当前位置：文档之家 › 北邮高等数学英文课件Lecture 11-1

北邮高等数学英文课件Lecture 11-1

  • 格式：pptx
  • 大小：1.99 MB
  • 文档页数：57

下载文档原格式

  / 57

合集下载

lecture 11 比较句式和定语从句的翻译

lecture 11 比较句式和定语从句的翻译

A is to B what C is to D A pen is to a writer what a gun to a soldier. 作家的笔就如同战士的枪一样。
May as well… as和Might as well… as One might as well expect the leopard to change its spots as expect him to change his hot temper.
我们的战士打起仗来既勇敢又善战。 To be uncourteous appears to him little less than a crime. 在他看来，不礼貌无异于犯罪。
As much as, As much, As many与 So many He was greatly respected as much as his brother was despised. 他非常受人尊敬，而他弟弟则受人鄙视。 I was not in the least surprised, for I had expected as much. 我一点都不吃惊，因为我早已了到如此。
他们都是那些抛妻弃子、身背大刀进入深 山的好汉们的后代。
In the early days of the clipper ships, sailors who went on long voyages would often get a dangerous disease called scurvy. 在使用快速帆船的初期，远洋海员们往 往会患一种叫做坏死病的危险疾病。
A causeless event or thing, we cannot think of any more than we can of a stick with only one end. 我们不可能设想有哪件事情是无缘无故产生 的，就象我们不可能设想有哪根棍子会只有 一头一样。 Nobody with any sense expects to find the whole truth in advertisement any more than he expects a man applying for a job to describe his short-coming, and more serious faults. 有头脑的人谁也不会指望广告里说的都是真 的，同样也不指望申请工作的人回说出自己 的缺点和严重过失。

北京邮电大学数学建模课程讲义.doc

北京邮电大学数学建模课程讲义.doc

前言数学模型与数学实验课在我国大多数高等院校中普遍开设，它是当前我国高等教育基础课程教学改革的前沿课程之一。

这一课程的成功开设，为学生正确理解数学教育的重要性以及数学学科与其它诸多专业课之间的内在联系有着非常重要的作用——它是学生在学期间弥合基础理论与各应用学科之间鸿沟的一座桥梁，同时，作为数学教育的一门重要的辅助课程，这门课程的开设也为整合大学数学教学过程中不同数学课程所可能留给初学者的各自孤立甚至极为琐碎的印象成为可能。

因此，如果在教学中处理得当，对这门课程在本科生教育中的重要意义作什么样的肯定性估价均不会过分。

北京邮电大学较早地将数学模型课作为全校公共选修课在本科生中开设，迄今已逾十载。

期间，有多位老师对本课程的教学以及课程建设做出了大量而又细致的工作，更兼期间校院领导对本课程教学的重视，应当说这门课程在我校的教学实践是相当成功的——愈来愈多的学生注册选修本课程和参加有关的数学建模竞赛活动。

特别，我校学生在最近几年参加全国大学生数学建模竞赛和美国大学生数学建模竞赛中连续取得了非常出色的成绩。

本课件的开发可以被看作是对我校最近几年在数学模型课课程教学以及数学建模活动的开展过程中所积累下来的资料所进行的一次较为全面的整理，其内容包括：1）贺祖国老师数学模型课教学以及组织培训参加数学建模竞赛队员过程中所编写的讲义、有关数学建模教学的交流文稿或科研论文等；2）1997年以来，我校学生参加全国大学生数学建模竞赛和美国大学生数学建模竞赛时所完成的参赛论文，甚至包括一些队员在学习期间完成的与数学建模有关的科研论文（这些论文有的在一些比较有影响的学术征文中获奖）；3）其它像数学建模竞赛题目、部分赛题的优秀参赛论文等学习资料。

本课件凝结了许多人的劳动和才智，其中王晓霞副教授（北方交通大学）在课程讲义的组织整理过程中做了大量的工作，孙洪祥、罗守山两位教授为课件的内容组成以及框架构思提供了许多宝贵的建议。

当然，应当特别题及的是在我们身边的许多参加数学建模学习和竞赛活动的同学，没有他们的劳动这项工作将黯然失色——这里资料的原创权有相当大部分是归他们的；网页是由董乘宇、袁楠、刘冰等几位同学合力设计完成的。

高数英文版课件

高数英文版课件

4 2
3 2 3 2
5(1) 3(1)
42((5252))3522
1 0
0 1
A1A 132
252
4 2
5 3
(3214)4(252)22
(3215)5(252)33
1 0
0 1
Determinant of a Matrix
• The quantity ad – bc that appears in the rule for calculating the inverse of a 2 x 2 matrix is called the determinant of the matrix.
• Here, we investigate division of matrices.
– With this operation, we can solve equations that involve matrices.
• The Inverse of a Matrix
Identity Matrices
b d
Matrices - Operations
When the original matrix is square, transposition does not affect the elements of the main diagonal
Aac
b d
AT
a b
c d
The identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal to their transpose since the diagonal is unaffected.

(高等数学英文课件)8.3 Infinite Series

(高等数学英文课件)8.3 Infinite Series
2 4 8 16
目录 上页 下页 返回 结束
目录 上页 下页 返回 结束
目录 上页 下页 返回 结束
Example 2. Consider the Geometric series
a ar ar2 ar3 arn1 arn1 arn a 0
n1
n0
Solution. Consider the partial sum
n1
n0
Solution.
Ifr1,Sna 1 rarn . 1 r
r 1,
lim
n
Sn
a 1 r
,
r 1,
lim
n
Sn
,
converge. diverge.
If r 1, a a a a
lim
n
Sn
lim na n
diverge.
目录 上页 下页 返回 结束
目录 上页 下页 返回 结束
1
10
1
2n
n1
n1 n
目录 上页 下页 返回 结束
Exercises
P637 2, 4, 11, 12, 14, 15, 21 25, 26, 27, 35, 45.
目录 上页 下页 返回 结束
Analysis.
ln n 1 ln n
1
1 n
Sn
1
1 2
1 3
1 n
n n1
ln 2 ln1 ln 3 ln 2 ln 4 ln 3
ln n 1 ln n ln n 1
lim
n
Sn
目录 上页 下页 返回 结束
目录 上页 下页 返回 结束
Example 9. Applying the Theorem 7.

北京邮电大学国际学院高等数学(下)幻灯片讲义(无穷级数)Le.

北京邮电大学国际学院高等数学(下)幻灯片讲义(无穷级数)Le.

1(x 2
−
2) +
1(x 4
−
2)2
=
3−
3x 2
+
x2 4
#
8
Convergence of Power Series
Abel’s Theorem (1)
(2)
∞
∑ Consider the series an xn. n=0
If it converges at x0 , x0 ≠ 0 , then it must converge
but converges for| x − x0 |< R. The series may or may not converge at either of the endpointsx = x0 − Randx = x0 + R. 2. The series converges for every x(R = ∞) . 3. The series converges at x = x0 and diverges elsewhere (R = 0).
n=1
13
Finding the Interval of Convergence Using the Ratio Test
∑ Solution Apply the Ratio Test to the series | un | , where un
is the nth term of the series in question.
divergence points is called the divergence domain of the series.
1
2
Series of Functions
Definition (Sum function and Convergence)

Lecture11

Lecture11
ÍIhÛÜ ÓÔ 3.1 h
CR
z →0
lim z ·
1 = 1, z (1 + z + z 2 )
8Ïh| ò R → ∞ h δ → 0 h ãC
δ →0
~ AB ªb ´C e ¶f w × z = 0 hæ z = 0 ª ªb hw ãTáC } ý $eqr p q h ifg de e f vhgh qr w e i u j efgm (kl 5 hh qr m e no m) X § epfq (0¢ hrs te uv fg) # h 2 ¤¥ fg w ¦ fghy xi § ¨© e ¥ fg hw y f f (x) z f (z) h§¥ e w xfg hfg d{ p q | a # $e 01 p q
δ1 →0
f (x)dx + lim
a
δ2 →0
lim
f (x)dx +
c −δ
f (x)dx
b
a
c+ δ
f (x)dx = lim
f (x)dx +
c+ δ
−1
R1 →∞
lim
−R1
dx + lim x(1 + x + x2 ) R2 →∞ dx + x(1 + x + x2 )
1
−δ
+ lim
δ →0
ngs
(−)k n k n − 2k 2
n−1
−∞
sin2 x dx, x2
1 − ei2z dz, z2
hi
.
sin3 x 3 dx = π; 3 x 4

北京邮电大学国际学院高等数学(下)幻灯片讲义(无穷级数)-Lecture 1

北京邮电大学国际学院高等数学(下)幻灯片讲义(无穷级数)-Lecture 1

is called the partial sum of the series. The partial sum of the series form a sequence
s1 = a1 , s2 = a1 + a2 ,
, sn = ∑ ak ,
k =1 n
n approaching infinite, we say that the series converges to the sum S, and we write
1+ 1 1 1 1 + + + + 2 4 8 16
Infinite Series
∑a
n =1
∞
n
=1 +
1 1 1 1 + + + + 2 4 8 16
Partial Sum First: Second: Third:
…
Value
s1 = 1
1 s2 = 1 + 2
21
2 2 1 2 1 4
The way to do so is not to try to add all the terms at once (we cannot) but rather to add the terms one at a time from the beginning and look for a pattern in how these "partial sum" grow.
k =1 n
Convergence and Divergence
Definition (Convergence and Divergence of a series) If the sequence of partial sums of a series

高等数学 第十一章 电子课件

高等数学 第十一章 电子课件

第一节
概率论
一、随机事件
（一）随机事件的概念
引例1 如果问“苹果从树上脱落，会往地上落吗？”，答案是“会”． 引例2 如果问“掷一枚骰子，能否出现7点？”，答案是“不能”． 引例3 抛掷一枚质地均匀的硬币，结果可能是正面朝上，也可能是反面朝上， 且事先无法确定抛掷的结果是什么． 引例4 在400 m短跑比赛前，运动员需通过抽签决定自己所在的跑道，且每 次抽签前都无法预测自己会在哪条跑道．
（二）概率的古典定义
在某些情况下，随机试验具有以下特征． 有限性：试验中所有可能出现的基本事件只有有限个． 等可能性：每个基本事件出现的可能性相等． 具有以上两个特点的概率模型是大量存在的，这种概率 模型称为古典概率模型，简称古典概型，也称等可能概型．
（二）概率的古典定义
定义 3 对于古典概型，设试验含有 n 个基本事件，若事件 A 包含 m 个基本事件，则事件 A
第十一章
概率统计基础
导学
概率论与数理统计是研究随机现象内在规律性的重要工具，其应用已 遍及自然科学、社会科学、工程技术、军事科学及生活实际等各领域，因 此掌握一定的概率统计知识十分必要．
本章主要介绍随机事件及其概率，随机变量及其分布，随机变量的期 望与方差，数理统计的基础知识，参数估计，假设检验及回归分析．
随机试验的一切可能结果所组成的集合称为样本空间，记作 ．随机试验的每
一个可能结果称为样本点，样本空间就是全体样本点的集合．
（一）随机事件的概念
定义1 随机试验的每一种可能的结果称为随机事件，简称事件．它通常用大写 英文字母A, B, C… 表示．
随机事件可分为基本事件和复合事件． 基本事件：在随机试验中，不可再分解的事件． 复合事件：在随机试验中，由若干个基本事件组合而成的事件．

《高等数学课件-全英文版(英语思维篇)》

《高等数学课件-全英文版(英语思维篇)》
Fundamental Theorem of Calculus
Discover the Fundamental Theorem of Calculus and its significance in integration.
Riemann Sums
Explore Riemann sums as a method for approximating definite integrals.
Functions and Graphs
Types of Functions
Discover the different types of functions and their graphical representations.
Graph Plotting
Learn how to plot and analyze functions using mathematical tools and software.
Differentiation
1
Derivative Definition
Learn the definition and basic rules
Chain Rule
2
of differentiation.
Discover how to differentiate
composite functions using the
Work and Energy
Explore how integration is used to calculate work and energy in various scenarios.
Differential Equations
1
Introduction to Differential

lecture11

lecture11
2
4
4
=
1 32
We’re not done yet, though. We still need to check the endpoints! At x = 0, A = 02 + At x = 1, � �2 1 1 A= + 02 = 4 16 � 1−0 4 �2 = 1 16
4
Lecture 11
Lecture 11
18.01 Fall 2006
Lecture 11: Max/Min Problems
Example 1. y = ln x (same function as in last lecture) x
1/e x0=e
Figure 1: Graph of y =
ln x . x
• What is the maximum value? Answer: y =
1
5
MIT OpenCourseWare
18.01 Single Variable Calculus
Fall 2006
For information about citing these materials or our Terms of Use, visit: /terms.
1
Lecture 11
18.01 Fall 2006
max
min
Figure 2: Search for max and min among critical points and endpoints
Example 3. Find the open-topped can with the least surface area enclosing a fixed volume, V.

section11北邮双语教学课件共69页

section11北邮双语教学课件共69页
2
The Concepts of Sets
Definition A set [集合] is a collection of all objects which are sharing some properties. Each of the objects belonging to a set is called an element [元素] of the set.
Example Set A {1, 2}, B { x | x2 3x 2 0} AB
Hale Waihona Puke Definition If A B and A B , then A is called a proper subset
[真子集] of B,
denoted by
A B
or
A B.
Example: the integer set [整数集] is denoted by Z. Z { x | x is an integer} {0, 1, 2, }
Example: the rational number [有理数集] is denoted by Q. Q { x | x is a rational number}
Remark (Notations) Sets are always denoted as capital letters, such as A, B, … . The elements of a set usually are denoted as small letters, such as a, b, … . The relation a A means that a is an element of set A, read as “ a belongs toA ”. The relation a A or a A means that a does not belong to A .

Lesson 11 The Future of the English.

Lesson 11 The Future of the English.
2019/6/1
urban: connected with a town or city 城市的；都市的；城镇的
e.g: urban areas 城市生活/urban life 城市生活
Paraphrase：
Now the contemporary English ,who have the same experience as me , can get bored easily .This explains why the English gamble and drink so much and enjoy dramatic changes in public life , any news that cause warm discussion. I think the urban English always get excited over a little thing.
why they gamble and booze so much and enjoy any dramatic
change in public life , any news that encourages excited talk: the urban English have always seemed to me a dramatic people.
Paraphrase：
The danger will always in front of us if we don’t get rid of boredom.
2019/6/1
6.Teenagers,who have not been able to use up enough energy during the day, turn at night to idiot vandalism.

交大数理逻辑课件11-1 函数共30页

交大数理逻辑课件11-1 函数共30页

一、有穷集之间的构造
例 A=P({1,2,3}), B={0,1}{1,2,3} 解： A={,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}.
B={ f0, f1, … , f7 }, 其中 f0={<1,0>,<2,0>,<3,0>}, f1={<1,0>,<2,0>,<3,1>}, f2={<1,0>,<2,1>,<3,0>}, f3={<1,0>,<2,1>,<3,1>}, f4={<1,1>,<2,0>,<3,0>}, f5={<1,1>,<2,0>,<3,1>}, f6={<1,1>,<2,1>,<3,0>}, f7={<1,1>,<2,1>,<3,1>}.
令 f：A→B,
f()=f0,
f({1})=f1, f({2})=f2, f({3})=f3,
f({1,2})=f4, f({1,3})=f5, f({2,3})=f6, f({1,2,3})=f7
构造从A到B的双射函数（续）
二、实数区间之间构造双射
构造方法：直线方程
例 A=[0,1]
B=[1/4,1/2] 构造双射 f :A→B
T(x) 1 0 1 0 0 1 1 1
自然映射
5. 设 R 是 A 上的等价关系, 令 g：A→A/R g(a) = [a], a∈A 称 g 是从 A 到商集 A/R 的自然映射.
➢ 恒等关系确定的自然映射是双射, 其他的自然映射一 般来说是满射.

北京邮电大学高等数学教材英文版

北京邮电大学高等数学教材英文版

北京邮电大学高等数学教材英文版Beijing University of Posts and Telecommunications Advanced Mathematics Textbook (English Version)IntroductionThe Beijing University of Posts and Telecommunications (BUPT) is renowned for its comprehensive and cutting-edge curriculum. As part of its commitment to providing a world-class education, BUPT offers an English version of its advanced mathematics textbook. This article will delve into the content, structure, and importance of the BUPT Advanced Mathematics Textbook (English Version).Content OverviewThe BUPT Advanced Mathematics Textbook covers a wide range of topics, including calculus, linear algebra, differential equations, and probability theory. Each chapter provides a comprehensive introduction to the topic, followed by detailed explanations and examples that illustrate the concepts. The textbook also includes exercises and practice problems to reinforce understanding and improve problem-solving skills.Structure of the TextbookThe textbook consists of several sections, each specifically designed to enhance comprehension and facilitate learning. Here is an overview of the main sections:1. Introduction: This section provides a brief overview of the topic, its relevance, and its applications in various fields.2. Theory and Concepts: In this section, the fundamental theory and concepts related to the topic are presented in a clear and concise manner. Definitions, theorems, and formulas are introduced, followed by in-depth explanations.3. Examples and Illustrations: To further enhance understanding, the textbook includes numerous examples and illustrations that demonstrate the application of the theory and concepts explained in the previous section. These examples provide real-life scenarios where the mathematical principles can be applied.4. Exercises and Problem Solving: A vital component of the textbook, this section offers a wide range of exercises and problems to test the reader's comprehension and analytical skills. Solutions and step-by-step explanations are provided to assist students in the learning process.Importance of the TextbookThe English version of the BUPT Advanced Mathematics Textbook serves a crucial role in promoting international education and collaborations. It enables international students to access and comprehend BUPT's advanced mathematics curriculum, fostering cross-cultural exchange and learning. Moreover, the textbook contributes to the overall improvement of mathematics education by providing a comprehensive resource for both students and instructors.ConclusionThe BUPT Advanced Mathematics Textbook (English Version) plays a significant role in Beijing University of Posts and Telecommunications'commitment to academic excellence and internationalization. With its comprehensive content, well-structured format, and emphasis on application and problem-solving, the textbook equips students with the necessary knowledge and skills to excel in advanced mathematics. It serves as a valuable resource for both domestic and international students, further reinforcing BUPT's reputation as a leading institution in mathematics education.。

  1. 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. 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. 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)。
A1 , A2 , , An . Choose a point ( xk , yk ) in each piece Ak , then
y
A
x
Sn f ( xk , yk )Ak .
k 1
16
n
Calculating Double Integrals Over Rectangles
on the xOy-plane and its density is a function of the point ( x , y ), f ( x , y ), then, how can we find the mass of this sheet metal? To find the mass, we suppose that f ( x , y )
( )
f ( x , y )d , f ( x , y )dA
( )
or
n
( )
f ( x , y )dxdy .
Thus,
( )
f ( x , y )d lim f ( x
A 0 k 1
k
, yk )Ak
As with functions of a single variable, the sums approach this limit no matter how the intervals [a , b] and [c , d ] that determine (σ) are partitioned, as long as the norms of the partitions both go to zero.
( ) ( ) ( ) ( )
( )
2. Additivity with respect to the domain of integration
Suppose that ( ) ( 1 ) ( 2 ) and ( 1 ),( 2 ) have no common part except for their
9
Geometric meaning of the double integral
Suppose that z f ( x , y ) 0,( x , y ) , then it can be think of a cylindrical body in three dimensional space.
17
Fubini’s Theorem for Calculating Double Integrals
Suppose that we wish to calculate the volume under the plane z 4 x y over the rectangular region ( ) : 0 x 2,0 y 1 on the xy-plane. If we denote the area of the cross-section at x as A( x ) , then the volume is
boundaries. Then
( )
( 1 )
( 2 )
f ( x , y )d
(
f ( x , y )d
1)
(

f ( x , y )d .
6
1)
Properties of Double Integral
3. Domination (1) f ( x , y )d 0 , if f ( x , y ) 0 on ( ).
denoted by k . Choose any point
Summation
Pk k and form the sum
பைடு நூலகம்
is
f ( P )
k 1 k n
n
k
.
If the limit lim f ( Pk ) k exist, where d 0 k 1 Precision n d max d ( k ) , we say that f is integrable
2 x
x
O
which is the area under the curve z 4 x y
l f ( x , y )d L .
( )
4. Mean Value Theorem Suppose that f C (( )) and ( ) is a closed bounded, and connected domain. Then there exists at least one point , ( ), such that
( ) : a x b, c y d .
Then we make a network of lines parallel to x- and y-axes. These lines divide (σ) into small pieces of area A xy . We number these in some order
V (4 x y )d
( )

as the integral
x2 x 0
A( x )dx .
z 4 x y
z
For each value of x, we may calculate A( x )
A( x )
y 1 y0
4
(4 x y )dy ,
is defined on a rectangular region Ω given by
: a x b, c y d .
We divide Ω into small pieces of area A and number them in some order A1 , A2 , , An .
Section 11.1
Concept and Properties of Double Integrals
School of Science, BUPT
1
Mass of a Thin Rectangular Sheet Metal A
k
Suppose a thin rectangular sheet metal lies
k 1
over the domain
( ).
4
Riemann, Bernhard (1826-1866), German mathematician
The Notation of the Double Integral
If function f is integral over the domain ( ), then the limit of Riemann sum is called the integral of the multivariable function f on the domain
2
Mass of a Thin Rectangular Sheet Metal A
k
We choose a point ( xk , yk ) in each piece
Ak and form the sum
Sn f ( xk , yk )Ak .
k 1 n
If f is continuous throughout Ω, then, as we refine the mesh (or two-dimensional partition) width to make the “norm” of each piece go to zero, we can expect that the sum should have a limit and the limit should be the mass of the thin rectangular steel
( )
(2) (3) (4)
( )
f ( x , y )d g ( x , y )d , if
( )
f ( x , y ) g ( x , y ) on ( ).
( )
f ( x , y )d
f ( x , y ) d
( )
If l f ( x , y ) L, ( x , y ) ( ), then
If f is continuous throughout (σ), then as we refined the mesh width to make both x and y go to zero, the sums approach a limit called the double integral of f over (σ). The notation for it is
Suppose f ( x , y ) and g( x , y ) are both integrable over the domain ( ), then 1. Linearity Property (1) (2)
kf ( x , y )d k f ( x , y )d , where k is a constant. f ( x , y ) g( x , y ) d f ( x , y )d g( x , y )d
( x k , yk )
metal.
3
The Concept of the Double Integral
Definition Double Integral
Page 256/definition 11.1.1
Suppose that a scalar function f pieces of area
is defined on a closed bounded
plane region ( ). Suppose, however, ( ) can be divided into pieces of