14_differential calculus 1

Differential（微分）的名词1. 引言微分（differential）是一种基本的数学概念，它在数学分析、物理学和工程学等领域中都扮演着重要的角色。

微分的概念最早由数学家和物理学家牛顿和莱布尼茨提出，通过对函数的变化率进行研究，微分使我们能够更好地理解和描述自然现象中的变化。

本文将深入讨论微分的概念、性质和应用，并探讨微分在数学和实际问题中的重要性。

2. 微分的概念与定义微分是一种用来描述函数变化率的数学工具。

当我们研究一个函数在某一点上的变化时，微分允许我们通过求取函数在该点的导数来获得更多的信息。

2.1 导数的定义函数在某一点的导数（derivative）描述了函数在该点的变化率。

对于函数f(x)，其在点x处的导数可以通过以下公式定义：f′(x)=limℎ→0f(x+ℎ)−f(x)ℎ这个公式表示了当自变量的增量h趋近于0时，函数在该点的变化率。

导数可以看作函数曲线在某一点上的切线斜率。

2.2 微分的定义微分是导数的近似值。

通过使用导数的定义，我们可以将函数在某一点x处微分为以下形式：df(x)=f′(x)⋅dx其中，dx表示自变量的增量，df(x)表示函数在点x的微小增量。

微分的概念可以让我们更好地研究函数的变化，特别是在一些计算问题中。

3. 微分的性质与应用微分在数学和实际问题中都有着广泛的应用。

它的一些性质和应用将在本节中进行讨论。

3.1 微分的基本性质微分具有一些基本的性质，如线性性、乘积法则和链式法则等。

这些性质使得微分成为解决复杂问题的强有力的工具。

3.1.1 线性性微分具有线性性质，即对于任意常数a和b以及两个可微函数f(x)和g(x)，有如下等式成立：d(af(x)+bg(x))=a⋅df(x)+b⋅dg(x)这个性质允许我们在进行微分计算时，对函数进行线性组合，从而简化计算过程。

3.1.2 乘积法则乘积法则是微分中的重要性质，用于计算两个函数的乘积的导数。

对于可微函数f(x)和g(x)，有如下等式成立：d(f(x)⋅g(x))=f′(x)⋅g(x)⋅dx+f(x)⋅g′(x)⋅dx乘积法则使得我们能够更方便地计算复杂函数的导数。

Calculus ICalculus, also known as mathematical analysis, is a branch of mathematics that deals with the study of rates of change and how things change over time. It is a fundamental mathematical tool that has become essential in many fields such as physics, engineering, economics, and biology. In this essay, we will explore Calculus I, which is the introductory course for Calculus.The study of Calculus is divided into two main branches: differential calculus and integral calculus. Differential calculus is concerned with the study of rates of change, while integral calculus is concerned with the study of accumulation. Calculus I focuses on the fundamental concepts of differential calculus.One of the key ideas in Calculus I is the concept of limit. A limit is the value that a function approaches as the independent variable approaches a certain value. Limits are an essential tool for studying the behavior of functions, especially at points where the function may not be defined.Another important concept in Calculus I is the derivative. The derivative of a function is the rate of change of the function at a particular point. It is defined as the limit of the difference quotient as the change in the independent variable approaches zero. The derivative is a fundamental concept in Calculus and is used extensively in many fields, including physics, engineering, and economics.The derivative has many important properties, including the power rule, product rule, quotient rule, and chain rule. These rules allowus to find the derivative of complicated functions quickly and efficiently.The derivative also has many applications, including optimization problems and finding the location of maximum and minimum values of a function. For example, in economics, the derivative is used to find the marginal cost and marginal revenue of a company. In physics, the derivative is used to find the instantaneous velocity and acceleration of an object.Another important concept in Calculus I is the notion of differentiation. Differentiation is the process of finding the derivative of a function. It is an integral part of Calculus and is used extensively in many fields.One of the most important applications of differentiation is in the study of optimization problems. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints. For example, in economics, firms try to maximize their profits subject to certain constraints, such as the cost of production.Integration is the second branch of Calculus, and it deals with finding the area under a curve. Integration is the inverse of differentiation, and it is used extensively in many fields, including physics and engineering.One of the most important applications of integration is in the study of volumes and areas. For example, in physics, the volume of a solid can be found by integrating the area under the curve of itscross-section. In engineering, the area of an irregular shape can be found by integrating the area under the curve of its boundary.Calculus I also covers important topics such as limits, continuity, and trigonometric functions. Limits are used extensively in Calculus to study the behavior of functions. Continuity is a fundamental concept in Calculus that ensures that a function is well-behaved and has no abrupt changes.Trigonometric functions are essential in Calculus because they are used extensively in the study of differential equations, which are equations that involve derivatives. Differential equations are used to model many real-world phenomena, such as the growth of a population and the spread of diseases.In conclusion, Calculus I is an essential course for any student studying mathematics, physics, engineering, or economics. It provides a solid foundation for more advanced courses in Calculus and other fields. The concepts of differential calculus, such as limits, derivatives, and differentiation, are fundamental in the study of many real-world problems. The concepts covered in Calculus I, such as optimization and integration, have many applications in numerous fields and are essential for solving problems in many areas of science and engineering.In addition to the topics mentioned above, Calculus I also covers related rates, which are useful in real-world scenarios where things are changing at different rates. For example, if you are filling a pool with water and you want to know how fast the water level is rising, you would use related rates. This involves finding the relationship between the rates of change of different variables and using this relationship todetermine one rate when the other rate is known.Another important concept in Calculus I is the Mean Value Theorem. This theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point in the interval where the derivative is equal to the average rate of change of the function over the interval. This theorem has applications in many areas, including economics, where it is used to prove the existence of equilibrium prices.Calculus I also covers curve sketching, which involves studying the behavior of a function as it approaches zero and infinity, finding its intercepts, and determining where it is increasing or decreasing. This is important in many fields as it allows us to understand the behavior of functions and predict their future values.One of the most important applications of Calculus I is in physics, where it is used extensively in studying motion. The concepts of calculus are used to determine the velocity, acceleration, and position of an object at any given point in time. Understanding these concepts is essential in fields such as aerospace engineering, where the motion of objects in space is critical to the success of missions.Calculus I is also used extensively in engineering, especially in the design and analysis of systems. For example, in electrical engineering, calculus is used to determine the power consumed by a circuit, while in civil engineering, it is used to calculate the stress on structures such as bridges and buildings. Calculus is also essential in chemical engineering, where it is used to determine therate of chemical reactions.In economics, calculus is used to model and analyze various economic phenomena, such as supply and demand, consumer behavior, and production optimization. The concepts of calculus are essential in understanding the dynamics of markets and the behavior of firms in different situations.Calculus I has numerous real-life applications, from modeling the growth of populations to understanding the spread of diseases. It is used in biostatistics to determine the probability of an individual developing a certain disease and in epidemiology to model the spread of infectious diseases. In ecology, calculus is used to study predator-prey relationships and competition between species.In the field of finance, calculus is used to determine the value of financial securities such as stocks and bonds. Understanding the concepts of calculus is essential in the field of quantitative finance, which involves using mathematical models to predict the behavior of financial markets.Overall, Calculus I is a fundamental course in mathematics that teaches students the basic concepts of differential calculus, including limits and derivatives, and their applications in various fields. It provides a solid foundation for more advanced courses in Calculus and other related fields. The concepts covered in Calculus I have numerous applications in many fields, including physics, engineering, economics, and biology, making it an essential tool for solving real-world problems.。

大学高等数学英文教材University Advanced Mathematics English TextbookChapter 1: Introduction to Calculus1.1 Basic ConceptsIn this chapter, we will introduce the fundamental ideas and principles of calculus. We will cover topics such as functions, limits, and continuity. Understanding these concepts is crucial for a solid foundation in calculus.1.2 DerivativesThe concept of derivatives is central to calculus. We will explore the definition of derivatives, as well as various rules and techniques for finding them. Additionally, applications of derivatives in real-world scenarios will be discussed.1.3 IntegrationIntegration is another important topic in calculus. We will delve into the concept of integration, techniques for finding antiderivatives, and various applications of integrals. The fundamental theorem of calculus will also be introduced.Chapter 2: Differential Calculus2.1 Limits and ContinuityBuilding upon the concepts introduced in Chapter 1, we will dive deeper into limits and continuity. We will examine different types of limits,including infinite limits and limits at infinity. The concept of continuity will be explored in detail.2.2 DifferentiationThis section focuses on the derivative, one of the key ideas in differential calculus. We will discuss the chain rule, product rule, and quotient rule, among other differentiation techniques. Various applications of derivatives, such as optimization and related rates, will also be covered.2.3 Higher-order Derivatives and ApplicationsIn this part, we will extend our understanding of derivatives to higher orders. We will explore concepts such as concavity, inflection points, and curve sketching. Furthermore, applications of higher-order derivatives in physics and economics will be discussed.Chapter 3: Integral Calculus3.1 Techniques of IntegrationExpanding upon the concepts introduced in Chapter 1, this section dives deeper into integration techniques. We will explore methods such as substitution, integration by parts, and partial fractions. Improper integrals and applications of integration will also be covered.3.2 Applications of IntegrationIntegration has various real-world applications, and we will explore some of them in this section. Topics such as area, volume, and arc length will be discussed, along with their practical applications in physics, engineering, and economics.3.3 Differential EquationsDifferential equations are a powerful tool in modeling natural phenomena. We will introduce different types of differential equations and discuss techniques for solving them. Applications of differential equations in science and engineering will also be explored.Chapter 4: Multivariable Calculus4.1 Functions of Several VariablesIn this chapter, we will extend our knowledge of calculus to functions of several variables. Topics covered include partial derivatives, gradients, and optimization techniques in multivariable calculus. Practical applications in physics and economics will be explored.4.2 Multiple IntegralsMultiple integrals allow us to calculate volumes, surface areas, and other quantities in higher dimensions. We will discuss double and triple integrals, as well as methods like polar coordinates and change of variables. Applications of multiple integrals in physics and engineering will also be covered.4.3 Vector CalculusVector calculus deals with vector fields and line integrals. We will discuss concepts such as divergence, curl, and Green's theorem. Applications of vector calculus in physics and engineering, particularly in the study of fluid flow and electrostatics, will be explored.ConclusionCompleting this textbook will equip students with a solid understanding of advanced mathematics concepts. Whether pursuing further studies in mathematics or applying mathematical principles in other fields, this textbook will provide a comprehensive foundation. Remember to practice regularly and seek clarification when facing challenges.。

数学专业英语－Differential CalculusHistorical IntroductionNewton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insur mountable problems could be solved by more or less routine methods.The succ essful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch o f calculus,differential calculus.The central idea of differential calculus is the notion of derivative.Like the inte gral,the derivative originated from a problem in geometry—the problem of find ing the tangent line at a point of a curve.Unlile the integral,however,the deriva tive evolved very late in the history of mathematics.The concept was not form ulated until early in the 17th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special func tions.Fermat’s idea,basically very simple,can be understood if we refer to a curve a nd assume that at each of its points this curve has a definite direction that ca n be described by a tangent line.Fermat noticed that at certain points where th e curve has a maximum or minimum,the tangent line must be horizontal.Thus t he problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.This raises the more general question of determining the direction of the tange nt line at an arbitrary point of the curve.It was the attempt to solve this gener al problem that led Fermat to discover some of the rudimentary ideas underlyi ng the notion of derivative.At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of findin g the tangent line at a point of a curve.The first person to realize that these t wo seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they explo ited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.Although the derivative was originally formulated to study the problem of tang ents,it was soon found that it also provides a way to calculate velocity and,mo re generally,the rate of change of a function.In the next section we shall consi der a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.A Problem Involving VelocitySuppose a projectile is fired straight up from the ground with initial velocity o f 144 feet persecond.Neglect friction,and assume the projectile is influenced onl y by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at ti me t we woule have f(t)=144 t.In actual practice,gravity causes the projectile t o slow down until its velocity decreases to zero and then it drops back to eart h.Physical experiments suggest that as the projectile is aloft,its height f(t) is gi ven by the formula(1)f(t)=144t –16 t2The term –16t2is due to the influence of gravity.Note that f(t)=0 when t=0 a nd when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.The problem we wish to consider is this:To determine the velocity of the proj ectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we int roduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient.Change in distance during time interval =f(t+h)-f(t)/hThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positiv e or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute v alue.The limit process by which v(t) is obtained from the difference quotient is wri tten symbolically as follows:V(t)=lim(h→0)[f(t+h)-f(t)]/hThe equation is used to define velocity not only for this particular example bu t,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.The example describe in the foregoing section points the way to the introducti on of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this in terval and introduce the difference quotient[f(x+h)-f(x)]/hwhere the number h,which may be positive or negative(but not zero),is such th at x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to a s the average rate of change of f in the interval joining x to x+h.Now we let h approach zero and see what happens to this quotient.If the quot ient.If the quotient approaches some definite values as a limit(which implies th at the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as “f prime of x”).Thus the formal defi nition of f’(x) may be stated as follows:Definition of derivative.The derivative f’(x)is defined by the equationf’(x)=lim(h→o)[f(x+h)-f(x)]/hprovided the limit exists.The number f’(x) is also called the rate of change of f at x.In general,the limit process which produces f’(x) from f(x) gives a way of ob taining a new function f’from a given function f.This process is called differ entiation,and f’is called the first derivative of f.If f’,in turn,is defined on an interval,we can try to compute its first derivative,denoted by f’’,and is calle d the second derivative of f.Similarly,the nth derivative of f denoted by f^(n),is defined to be the first derivative of f^(n-1).We make the convention that f^(0) =f,that is,the zeroth derivative is the function itself.Vocabularydifferential calculus微积分differentiable可微的intergral calculus 积分学differentiate 求微分hither to 迄今 integration 积分法insurmountable 不能超越 integral 积分routine 惯常的integrable 可积的fuse 融合integrate 求积分originate 起源于sign-preserving保号evolve 发展，引出 axis 轴（单数）tangent line 切线 axes 轴（复数）direction 方向 contradict 矛盾horizontal 水平的contradiction 矛盾vertical 垂直的 contrary 相反的rudimentary 初步的，未成熟的composite function 合成函数，复合函数area 面积composition 复合函数intimately 紧密地interior 内部exploit 开拓，开发 interior point 内点inaugurate 开始 imply 推出，蕴含projectile 弹丸 aloft 高入云霄friction摩擦initial 初始的gravity 引力 instant 瞬时rate of change 变化率integration by parts分部积分attain 达到definite integral 定积分defferential 微分indefinite integral 不定积分differentiation 微分法 average 平均Notes1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.意思是：在很大程度上是牛顿和莱伯尼，他们相互独立地把积分学的思想发展到这样一种程度，使得迄今一些难于超越的问题可以或多或少地用通常的方法加以解决。

极限思想外文翻译pdfBSHM Bulletin, 2014Did Weierstrass’s differential calculus have a limit-avoiding character? His,,,definition of a limit in styleMICHIYO NAKANENihon University Research Institute of Science & Technology, Japan In the 1820s, Cauchy founded his calculus on his original limit concept and,,,developed his the-ory by using inequalities, but he did not apply theseinequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distin-guished bytheir limit-avoiding character. Dugac’s partial publication of the 1861 lecturesmakes these differences clear. But in the unpublished portions of the lectures,,,,Weierstrass actu-ally defined his limit in terms ofinequalities. Weierstrass’slimit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not providethe basic structure for the modern e d style analysis. Thus it was Dini’s 1878 text-book that introduced the,,,definition of a limit in terms of inequalities.IntroductionAugustin Louis Cauchy and Karl Weierstrass were two of the most important mathematicians associated with the formalization of analysis on the basis of the e d doctrine. In the 1820s, Cauchy was the first to give comprehensive statements of mathematical analysis that were based from the outset on a reasonably clear definition of the limit concept (Edwards 1979, 310). He introduced various definitions and theories that involved his limit concept. His expressions were mainly verbal, but they could be understood in terms of inequalities: given an e, find n or d (Grabiner 1981, 7). As we show later, Cauchy actually paraphrased his limit concept in terms of e, d, and n0 inequalities, in his more complicated proofs. But it was Weierstrass’s 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen? 2003, 185–186).Weierstrass’s adoption of full epsilonic arguments, however, didnot mean that he attained a prototype of the modern theory. Modern analysis theory is founded on limits defined in terms of e d inequalities. His lectures were not founded on Cauchy’s limit or his own original definition of limit (Dugac 1973). Therefore, in order to clarify the formation of the modern theory, it will be necessary toidentify where the e d definition of limit was introduced and used as a foundation.We do not find the word ‘limit’ in the pu blished part of the 1861 lectures.Accord-ingly, Grattan-Guinness (1986, 228) characterizesWeierstrass’s analysis aslimit-avoid-ing. However, Weierstrass actually defined his limit in terms of epsilonics in the unpublished portion of his lectures. Histheory involved his limit concept, although the concept did not function as the foundation of his theory. Based on this discovery, this paper re-examines the formation of e d calculus theory, noting mathematicians’ treat-ments of their limits. We restrict ourattention to the process of defining continuity and derivatives. Nonetheless, this focus provides sufficient information for our purposes.First, we confirm that epsilonics arguments cannot representCauchy’s limit,though they can describe relationships that involved his limit concept. Next, we examine how Weierstrass constructed a novel analysis theory which was not based2013 British Society for the History of Mathematics52 BSHM Bulletinon Cauchy’s limits but could have involved Cauchy’s resu lts. Thenwe confirmWeierstrass’s definition of limit. Finally, we note that Dini organized his analysis textbook in 1878 based on analysis performed inthe e d style.Cauchy’s limit and epsilonic argumentsCauchy’s series of textbooks on calculus, Cours d’analyse (1821), Resume deslecons? donnees a l’Ecole royale polytechnique sur le calcul infinitesimal tomepremier (1823), and Lecons? sur le calcul differentiel (1829), are often considered as the main referen-ces for modern analysis theory, the rigour of which is rooted more in the nineteenth than the twentieth century.At the beginning of his Cours d’analyse, Cauchy defined the limit concept as fol-lows: ‘When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all theothers’ (1821, 19; English translation fromGrabiner 1981, 80). Starting from this concept, Cauchy developed a theory of continuous func-tions, infinite series, derivatives, and integrals, constructing an analysis based on lim-its (Grabiner 1981, 77).When discussing the evolution of the limit concept, Grabiner writes:‘This con-cept, translated into the algebra of inequalities, was exactly what Ca uchy needed for his calculus’ (1981, 80). From the present-day point of view, Cauchy described rather than defined his kinetic concept of limits. According to his ‘definition’—which has the quality of a translation or description—he could develop any aspectof the theory by reducing it to the algebra of inequalities.Next, Cauchy introduced infinitely small quantities into his theory. ‘When the suc-cessive absolute values of a variable decrease indefinitely, in such a way as to become less than any given quantity, that variable becomes what is called an infinitesimal. Such a variable has zero for its limit’ (1821, 19; English translationfrom Birkhoff and Merzbach 1973, 2). That is to say, in Cauchy’s framework ‘thelimit of variable x is c’ is intuitively understood as ‘x indefinitely approaches c’,and is represented as ‘jx cj is as little as desired’ or ‘jx cj is infinitesimal’.Cauchy’s idea of defining infinitesimals as variables of a special kind was original, because Leibniz and Euler, for example, had treated them as constants (Boyer 1989, 575; Lutzen? 2003, 164).In Cours d’analyse Cauchy at first gave a verbal definition of a continuous func-tion. Then, he rewrote it in terms of infinitesimals:[In other words,] the function f ðxÞ will rema in continuous relative to x in a given interval if (in this interval) an infinitesimalincrement in the variable always pro-duces an infinitesimal increment in the function itself. (1821, 43; English transla-tion from Birkhoff and Merzbach 1973, 2).He introduced the infinitesimal-involving definition and adopted a modified version of it in Resume (1823, 19–20) and Lecons? (1829, 278).Following Cauchy’s definition of infinitesimals, a continuous function can be defined as a function f ðxÞ in which ‘the variable f ðx þ aÞ f ðxÞ is an infinitelysmall quantity (as previously defined) whenever the variable a is, that is, that f ðx þ aÞ f ðxÞ approaches to zero as a does’, as notedby Edwards (1979, 311). Thus,the definition can be translated into the language of e dinequalities from a modern viewpoint. Cauchy’s infinitesimals are variables, and we can also takesuch an interpretation.Volume 29 (2014) 53Cauchy himself translated his limit concept in terms of e d inequalities. He changed ‘If the difference f ðx þ 1Þ f ðxÞ converges towards a certain limit k, for increasing values of x, (. . .)’ to‘First suppose that the quantity k has a finitevalue, and denote by e a number as small as we wish. . . . we cangive the number h a value large enough that, when x is equal to orgreater than h, the difference in question is always contained between the limits k e; k þ e’ (1821, 54; Englishtranslation from Bradley and Sandifer 2009, 35).In Resume , Cauchy gave a definition of a derivative: ‘if f ðxÞ is continuous, thenits derivative is the limit of the difference quotient,,yf(x，i),f(x), ,xias i tends to 0’ (1823, 22–23). He also translated the concept of derivative asfollows: ‘Designate by d and e two very small numbers; the first being chosen in such a way that, for numerical values of i less than d, [. . .], the ratio f ðx þ iÞ fðxÞ=i always remains greater than f ’ðx Þ e and less than f ’ðxÞ þ e’ (1823,44–45; English transla-tion from Grabiner 1981, 115).These examples show that Cauchy noted that relationships involving limits or infinitesimals could be rewritten in term of inequalities. Cauchy’s argumentsabout infinite series in Cours d’analyse, which dealt with the relationship betweenincreasing numbers and infinitesimals, had such a character. Laugwitz (1987, 264; 1999, 58) and Lutzen? (2003, 167) have noted Cauchy’s strict use of the e Ncharacterization of convergence in several of his proofs. Borovick and Katz (2012) indicate that there is room to question whether or not our representation using e d inequalities conveys messages different from Cauchy’s original intention. Butthis paper accepts the inter-pretations of Edwards, Laugwitz, and Lutzen?.Cauchy’s lectures mainly discussed properties of series and functions in the limit process, which were represented as relationships between his limits or his infinitesi-mals, or between increasing numbers and infinitesimals. His contemporaries presum-ably recognized the possibility of developing analysis theory in terms of only e, d, and n0 inequalities. With a few notable exceptions, all of Cauchy’s lectures could be rewrit-ten in terms of e d inequalities. Cauchy’s limits and hisinfinitesimals were not func-tional relationships,1 so they were not representable in terms of e d inequalities.Cauchy’s limit concept was the foundation of his theory. Thus, Weierstrass’s fullepsilonic analysis theory has a different foundation from that of Cauchy.Weierstrass’s 1861 lecturesWeierstrass’s consistent use of e d argumentsWeierstrass delivered his lectures ‘On the differential calculus’ at the GewerbeInsti-tut Berlin2 in the summer semester of 1861. Notes of these lectures were taken by1Edwards (1979, 310), Laugwitz (1987, 260–261, 271–272), andFisher (1978, 16–318) point out tha t Cauchy’s infinitesimals equate to a dependent variablefunction or aðhÞ that approaches zero as h ! 0. Cauchy adopted the latter infinitesimals, which can be written in terms of e d arguments, when he intro-duced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchy’s is a vari-able in the parts that the present paperdiscusses.2A forerunner of the Technische Universit?at Berlin.54 BSHM BulletinHerman Amandus Schwarz, and some of them have been published in the original German by Dugac (1973). Noting the new aspects related to foundational concepts in analysis, full e d definitions of limit and continuous function, a new definition of derivative, and a modern definition of infinitesimals, Dugac considered that the nov-elty of Weierstrass’s lectures was incontestable (1978, 372, 1976, 6–7).3 After beginning his lectures by defining a variable magnitude, Weierstrass gave the definition of a function using the notion of correspondence. This brought him to the following important definition, which did not directly appear in Cauchy’s theory:(D1) If it is now possible to determine for h a bound d such thatfor all values of h which in their absolute value are smaller than d, f ðx þ hÞ f ðxÞ becomes smaller than any magnitude e, however small, then one says that infinitely small changes of the argument correspond to infinitely small changes of the function. (Dugac 1973, 119; English translation from Calinger 1995, 607)That is, Weierstrass defined not infinitely small changes of variables but ‘infinitelysmall changes of the arguments correspond(ing) to infinitely small changes of function’ that were presented in terms of e d inequalities. He founded his theory on this correspondence.Using this concept, he defined a continuous function as follows: (D2) If now a function is such that to infinitely small changes of the argument there correspond infinitely small changes of the function, one then says that it is a continuous function of the argument, or that it changes continuously with this argument. (Dugac 1973, 119–120; English translation from Calinger 1995, 607)So we see that in accordance with his definition of correspondence, Weierstrass actually defined a continuous function on an interval in terms of epsiloni cs. Since (D2) is derived by merely changing Cauchy’s term ‘produce’ to, it seems that Weierstrass took the idea of this definition from‘correspond’Cauchy. However, Weierstrass’s definition was given in terms of epsilonics, whileCauchy’s definition c an only be interpreted in these terms. Furthermore, Weierstrass achieved it without Cauchy’s limit.Luzten? (2003, 186) indicates that Weierstrass still used the concept of ‘infinitelysmall’ in his lectures. Until giving his definition of derivative, Weierstrass actuallya function continued to use the term ‘infinitesimally small’ and often wrote of ‘which becomes infinitely small with h’. But several instances of‘infinitesimallysmall’ appeared in forms of the relationships involving them. Definition (D1) gives the rela-tionship in terms of e d inequalities. We may therefore assume that Weierstrass’s lectures consistently used e d inequalities, even though his definitions were not directly written in terms of these inequalities.Weierstrass inserted sentences confirming that the relationships involving the term ‘infinitely small’ were defined in terms of e d inequalities as follows:ðhÞ is an (D3) If h denotes a magnitude which can assume infinitely small values, ’arbitrary function of h with the property that for an infinitely small value of h it3The present paper also quotes Kurt Bing’s translation included in Calinger’sClassics of mathematics.Volume 29 (2014) 55also becomes infinitely small (that is, that always, as soon as a definite arbitrary small magnitude e is chosen, a magnitude d can be determined such that for all values of h whose absolute value is smaller than d, ’ðhÞ becomes smaller than e).(Dugac 1973, 120; English translation from Calinger, 1995, 607)As Dugac (1973, 65) in dicates, some modern textbooks describe ’ðhÞ as infinitelysmall or infinitesimal.Weierstrass argued that the whole change of function can in general be decom-posed asDf ðxÞ ? f ðx þ hÞ f ðxÞ ? p:h þ hðhÞ; ð 1Þwhere the factor p is independent of h and ðh Þ is a magnitude that becomes infinitely small with h.4 However, he overlooked that such decomposition is not possible for all functions and inserted the term‘in general’. He rewrote h as dx.One can make the difference between Df ðxÞ and p:dx s maller than any magnitude with decreasing dx. Hence Weierstrass defined ‘differential’ as the changewhich a function undergoes when its argument changes by an infinitesimally small magnitude and denoted it as df ðxÞ. Then, df ðxÞ ? p:dx. Weierstrass pointed outthat the differential coefficient p is a function of x derived from f ðxÞ and called it a derivative (Dugac 1973, 120–121; English translation from Calinger 1995, 607–608). In accordance with Weierstrass’s definitions (D1) and (D3),he largelydefined a derivative in terms of epsilonics.Weierstrass did not adopt the term ‘infinitely small’ but directly used e dinequalities when he discussed properties of infinite seriesinvolving uniform conver-gence (Dugac 1973, 122–124). It may beinferred from the publishedportion of his notes that Cauchy’s limit has no place in Weierstrass’s lectures.Grattan-Guinness’s (1986, 228) description of the limit-avoiding character of his analysis represents this situation well.However, Weierstrass thought that his theory included most of the content of Cauchy’s theory. Cauchy first gave the definition of limits of variables andinfinitesi-mals. Then, he demonstrated notions and theorems that were written in terms of the relationships involving infinitesimals. From Weierstrass’s viewpoint,they were writ-ten in terms of e d inequalities. Analytical theory mainly examines properties of functions and series, which were described in the relationships involving Cauchy’s limits and infinitesimals. Weierstrass recognized this fact and had the idea of consis-tently developing his theory in terms of inequalities. Hence Weierstrass atfirst defined the relationships among infinitesimals in terms of e d inequalities. In accor-dance with this definition, Weierstrass rewrote Cauchy’sresults and naturally imported them into his own theory. This is a process that may be described as fol-lows: ‘Weierstrass completed the transformation away fromthe use of terms such as “infinitely small”’ (Katz 1998, 728).Weierstrass’s definition of limitDugac (1978, 370–372; 1976, 6–7) read (D1) as the first definition of limit withthe help of e d. But (D1) does not involve an endpoint thatvariables or functions4Dugac (1973, 65) indicated that ðhÞ corresponds to the modernnotion of oð1Þ. In addition, hðhÞ corre-sponds to the function that was introduced as ’ðhÞ in theformer quotation from Weierstrass’s sentences.。