决实际问题，推动科学技术的发展。
希望本文能够带给您对几何级数的全面了解，并在您的学习和工作中发挥积极的作用。
微积分常用公式
decimal小數
decreasing遞減
dimensional維數
dot product點積
direction angle方向角
direction cosine方向餘弦
determinant行列式
direction number方向數
density密度
double integrals雙重積分
微積分常用公式及常用述語中英文對照
微積分常用述語中英文對照
AChinese
absolute maximum絕對最大值
absolute minimum絕對最小值
absolutely convergent絕對收斂
absolute value絕對值
acceleration加速度
alternating series交錯級數
intercept截距
interval of convergence收斂區間
inverse function反函數
Inverse trigonometric function反三角函數
infinity無限大
irrational number無理數
inference推理
interval區間
integral exponent整數指數
cardioid心臟線
center of mass質心
Chain Rule鏈式法則
closed interval閉區間
coefficient係數
Complex numbers複數
Conclusion(delete)結論(delete)
conditional inequality條件不等式
coordinate座標
一桶水倒掉一半再倒掉一半的题目
一桶水倒掉一半再倒掉一半的题目One of the most famous and intriguing problems in mathematics is the "bucket problem." It goes as follows:you have a bucket filled with water, and you pour out halfof the water. Then, you pour out half of the remaining water. This process continues indefinitely. The question is, how much water is left in the bucket after an infinite number of pours?From a mathematical perspective, this problem is a classic example of a geometric series. Each time you pourout half of the water, you are left with half of what you had before. So, if we let x represent the original amountof water in the bucket, the amount of water left after the first pour is x/2, after the second pour is (x/2)/2 = x/4, and so on.To determine the amount of water left after an infinite number of pours, we need to find the sum of this infinite geometric series. The formula for the sum of an infinitegeometric series is S = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = x and r = 1/2.Using the formula, we can calculate the sum of the series as S = x / (1 - 1/2) = x / (1/2) = 2x. Therefore, after an infinite number of pours, there will be 2 times the original amount of water left in the bucket.Now, let's look at this problem from a more intuitive perspective. Imagine you have a bucket filled with water, and you start pouring out half of the water. It's easy to see that after the first pour, the water level is reduced by half. But what happens after the second pour? It seems like there should be even less water left, right?As you continue pouring out half of the water, it becomes clear that the amount of water left is getting closer and closer to zero. However, it never actually reaches zero. No matter how many times you pour out half of the water, there will always be a tiny amount left.This concept is known as an "infinitesimal" in mathematics. It represents an infinitely small quantitythat is greater than zero but smaller than any positive number. In the case of the bucket problem, the amount of water left after an infinite number of pours is an infinitesimal.From a philosophical standpoint, this problem raises interesting questions about the nature of infinity and the limits of human understanding. Our intuition tells us that if we keep pouring out half of the water, eventually there will be nothing left. But mathematics shows us that even after an infinite number of pours, there will always be some water remaining.This paradox challenges our preconceived notions and reminds us that sometimes reality is stranger than we can imagine. It highlights the limitations of our intuition and the importance of relying on rigorous mathematical reasoning to uncover the truth.In conclusion, the bucket problem is a fascinatingmathematical puzzle that explores the concept of an infinite series. From a mathematical perspective, the sum of the series reveals that there will be 2 times the original amount of water left after an infinite number of pours. However, from an intuitive and philosophical standpoint, this problem challenges our understanding of infinity and raises thought-provoking questions about the nature of reality. It serves as a reminder that sometimes the answers to our questions lie beyond our intuition and require us to delve into the realm of mathematics to uncover the truth.。
美国大学生数学建模MCM 数学专用名词
美国大学生数学建模MCM 数学专用名词augmented matrix增广矩阵asymptotic渐进的asymptote渐进线asymmetrical非对称的associative law结合律ascending上升的arrangement排列arithmetic算术argument幅角，幅度，自变量，论证area面积arc length弧长apothem边心距apex顶点aperiodic非周期的antisymmetric反对称的antiderivative原函数anticlockwise逆时针的annihilator零化子angular velocity角速度angle of rotation旋转角angle of incidence入射角angle of elevation仰角angle of depression俯角angle of circumference圆周角analytic space复空间analytic geometry解析几何analytic function解析函数analytic extension解析开拓amplitude幅角，振幅alternative互斥的alternate series交错级数almost everywhere几乎处处algebraic topology代数拓扑algebraic expression代数式algebraic代数的affine仿射(几何学)的admissible error容许误差admissible容许的adjugate伴随转置的adjoint operator伴随算子adjoint伴随的adjacency邻接additive加法，加性acute angle锐角accumulation point聚点accidential error偶然误差accessible point可达点abstract space抽象空间abstract algebra抽象代数absolute value绝对值absolute integrable绝对可积absolute convergent绝对收敛Abelian阿贝尔的，交换的balance equation平衡方程bandwidth带宽barycenter重心base基base vectors基向量biased error有偏误差biased statistic有偏统计量bilinear双线性的bijective双射的bilateral shift双侧位移的binomial二项式bisector二等分线，平分线boundary边界的，边界bounded有界的broken line折线bundle丛，把，卷calculus微积分calculus of variations变分法cancellation消去canonical典型的，标准的canonical form标准型cap交，求交运算capacity容量cardinal number基数Cartesian coordinates笛卡尔坐标category范畴，类型cell单元，方格，胞腔cell complex胞腔复形character特征标characterization特征circuit环路，线路，回路circular ring圆环circulating decimal循环小数clockwise顺时针方向的closed ball闭球closure闭包cluster point聚点coefficient系数cofinal共尾的cohomology上同调coincidence重合，叠和collinear共线的collective集体的columnar rank列秩combinatorial theory组合理论common tangent公切线commutative交换的compact紧的compact operator紧算子compatibility相容性compatible events相容事件complementary余的，补的complete完全的，完备的complex analysis复变函数论complex potential复位势composite复合的concave function凹函数concentric circles同心圆concurrent共点conditional number条件数confidence interval置信区间conformal共形的conic圆锥的conjugate共轭的connected连通的connected domain连通域consistence相容，一致constrained约束的continuable可延拓的continuity连续性contour周线，回路，轮廓线convergence收敛性convexity凸形convolution对和，卷积coordinate坐标coprime互质的，互素的correspondence对应coset陪集countable可数的counterexample反例covariance协方差covariant共变的covering覆盖critical临界的cubic root立方根cup并，求并运算curl旋度curvature曲率curve曲线cyclic循环的decade十进制的decagon十边形decimal小数的，十进制的decision theory决策论decomposable可分解的decreasing递减的decrement减量deduction推论，归纳法defect亏量，缺陷deficiency亏格definition定义definite integral定积分deflation压缩deflection挠度，挠率，变位degenerate退化的deleted neighborhood去心邻域denominator分母density稠密性，密度density function密度函数denumerable可数的departure偏差，偏离dependent相关的dependent variable因变量derangement重排derivation求导derivative导数descent下降determinant行列式diagram图，图表diameter直径diamond菱形dichotomy二分法diffeomorphism微分同胚differentiable可微的differential微分differential geometry微分几何difference差，差分digit数字dimension维数directed graph有向图directed set有向集direct prodect直积direct sum直和direction angle方向角directional derivative方向导数disc圆盘disconnected不连通的discontinuous不连续的discrete离散的discriminant判别式disjoint不相交的disorder混乱，无序dissection剖分dissipation损耗distribution分布，广义函数divergent发散的divisor因子，除数division除法domain区域，定义域dot product点积double integral二重积分dual对偶dynamic model动态模型dynamic programming动态规划dynamic system动力系统eccentricity离心率econometrics计量经济学edge棱，边eigenvalue特征值eigenvector特征向量eigenspace特征空间element元素ellipse椭圆embed嵌入empirical equation经验公式empirical assumption经验假设endomorphism自同态end point端点entropy熵entire function整函数envelope包络epimorphism满同态equiangular等角equilateral等边的equicontinuous等度连续的equilibrium平衡equivalence等价error estimate误差估计estimator估计量evaluation赋值，值的计算even number偶数exact sequence正合序列exact solution精确解excenter外心excision切割，分割exclusive events互斥事件exhaustive穷举的expansion展开，展开式expectation期望experimental error实验误差explicit function显函数exponent指数extension扩张，外延face面factor因子factorial阶乘fallacy谬误fiducial置信field域，场field theory域论figure图形，数字finite有限的finite group有限群finite iteration有限迭代finite rank有限秩finitely covered有限覆盖fitting拟合fixed point不动点flag标志flat space平旦空间formula公式fraction分数，分式frame架，标架free boundary自由边界frequency频数，频率front side正面function函数functional泛函functor函子，算符fundamental group基本群fuzzy模糊的gain增益，放大率game对策gap间断，间隙general topology一般拓扑学general term通项generalized普遍的，推广的generalized inverse广义逆generalization归纳，普遍化generating line母线genus亏格geodesic测地线geometrical几何的geometric series几何级数golden section黄金分割graph图形，网格half plane半平面harmonic调和的hexagon六边形hereditary可传的holomorphic全纯的homeomorphism同胚homogeneous齐次的homology同调homotopy同伦hyperbola双曲线hyperplane超平面hypothesis假设ideal理想idempotent幂等的identical恒等，恒同identity恒等式，单位元ill-condition病态image像点，像imaginary axis虚轴imbedding嵌入imitation模仿，模拟immersion浸入impulse function脉冲函数inclination斜角，倾角inclined plane斜面inclusion包含incomparable不可比的incompatible不相容的，互斥的inconsistent不成立的indefinite integral不定积分independence无关（性），独立（性）index指数，指标indivisible除不尽的inductive归纳的inductive definition归纳定义induced诱导的inequality不等式inertia law惯性律inference推理，推论infimum下确界infinite无穷大的infinite decimal无穷小数infinite series无穷级数infinitesimal无穷小的inflection point拐点information theory信息论inhomogeneous非齐次的injection内射inner point内点instability不稳定integer整数integrable可积的integrand被积函数integral积分intermediate value介值intersection交，相交interval区间intrinsic内在的，内蕴的invariant不变的inverse circular funct反三角函数inverse image逆像，原像inversion反演invertible可逆的involution对合irrational无理的，无理数irreducible不可约的isolated point孤立点isometric等距的isomorphic同构的iteration迭代joint distribution联合分布kernel核keyword关键词knot纽结known已知的large sample大样本last term末项lateral area侧面积lattice格子lattice point格点law of identity同一律leading coefficient首项系数leaf蔓叶线least squares solution最小二乘解lemma引理Lie algebra李代数lifting提升likelihood似然的limit极限linear combination线性组合linear filter线性滤波linear fraction transf线性分linear filter线性滤波式变换式变换linear functional线性泛函linear operator线性算子linearly dependent线性相关linearly independent线性无关local coordinates局部坐标locus(pl.loci)轨迹logarithm对数lower bound下界logic逻辑lozenge菱形lunar新月型main diagonal主对角线manifold流形mantissa尾数many-valued function多值函数map into映入map onto映到mapping映射marginal边缘master equation主方程mathermatical analysis数学分析mathematical expectati数学期望matrix(pl. matrices)矩阵maximal极大的，最大的maximum norm最大模mean平均，中数measurable可测的measure测度mesh网络metric space距离空间midpoint中点minus减minimal极小的，最小的model模型modulus模，模数moment矩monomorphism单一同态multi-analysis多元分析multiplication乘法multipole多极mutual相互的mutually disjoint互不相交natural boundary自然边界natural equivalence自然等价natural number自然数natural period固有周期negative负的，否定的neighborhood邻域nil-factor零因子nilpotent幂零的nodal节点的noncommutative非交换的nondense疏的，无处稠密的nonempty非空的noncountable不可数的nonlinear非线性的nonsingular非奇异的norm范数normal正规的，法线normal derivative法向导数normal direction法方向normal distribution正态分布normal family正规族normal operator正规算子normal set良序集normed赋范的n-tuple integral重积分number theory数论numerical analysis数值分析null空，零obtuse angle钝角octagon八边形octant卦限odd number奇数odevity奇偶性off-centre偏心的one-side单侧的open ball开球operations reserach运筹学optimality最优性optimization最优化optimum最佳条件orbit轨道order阶，级，次序order-preserving保序的order-type序型ordinal次序的ordinary寻常的，正常的ordinate纵坐标orient定方向orientable可定向的origin原点original state初始状态orthogonal正交的orthonormal规范化正交的outer product外积oval卵形线overdetermined超定的overlaping重叠，交迭pairity奇偶性pairwise两两的parabola抛物线parallel平行parallel lines平行线parallelogram平行四边形parameter参数parent population母体partial偏的，部分的partial ordering偏序partial sum部分和particle质点partition划分，分类path space道路空间perfect differential全微分period周期periodic decimal循环小数peripheral周界的，外表的periphery边界permissible容许的permutable可交换的perpendicular垂直perturbation扰动，摄动phase相，位相piecewise分段的planar平面的plane curve平面曲线plane domain平面区域plane pencil平面束plus加point of intersection交点pointwise逐点的polar coordinates极坐标pole极，极点polygon多边形polygonal line折线polynomial多项式positive正的，肯定的potency势，基数potential位势prime素的primitive本原的principal minor主子式prism棱柱proof theory证明论probability概率projective射影的，投影proportion比例pure纯的pyramid棱锥，棱锥体quadrant像限quadratic二次的quadric surface二次曲面quantity量，数量quasi-group拟群quasi-norm拟范数quasi-normal拟正规queuing theory排队论quotient商radial径向radical sign根号radication开方radian弧度radius半径ramified分歧的random随机randomize随机化range值域，区域，范围rank秩rational有理的raw data原始数据real function实函数reciprocal倒数的，互反的reciprocal basis对偶基reciprocity互反性rectangle长方形，矩形rectifiable可求长的recurring decimal循环小数reduce简化，化简reflection反射reflexive自反的region区域regular正则regular ring正则环related function相关函数remanent剩余的repeated root重根residue留数，残数resolution分解resolvent预解式right angle直角rotation旋转roundoff舍入row rank行秩ruled surface直纹曲面runs游程，取遍saddle point鞍点sample样本sampling取样scalar field标量场scalar product数量积，内积scale标尺，尺度scattering散射，扩散sectorial扇形self-adjoint自伴的semicircle半圆semi-definite半定的semigroup半群semisimple半单纯的separable可分的sequence序列sequential相继的，序列的serial序列的sheaf层side face侧面similar相似的simple curve简单曲线simplex单纯形singular values奇异值skeleton骨架skewness偏斜度slackness松弛性slant斜的slope斜率small sample小样本smooth manifold光滑流形solid figure立体形solid geometry立体几何solid of rotation旋转体solution解solvable可解的sparse稀疏的spectral theory谱论spectrum谱sphere球面，球形spiral螺线spline function样条函数splitting分裂的statistics统计，统计学statistic统计量stochastic随机的straight angle平角straight line直线stream-line流线subadditive次可加的subinterval子区间submanifold子流形subset子集subtraction减法sum和summable可加的summand被加数supremum上确界surjective满射的symmetric对称的tabular表格式的tabulation列表，造表tangent正切，切线tangent space切空间tangent vector切向量tensor张量term项terminal row末行termwise逐项的tetrahedroid四面体topological拓扑的torsion挠率totally ordered set全序集trace迹trajectory轨道transcendental超越的transfer改变，传transfinite超限的transformation变换式transitive可传递的translation平移transpose转置transverse横截、trapezoid梯形treble三倍，三重trend趋势triad三元组triaxial三轴的，三维的trigon三角形trigonometric三角学的tripod三面角tubular管状的twist挠曲，扭转type类型，型，序型unbiased无偏的unbiased estimate无偏估计unbounded无界的uncertainty不定性unconditional无条件的unequal不等的uniform一致的uniform boundness一致有界uniformly bounded一致有界的uniformly continuous一致连续uniformly convergent一致收敛unilateral单侧的union并，并集unit单位unit circle单位圆unitary matrix酉矩阵universal泛的，通用的upper bound上界unrounded不舍入的unstable不稳定的valuation赋值value值variation变分，变差variety簇vector向量vector bundle向量丛vertex顶点vertical angle对顶角volume体积，容积wave波wave form波形wave function波函数wave equation波动方程weak convergence弱收敛weak derivatives弱导数weight权重，重量well-ordered良序的well-posed适定的zero零zero divisor零因子zeros零点zone域，带</Words>。
解析数论是使用数学分析作为工具来解决数论问题的分支
解析数论是使用数学分析作为工具来解决数论问题的分支。
微积分和复变函数论发展以后，产生了解析数论。
该学科的第一个主要成就是狄利克雷用解析方法证明了Dirichlet's theorem on arithmetic progressions。
依靠黎曼zeta函数对素数定理的证明是另一个里程碑。
解析数论是解决数论中艰深问题的重要工具，数论中有些问题必须由解析方法才能提出或解决。
中国的华罗庚、王元、陈景润等人在“哥德巴赫猜想”、“华林问题”等解析数论问题上取得世界公认的成就。
黎曼ζ函数Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics. The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.[1]DefinitionThe Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that Re(s) > 1 and diverges on the open half-plane of s such that Re(s) < 1. The function defined by the series on the half-plane of convergence can however be continued analytically to all complex s≠ 1. For s= 1 the series is formally identical to the harmonic series which diverges to infinity. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region {s∈ C : s≠1} of the complex plane and has a simple pole at s= 1 with residue 1.Specific valuesThe values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.this is the harmonic series.this is employed in calculating the critical temperature for a Bose–Einstein condensate in physics, and for spin-wave physics in magnetic systems.the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime? [2]this is called Apéry's constant.Stefan–Boltzmann law and Wien approximation in physics.Euler product formulaThe connection between the zeta function and prime numberswas discovered by Leonhard Euler, who proved the identitywhere, by definition, the left hand side is ζ(s) and the infiniteproduct on the right hand side extends over all prime numbers p(such expressions are called Euler products):Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s= 1, diverges, Euler's formula implies that there are infinitely many primes. For s an integer number, the Euler product formula can be used to calculate the probability that s randomly selected integers are relatively prime. It turns out that this probability is indeed 1/ζ(s). The functional equationThe Riemann zeta function satisfies the functional equationvalid for all complex numbers s, which relates its values at points s and 1 −s. Here, Γ denotes the gamma function. This functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place. An equivalent relationship was conjectured by Euler in 1749 for the functionAccording to André Weil, Riemann seems to have been very familiar with Euler's work on the subject.[3]The functional equation given by Riemann has to be interpreted analytically if any factors in the equation have a zero or pole. For instance, when s is 2, the right side has a simple zero in the sine factor and a simple pole in the Gamma factor, which cancel out and leave a nonzero finite value. Similarly, when s is 0, the right side has a simple zero in the sine factor and a simple pole in the zeta factor, which cancel out and leave a finite nonzero value. When s is 1, the right side has a simple pole in the Gamma factor that is not cancelled out by a zero in any other factor, which is consistent with the zeta-function on the left having a simple pole at 1.There is also a symmetric version of the functional equation, given by first definingThe functional equation is then given by(Riemann defined a similar but different function which he called ξ(t).)The functional equation also gives the asymptotic limit(GergőNemes, 2007)Zeros, the critical line, and the Riemann hypothesisThe functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s∈ C: 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set{s∈ C: Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that allnon-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result[4]is that ζ(σ+ i t) ≠ 0 whenever |t| ≥ 3 andThe strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, thenThe critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + i14.13472514... Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis Re(s) = 1/2. Furthermore, the fact that ζ(s) = ζ(s*)* for all complex s≠ 1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zero distribution has been studied using rescaled range analysis.[5] The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude, and also for the zeros of other L-functions.The properties of the Riemann zeta function in the complex plane, specifically along parallels to the imaginary axis, has also been studied, by the relation to prime numbers, in recentphysical interference experiments, by decomposing the sum into two parts with opposite phases, ψ and ψ*, which then are brought to interference. [6]For sums involving the zeta-function at integer and half-integer values, see rational zeta series.[O] ReciprocalThe reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius functionμ(n):for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2. [O] UniversalityThe critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.[O] Representations[O] Mellin transformThe Mellin transform of a function f(x) is defined asin the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we havewhere Γ denotes the Gamma function. By subtracting off the first terms of the power series expansion of 1/(exp(x) −1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we haveand when the real part of s is between −1 and 0,We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, thenfor values with We can relate this to the Mellin transform of π(x) bywhereconverges forA similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers p n with a weight of 1/n,so that Now we haveThese expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.Also, from the above (specifically, the second equation in this section), we can write the zeta function in the commonly seen form:[O] Laurent seriesThe Riemann zeta function is meromorphic with a single pole of order one at s= 1. It can therefore be expanded as a Laurent series about s= 1; the series development then isThe constants γn here are called the Stieltjes constants and can be defined by the limitThe constant term γ0 is the Euler-Mascheroni constant.[O] Rising factorialAnother series development valid for the entire complex plane iswhere is the rising factorial This can be used recursively to extend the Dirichlet series definition to all complex numbers.The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss-Kuzmin-Wirsing operator acting on x s−1; that context gives rise to a series expansion in terms of the falling factorial.[O] Hadamard productOn the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansionwhere the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler-Mascheroni constant. A simpler infinite product expansion isThis form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ.[O] Globally convergent seriesA globally convergent series for the zeta function, valid for all complex numbers s except s = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930:The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.[O] ApplicationsAlthough mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum 1 + 2 + 3 + 4 + · · ·, but we can rewrite it as a sum of reciprocals:The sum S appears to take the form of However, −1 lies outside of the domain for which the Dirichlet series for thezeta-function converges. However, a divergent series of positive terms such as this one can sometimes be represented in a reasonable way by the method of Ramanujan summation (see Hardy, Divergent Series.) Ramanujan summation involves an application of the Euler–Maclaurin summation formula, and when applied to the zeta-function, it extends its definition to the whole complex plane. In particularwhere the notation indicates Ramanujan summation.[7]For even powers we have:and for odd powers we have a relation with the Bernoulli numbers:Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.[O] GeneralizationsThere are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. These include the Hurwitz zeta functionwhich coincides with Riemann's zeta-function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function andL-function.The polylogarithm is given bywhich coincides with Riemann's zeta-function when z = 1.The Lerch transcendent is given bywhich coincides with Riemann's zeta-function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).The Clausen function that can be chosen as the real orimaginary part ofThe multiple zeta functions are defined byOne can analytically continue these functions to then-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.[O] Zeta-functions in fiction。
Geometric Sequences and Sums
The Rule
We can also calculate any term using the Rule:
Example:
The Rule for any term is:
xn = 10 × 3(n-1)
So, the 4th term is: x4 = 10×3(4-1) = 10×33 = 270 And the 10th term is: x10 = 10×3(10-1) = 10×39 = 196830
Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing. First, call the whole sum "S": S = a + ar + ar2 + ... + ar(n−2)+ ar(n−1) Next, multiply S by r: S· r = ar + ar2 + ar3 + ... + ar(n−1) + arn
What is that funny Σ symbol? It is called Sigma Notation
And below and above it are shown the starting and ending values:
The formula is easy to use ... just "plug in" the values of a, r and n
a + ar + ar2 + ... + ar(n-1)
AP Calculus BC AP微积分BC考试内容授课大纲
AP Calculus BC AP微积分BC考试内容授课大纲AP微积分BC包含AB的内容，难度也远高于AB，大多数学校也更愿意接受BC的成绩。
一般需要花费1年的时间来完成。
微积分是除了部分文科和艺术类专业外，绝大多数专业都要学的基础课程，专业适用面很广。
AP微积分BC考试将测试你对课程单元所涵盖的数学概念的理解，以及你解决问题所用的适当公式和程序的能力，以及用正确的符号交流工作的能力。
部分考试允许使用绘图计算器。
注意：你不能在同一年内同时参加AP微积分AB和微积分BC考试。
考试时长：3hrs 15mins考试时间：Tue, May 4, 2021, 8 AM Local考试分布：Section 1:Multiple Choice多选题，45道题，时长1hr45mins，占比50%Section 2:Free Response自由回答，6道题，时常1hr30mins，占比50%在此之前学生最好掌握以下的知识点：代数、几何、三角学、解析几何、初等函数的课程，特别是理解线性、多项式、有理函数、指数函数、对数函数、三角函数、反三角函数、分段函数，以及数列、级数、极方程。
你应该知道如何画这些函数的图和解。
你还应该熟悉一般函数的代数变换、组合、综合和逆运算。
课程安排：1、Unit 1: 极限和连续性You’ll start to explore how limits will allow you to solve problems involving change学习用极限解决变化问题，并更好地理解函数的数学推理。
Topics may include:•How limits help us to handle change at an instant 极限如何帮助我们处理瞬间的变化•Definition and properties of limits in variousrepresentations各种表示法中极限的定义和性质•Definitions of continuity of a function at apoint and over a domain函数在一点上和区间上连续性的定义•Asymptotes and limits at infinity渐近线和无穷极限•Reasoning using the Squeeze theorem and theIntermediate Value Theorem用夹逼定理和介值定理进行推理考试占比4%–7%Unit 2:微分：定义与基本性质Unit 2: Differentiation: Definition and Fundamental PropertiesTopics may include:•Defining the derivative of a function at a point and as a function定义在一点处函数的导数和导函数•Connecting differentiability and continuity可微性与连续性•Determining derivatives for elementary functions微分运算法则•Applying differentiation rules微分运算法则考试占比4%–7%Unit 3: Differentiation: Composite, Implicit, and Inverse Functions微分：复合函数、隐函数和反函数掌握使用链式法则，学习新的微分技巧和高阶导数Topics may include:· The chain rule for differentiating posite functions 微分复合函数的链式法则· Implicit differentiation隐函数微分· Differentiation of general and particular inverse functions区分一般反函数和特殊反函数· Determining higher-order derivatives of functions 函数的高阶导数求法考试占比4%–7%Unit 4: Contextual Applications of Differentiation微分的实境应用利用导数设置和解决瞬时变化率的实际问题，利用数学推理求不定式的极限。
Geometric Modeling
Geometric ModelingGeometric modeling is a fundamental concept in the field of computer graphics and design. It involves the creation and manipulation of digital representations of objects and environments using geometric shapes and mathematical equations. This process is essential for various applications, including animation, virtual reality, architectural design, and manufacturing. Geometric modeling plays a crucial role in bringing creative ideas to life and enabling the visualization of complex concepts. In this article, we will explore the significance of geometric modeling from multiple perspectives, including its technical aspects, creative potential, and real-world applications. From a technical standpoint, geometric modeling relies on mathematical principles to define and represent shapes, surfaces, and volumes in a digital environment. This involves the use of algorithms to generate and manipulate geometric data, enabling the creation of intricate and realistic 3D models. The precision and accuracy of geometric modeling are essential for engineering, scientific simulations, and industrial design. Engineers and designers utilize geometric modeling software to develop prototypes, analyze structural integrity, and simulate real-world scenarios. The ability to accurately model physical objects and phenomena in a virtual space is invaluable for testing and refining concepts before they are realized in the physical world. Beyond its technical applications, geometric modeling also offers immense creative potential. Artists and animators use geometric modeling tools to sculpt, texture, and animate characters and environments for films, video games, and virtual experiences. The ability to manipulate geometric primitives and sculpt organic forms empowers creatives to bring their imaginations to life in stunning detail. Geometric modeling software provides a canvas for artistic expression, enabling artists to explore new dimensions of creativity and visual storytelling. Whether it's crafting fantastical creatures or architecting futuristic cityscapes, geometric modeling serves as a medium for boundless creativity and artistic innovation. In the realm of real-world applications, geometric modeling has a profound impact on various industries and disciplines. In architecture and urban planning, geometric modeling software is used to design and visualize buildings, landscapes, and urban developments. This enables architects and urban designers toconceptualize and communicate their ideas effectively, leading to the creation of functional and aesthetically pleasing spaces. Furthermore, geometric modelingplays a critical role in medical imaging and scientific visualization, allowing researchers and practitioners to study complex anatomical structures and visualize scientific data in meaningful ways. The ability to create accurate and detailed representations of biological and physical phenomena contributes to advancementsin healthcare, research, and education. Moreover, geometric modeling is integral to the manufacturing process, where it is used for product design, prototyping,and production. By creating digital models of components and assemblies, engineers can assess the functionality and manufacturability of their designs, leading tothe development of high-quality and efficient products. Geometric modeling also facilitates the implementation of additive manufacturing technologies, such as 3D printing, by providing the digital blueprints for creating physical objects layer by layer. This convergence of digital modeling and manufacturing technologies is revolutionizing the production landscape and enabling rapid innovation across various industries. In conclusion, geometric modeling is a multifaceteddiscipline that intersects technology, creativity, and practicality. Its technical foundations in mathematics and algorithms underpin its applications in engineering, design, and scientific research. Simultaneously, it serves as a creative platform for artists and animators to realize their visions in virtual spaces. Moreover,its real-world applications extend to diverse fields such as architecture, medicine, and manufacturing, where it contributes to innovation and progress. The significance of geometric modeling lies in its ability to bridge the digital and physical worlds, facilitating the exploration, creation, and realization of ideas and concepts. As technology continues to advance, geometric modeling will undoubtedly play an increasingly pivotal role in shaping the future of design, visualization, and manufacturing.。
ch 14.2 Geometric_Series_-_Sum_to_infinity几何级数的和解析
a, ar , ar ,
2
ar ,
3
.
.
.
n 1
The nth term of an G.P. is
The sum of n terms is
Sn a (1 r 1 r
1m
1 2
. . .
m
1 4
m
1 8
m
In theory, we could continue for ever, but the total length would still be 2 metres, so
1 1 1 1
2 4 8
. . . 2
This is an example of an infinite series.
(a ) 2 3
9 2 3
S a 1 r
for
1 r 1
r 1
This can also be written as
e.g. 1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity.
IFY Maths
Geometric Series Sum to Infinity
Suppose we have a 2 metre length of string . . . . . . which we cut in half
数学新词汇的中英翻译
数学新词汇的中英翻译数学新词汇的中英翻译数学新词汇的中英翻译algebraic equation代数方程elementary operations-addition基础混算-加法elementary operations-subtaction基础混算-减法elementary operations-multiplication基础混算-乘法elementary operations-division基础混算-除法elementary operation基础四则混算decimal operations小数混算fractional operations分数混算convert fractional no. into decimal no.分数转小数convert fractional no. into percentage. 分数转百分数convert decimal no. into percentage. 小数转百分数convert percentage into decimal no. 百分数转小数percentage百分数numerals数字符号common factors and multiples公因子及公倍数sorting数字排序area图形面积perimeter图形周界change units : time单位转换-时间change units : weight单位转换-重量change units :length单位转换-长度directed numbers有向数fractional operations分数混算decimal operations小数混算convert fractional no. into decimal no. 分数转小数convert fractional no. into percentage. 分数转百分数convert decimal no. into percentage. 小数转百分数convert percentage into decimal no. 百分数转小数percentage百分数indices指数algebraic substitution代数代入polynomials多项式co-geometry坐标几何学solving linear equation解一元线性方程solving simultaneous equation 解联立方程slope直线斜率equation of straight line直线方程x-intercept ( equation of st. line ) 直线x轴截距y-intercept ( equation of st. line ) 直线y轴截距factorization因式分解quadratic equation二次方程x-intercept ( quadratic equation )二次曲线x轴截距geometry几何学inequalities不等式rate and ratio比和比例bearing方位角trigonometry三角学probability概率statistics-graph统计学-统计图表statistics-measure of central tendency 统计学-量度集中趋势salary tax薪俸税bridging game汉英对对碰indices指数function函数rate and ratio比和比例trigonometry三角学inequalities不等式linear programming线性规划co-geometry坐标几何学slope直线斜率equation of straight line直线方程x-intercept ( equation of st. line ) 直线x轴截距y-intercept ( equation of st. line )直线y轴截距factorization因式分解quadratic equation二次方程x-intercept ( quadratic equation )二次曲线x轴截距method of bisection分半方法polynomials多项式probability概率statistics-graph统计学-统计图表statistics-measure of central tendency 统计学-量度集中趋势statistics-measure of dispersion统计学-量度分布statistics-normal distribution统计学-正态分布surds根式probability概率statistics-measure of dispersion 统计学-量度离差statistics-normal distribution统计学-正态分布statistics-binomial distribution 统计学statistics-poisson distribution 统计学statistics-geometric distribution 统计学co-geometry坐标几何学sequence序列十万hundred thousand三位数3-digit number千thousand千万ten million小数decimal分子numerator分母denominator分数fraction五位数5-digit number公因子common factor公倍数common multiple中国数字chinese numeral平方square平方根square root古代计时工具ancient timing device古代记时工具ancient time-recording device古代记数方法ancient counting method古代数字ancient numeral包含grouping四位数4-digit number四则计算mixed operations (the four operations)加plus加法addition加法交换性质commutative property of addition未知数unknown百分数percentage百万million合成数composite number多位数large number因子factor折扣discount近似值approximation阿拉伯数字hindu-arabic numeral定价marked price括号bracket计算器calculator差difference真分数proper fraction退位decomposition除divide除法division除数divisor乘multiply乘法multiplication乘法交换性质commutative property of multiplication乘法表multiplication table乘法结合性质associative property of multiplication被除数dividend珠算computation using chinese abacus倍数multiple假分数improper fraction带分数mixed number现代计算工具modern calculating devices售价selling price万ten thousand最大公因子highest common factor (h.c.f.)最小公倍数lowest common multiple (l.c.m.)减minus / subtract减少decrease减法subtraction等分sharing等于equal进位carrying短除法short division单数odd number循环小数recurring decimal零zero算盘chinese abacus亿hundred million增加increase质数prime number积product整除性divisibility双数even number罗马数字roman numeral数学mathematics, maths(bre), math(ame)公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.) 被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.) 被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator平均数mean算术平均数arithmatic mean几何平均数geometric mean n个数之积的n次方根倒数（reciprocal）x的倒数为1/x有理数rational number无理数irrational number实数real number虚数imaginary number数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system 权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth power n次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth root n次方根the root of n, the nth rootsqrt(2)=1.414sqrt(3)=1.732sqrt(5)=2.236常量constant变量variable坐标系coordinates坐标轴x-axis, y-axis, z-axis横坐标x-coordinate纵坐标y-coordinate原点origin象限quadrant截距(有正负之分)intercede（方程的）解solution几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理pythagorean theorem 钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle 等边三角形equilateral triangle 四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width周长perimeter面积area相似similar全等congruent三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant补充：集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field of definition 值域range单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.) 无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral 复数complex number矩阵matrix行列式determinant圆circle圆心centre(bre), center(ame) 半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference轨迹locus, loca(pl.)平行六面体parallelepiped 立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron 十二面体dodecahedron 二十面体icosahedron多面体polyhedron旋转rotation轴axis球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space双曲线hyperbola抛物线parabola四面体tetrahedron五面体pentahedron六面体hexahedron菱形rhomb, rhombus, rhombi(pl.), diamond正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon相位phase周期period振幅amplitude内心incentre(bre), incenter(ame)外心excentre(bre), excenter(ame)旁心escentre(bre), escenter(ame)垂心orthocentre(bre), orthocenter(ame)重心barycentre(bre), barycenter(ame)内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation 比例propotion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph 曲线统计图curve diagram扇形统计图pie diagram数学新词汇的中英翻译相关内容:31。
edexcel A-level P1-P4公式
P1Index laws:a m ×a n =a m+n a m ÷a n =a m−n (a m )n =a mn (ab)n =a nb n a 1m=√a ma n m=√a n m=(√a m)na−m=1am a 0=1 (a b )n =a n bn Algorithm of surds:√ab =√a ×√b √a b =√a √bDifference of two squares:x 2−y 2=(x +y )(x −y)Quadratic formula:x =−b ±√b 2−4ac 2aCompleting the square:x2+bx=(x+b2)2−(b2)2ax2+bx+c=a(x+b2a)2+(c−b24a)The gradient of a straight line:m=y2−y1 x2−x1Equations of straight line:y=mx+cax+by+c=0 ,where a,b and c are integersy−y1=m(x−x1)Distance between two points:d=√(x2−x1)2+(y2−y1)2The cosine rule:a2=b2+c2−2bc cos Acos A=b2+c2−a22bcThe sine rule:a sin A =bsin B=csin CAreas of triangles:Area=12ab sin CRadians:2π radians=360°π radians=180°1 radian=180°πArc length:l=rθArea of sectors:A=12r2θArea of segment:A=12r2(θ−sinθ)Differentiating from first principles:f′(x)=limh→0f(x+h)−f(x)hDifferentiating x n:if f(x)=x n,then f′(x)=nx n−1if f(x)=ax n,then f′(x)=anx n−1 Differentiating functions with two or more terms:if y=f(x)±g(x) ,then dydx=f′(x)±g′(x)Integrating x n:if dydx=x n,then y=1n+1x n+1+c,n≠−1if dydx=kx n,then y=kn+1x n+1+c,n≠−1Indefinite integrals:∫x n dx=x n+1n+1+c , n≠−1∫(f′(x)+g′(x))dx=∫f′(x)dx+∫g′(x)dxP2 The midpoint of a line segment:(x1+x22,y1+y22)Equation of a circle:x2+y2=r2(x−a)2+(y−b)2=r2Laws of logarithms:the multiplication law∶log a x+log a y=log a xythe division law∶log a x−log a y=log a(x y )the power law∶log a(x k)=k log a xlog a(1x)=log a(x−1)=−log a xlog a a=1,a>0,a≠1log a1=0, a>0,a≠1Using logarithms to solve equations:whenever f(x)=g(x),log a f(x)=log a g(x) Changing the base of a logarithm:log a x=log b x log b alog a b=1 log b aFactorial notation:n!=n ×(n −1)×(n −2)×⋯×3×2×10!=1The binomial expansion:(a +b)n =a n +(n 1)a n−1b +(n 2)a n−2b 2+⋯+(nr)a n−r b r +⋯+b n ,(n ∈N,N is positive integers)(nr )=nCr =n!r!(n −r)!The formula for the nth term of an arithmetic sequence:u n =a +(n −1)dThe formula for the sum of the first n terms of an arithmetic series:S n =n2(2a +(n −1)d)S n =n2(a +l)The formula for the nth term of an geometric sequence:u n =ar n−1The formula for the sum of the first n terms of an geometric series:S n =a(1−r n )1−r ,r ≠1 S n =a(r n −1)r −1,r ≠1The formula for the sum to infinity of a convergent geometric series:S ∞=a1−r ,|r |<1The formula for trigonometric:sin(180°−θ)=sinθcos(180°−θ)=−cosθtan(180°−θ)=−tanθsin(360°−θ)=−sinθcos(360°−θ)=cosθtan(360°−θ)=−tanθsin(180°+θ)=−sinθcos(180°+θ)=−cosθtan(180°+θ)=tanθTrigonometric identities:(sinθ)2+(cosθ)2=1tanθ=sinθcosθDefinite integrals:∫f′(x)dx=ba[f(x)]a b=f(b)−f(a) Area under curves:A=∫ydxbaThe trapezium rule:∫ydx≈12h(y0+2(y1+y2+⋯+y n−1)+y n)b aP3 and P4 Secant:sec x=1 cos xCosecant:cosec x=1 sin xCotangent:cot x=1 tan xcot x=cos x sin xTrigonometric identities:1+(tan x)2≡(sec x)21+(cot x)2≡(cosec x)2 Compound-angle formulae:sin(A+B)≡sin A cos B+cos A sin Bsin(A−B)≡sin A cos B−cos A sin Bcos(A+B)≡cos A cos B−sin A sin Bcos(A−B)≡cos A cos B+sin A sin Btan(A+B)≡tan A+tan B 1−tan A tan Btan(A−B)≡tan A−tan B 1+tan A tan BDouble-angle formulae:sin2A≡2sin A cos Acos2A≡(cos A)2−(sin A)2≡2(cos A)2−1≡1−2(sin A)2tan2A≡2tan A 1−(tan A)2Trigonometric expressions:asin x±b cos x=R sin(x±α)a cos x±b sin x=R cos(x∓α)a=R cosα,b=R sinα,and R=√a2+b2 Natural logarithms:e ln x=ln(e x)=xe x ln a=a xLogarithms and non-linear data:if y=ax n,then log y=n log x+log aif y=ab x,then log y=log b x+log a Differentiation:if y=e x,then dydx=e xif y=e kx,then dydx=ke kxif y=ln x,then dydx=1xif y=a x,then dydx=a x ln aif y=a kx,then dydx=a kx k ln aif y=sin x,then dydx=cos xif y=sin kx,then dydx=k cos kxif y=cos x,then dydx=−sin xif y=cos kx,then dydx=−k sin kxif y=tan x,then dydx=(sec x)2if y=tan kx,then dydx=k(sec kx)2if y=cosec x,then dydx=−cosec x cot xif y=cosec kx,then dydx=−kcosec kx cot kxif y=cot x,then dydx=−(cosec x)2if y=cot kx,then dydx=−k(cosec kx)2if y=sec x,then dydx=sec x tan xif y=sec kx,then dydx=ksec kx tan kxif y=arcsin x,then dydx=1√1−x2if y=arccos x,then dydx=1√1−x2if y=arctan x,then dydx=11+x2if y=uv,then dydx=udvdx+vdudxif y=uv,thendydx=v dudx−u dvdxv2Parametric differentiation:dy dx =dydtdxdtImplicit differentiation:d dx (f(y))=f′(y)dydxd dx (y n)=ny n−1dydxd dx (xy)=xdydx+yV olume of revolution around the x-axis:Volume=π∫y2dxbaThe chain rule:dy dx =dydu×dudx dydx=1dxdyIntegration:∫x n dx=x n+1n+1+c∫e x dx=e x+c∫1xdx=ln|x|+c∫cos x dx=sin x+c ∫sin x dx=−cos x+c ∫tan x dx=ln|sec x|+c ∫cot x dx=ln|sin x|+c∫cosec x dx=−ln|cosec x+cot x|+c=ln|tan(12x)|+c∫sec x dx=ln|sec x+tan x|+c=ln|tan(12x+14π)|+c∫(sec x)2dx=tan x+c∫(sec kx)2dx=1ktan kx+c∫cosec x cot x dx=−cosec x+c ∫(cosec x)2dx=−cot x+c∫sec x tan x dx=sec x+c∫f′(ax+b)dx=1af(ax+b)+c∫u dvdxdx=uv−∫vdudxdxSeparating the variables to solve differential equations:if dydx=f(x)g(y),then ∫1g(y)dy=∫f(x)dxBinomial series:(1+x)n=1+nx+n(n−1)2!x2+⋯+n(n−1)⋯(n−r+1)r!x r+⋯ (|x|<1,n∈R)(a+bx)n=(a(1+bax))n=a n(1+bax)nMultiply a column vector by a scalar:λ(pq)=(λpλq)Add two column vectors:(pq)+(rs)=(p+rq+s)Magnitude of the vector:|a|=√x2+y2 Equation of a line in three dimensions:r=a+λb The scalar product of two vectors:a∙b=|a||b|cosθa∙b=(a1a2 a3)∙(b1b2b3)=a1b1+a2b2+a3b3。
丌的发展史及其证明方法
丌的发展史及其证明方法The development of mathematics is a vast and intricate journey, spanning millennia and encompassing numerous civilizations. The concept of infinity, a fundamental aspect of mathematics, has been a subject of intense scrutiny and debate throughout this history. The earliest recorded discussions on infinity date back to ancient Greek philosophers, such as Zeno of Elea, who posed paradoxes that challenged the conventional understanding of infinity.数学的发展史是一部跨越千年、涵盖众多文明的广阔而复杂的旅程。
无穷大的概念，作为数学的一个基本方面，在历史上一直是受到密切关注和辩论的主题。
关于无穷大的最早记录讨论可以追溯到古希腊哲学家，如埃利亚的芝诺，他提出了悖论，挑战了人们对无穷大的传统理解。
Over time, mathematicians have devised various methods to tackle the intricacies of infinity. One such method is the use of limits, a concept introduced by mathematicians in the 17th century. Limits allow us to approach infinity without actually reaching it, providing a rigorous foundation for studying infinite sequences and series. For instance, the concept of a limit can be used to determine the sum of an infinite geometric series, a problem that would otherwise be intractable.随着时间的推移，数学家们设计了各种方法来处理无穷大的复杂性。
各种数学名词的英语翻译
各种数学名词的英语翻译各种数学名词的英语翻译数学 mathematics,maths(BrE), math(AmE)公理 axiom定理 theorem计算 calculation运算 operation证明 prove假设 hypothesis, hypotheses(pl.)命题 proposition算术 arithmetic加 plus(prep.), add(v.), addition(n.)被加数 augend, summand加数 addend和 sum减 minus(prep.),subtract(v.),subtraction(n.)被减数 minuend减数 subtrahend差 remainder乘 times(prep.),multiply(v.),multiplication(n.)被乘数 multiplicand,faciend乘数 multiplicator积 product除 divided by(prep.),divide(v.), division(n.)被除数 dividend除数 divisor商 quotient等于 equals, is equal to, is equivalent to大于 is greater than小于 is lesser than大于等于 is equal or greater than小于等于 is equal or lesser than 运算符 operator平均数mean算术平均数arithmatic mean几何平均数geometric mean n个数之积的n次方根倒数（reciprocal） x的倒数为1/x有理数 rational number无理数 irrational number实数 real number虚数 imaginary number数字 digit数 number自然数 natural number整数 integer小数 decimal小数点 decimal point分数 fraction分子 numerator分母 denominator比 ratio正 positive负 negative零 null, zero, nought, nil十进制 decimal system二进制 binary system十六进制 hexadecimal system权 weight, significance进位 carry截尾 truncation四舍五入 round下舍入 round down上舍入 round up有效数字 significant digit无效数字 insignificantdigit代数 algebra公式 formula, formulae(pl.)单项式 monomial多项式 polynomial,multinomial系数 coefficient未知数 unknown, x-factor,y-factor, z-factor等式，方程式 equation一次方程 simple equation二次方程 quadratic equation三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm指数，幂 exponent乘方 power二次方，平方 square三次方，立方 cube四次方 the power of four, thefourth powern次方 the power of n, the nthpower开方 evolution, extraction二次方根，平方根 square root三次方根，立方根 cube root四次方根 the root of four,the fourth rootn次方根 the root of n, thenth rootsqrt(2)=1.414sqrt(3)=1.732sqrt(5)=2.236常量 constant变量 variable坐标系 coordinates坐标轴 x-axis, y-axis,z-axis横坐标 x-coordinate纵坐标 y-coordinate原点 origin象限quadrant截距(有正负之分)intercede（方程的）解solution几何geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect角 angle角度 degree弧度 radian锐角 acute angle直角 right angle钝角 obtuse angle平角 straight angle周角 perigon底 base边 side高 height三角形 triangle锐角三角形 acute triangle 直角三角形 right triangle 直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem钝角三角形 obtuse triangle 不等边三角形 scalene triangle等腰三角形 isosceles triangle等边三角形 equilateral triangle四边形 quadrilateral平行四边形 parallelogram 矩形 rectangle长 length宽 width周长 perimeter面积 area相似 similar全等 congruent三角 trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦 arc sine 反余弦 arc cosine反正切 arc tangent反余切 arc cotangent反正割 arc secant反余割 arc cosecant补充：集合aggregate元素 element空集 void子集 subset交集 intersection并集 union补集 complement映射 mapping函数 function定义域 domain, field ofdefinition值域 range单调性 monotonicity奇偶性 parity周期性 periodicity图象 image数列，级数 series微积分 calculus微分 differential导数 derivative极限 limit无穷大 infinite(a.)infinity(n.)无穷小 infinitesimal积分 integral定积分 definite integral不定积分 indefiniteintegral复数 complex number矩阵 matrix行列式 determinant圆 circle圆心 centre(BrE),center(AmE)半径 radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆 ellipse圆周 circumference轨迹 locus, loca(pl.)平行六面体 parallelepiped立方体 cube七面体 heptahedron八面体 octahedron九面体 enneahedron十面体 decahedron十一面体 hendecahedron十二面体 dodecahedron二十面体 icosahedron多面体 polyhedron旋转 rotation轴 axis球 sphere半球 hemisphere底面 undersurface表面积 surface area体积 volume空间 space双曲线 hyperbola抛物线 parabola四面体 tetrahedron五面体 pentahedron六面体 hexahedron菱形 rhomb,rhombus, rhombi(pl.),diamond正方形 square梯形 trapezoid直角梯形 right trapezoid等腰梯形 isoscelestrapezoid五边形 pentagon六边形 hexagon七边形 heptagon八边形 octagon九边形 enneagon十边形 decagon十一边形 hendecagon十二边形 dodecagon多边形 polygon正多边形 equilateralpolygon相位 phase周期 period振幅 amplitude内心 incentre(BrE),incenter(AmE)外心 excentre(BrE),excenter(AmE)旁心 escentre(BrE),escenter(AmE)垂心 orthocentre(BrE), orthocenter(AmE)重心 barycentre(BrE), barycenter(AmE)内切圆 inscribed circle外切圆 circumcircle统计 statistics平均数 average加权平均数 weighted average 方差 variance标准差 root-mean-square deviation, standarddeviation比例 propotion百分比 percent百分点 percentage百分位数 percentile排列 permutation组合 combination概率，或然率 probability分布 distribution正态分布 normaldistribution非正态分布 abnormal distribution图表 graph条形统计图 bar graph柱形统计图 histogram折线统计图 broken line graph 曲线统计图 curve diagram 扇形统计图 pie diagram--------------数学中常用单词术语abscissa 横坐标absolute value 绝对值acute angle 锐角adjacent angle 邻角addition 加algebra 代数altitude 高angle bisector 角平分线arc 弧area 面积arithmetic mean 算术平均值（总和除以总数）arithmetic progression等差数列（等差级数）arm 直角三角形的股at 总计（乘法）average 平均值base 底be contained in 位于...上bisect 平分center 圆心chord 弦circle 圆形circumference 圆周长circumscribe 外切，外接clockwise 顺时针方向closest approximation最相近似的combination 组合common divisor 公约数，公因子common factor 公因子complementary angles余角（二角和为90度）composite number 合数（可被除1及本身以外其它的数整除）concentric circle 同心圆cone 圆锥（体积＝1/3*pi*r*r*h）congruent 全等的consecutive integer连续的整数coordinate 坐标的cost 成本counterclockwise 逆时针方向cube 1.立方数2.立方体（体积＝a*a*a 表面积＝6*a*a）cylinder 圆柱体decagon 十边形decimal 小数decimal point 小数点decreased 减少decrease to 减少到decrease by 减少了degree 角度define 1.定义2.化简denominator 分母denote 代表，表示depreciation 折旧distance 距离distinct 不同的dividend 1. 被除数 2.红利divided evenly 被除数divisible 可整除的division 1.除 2.部分divisor 除数down payment 预付款，定金equation 方程equilateral triangle等边三角形even number 偶数expression 表达exterior angle 外角 face （立体图形的）某一面factor 因子fraction 1.分数2.比例geometric mean 几何平均值（N个数的乘积再开N次方）geometric progression等比数列（等比级数）have left 剩余height 高hexagon 六边形hypotenuse 斜边improper fraction 假分数increase 增加increase by 增加了 increase to 增加到 inscribe 内切，内接intercept 截距integer 整数interest rate 利率in terms of... 用...表达interior angle 内角 intersect 相交irrational 无理数 isosceles triangle 等腰三角形least common multiple 最小公倍数 least possible value 最小可能的值leg 直角三角形的股length 长list price 标价margin 利润mark up 涨价mark down 降价maximum 最大值median, medium 中数（把数字按大小排列，若为奇数项，则中间那项就为中数，若为偶数项，则中间两项的算术平均值为中数。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Sum
Sn
Number of terms, n
Geometric series – Sum to Infinity We will find a formula for the sum of an infinite number of terms of a G.P. This is called “the sum to infinity”, S
u n ar
n 1
1 r
The sum to infinity is
a ( r n 1 ) Sn r 1
r 1
a S ; 1 r 1 or 1 r
Geometric series – Sum to Infinity Exercises
1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity. 27 . . . (a ) 2 3 9 2 4 1 . . . (b) 3 1 1 3 9
If r = -1, the terms have the same magnitude but they alternate in sign. e.g. 2, -2, 2, -2, . . . A Geometric Series converges only if the common ratio r lies between 1 and 1.
SUMMARY
A geometric sequence or geometric progression (G.P.) is of the form
a , ar , ar 2 , ar 3 , . . .
The nth term of an G.P. is
The sum of n terms is n a (1 r ) or Sn
IFY Maths 1
Geometric Series Sum to Infinity
Geometric series – Sum to Infinity Suppose we have a 2 metre length of string . . . . . . which we cut in half
1
1 2
1 4
1 8
...
1 S 2 1 1 2
Geometric series – Sum to Infinity However, if, for example r = 2, As n increases, 2
As
n
r n 2n
also increases. In fact,
Geometric series – Sum to Infinity As n approaches infinity, 1 2 We write:
As
n ,
12 n 0
n0
n approaches zero.
So, for
r
1, 2
For the series
a (1 r ) Sn 1 r a S 1 r
a S 1 r
for
1 r 1
r 1
This can also be written as
Geometric series – Sum to Infinity e.g. 1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity.
2
Solution:
1 2
1 8
1 32
. . .
r
1 2
1 2 4
so r does satisfy 1 < r < 1
a 2 S S 1 r 1 1e series converges to
1 6
Geometric series – Sum to Infinity
e.g. For the G.P.
1
1 2
1 4
1 8
...
we know that the sum of n terms is given by 1 1 a (1 r n ) Sn Sn
1 n 2 1 1 2 1 n. 2
1 r
As n varies, the only part that changes is This term gets smaller as n gets larger.
1m
1m
We leave one half alone and cut the 2nd in half again
1m
1 2
m
1 4
1 2
m
1 4
. . . and again cut the last piece in half
m
m
1m
1 2
m
Geometric series – Sum to Infinity Continuing to cut the end piece in half, we would have in total
1 2
1 4
1 8
... 2
Geometric series – Sum to Infinity
The series 1 1 1 1 . . . 2
2 4 8
is a G.P. with the common ratio r 1 .
2
Even though there are an infinite number of terms, this series converges to 2.
2n
n ,
The geometric series with r 2 diverges There is no sum to infinity
Geometric series – Sum to Infinity Convergence If r is any value greater than 1, the series diverges. Also, if r 1, ( e.g. r = 2 ), r n as n So, again the series diverges. If r = 1, all the terms are the same.
1
1m
1 2
1 4
1 8
...
1 2
m
1 4
m
1 8
m
In theory, we could continue for ever, but the total length would still be 2 metres, so
1
This is an example of an infinite series.