微积分知识点英语版

简介微积分

数学就像一种奇妙的幻想，但这种奇妙幻想最终还是会真实的体现在现实中。做数学运算有一种在做一个想象的发明的感觉，但它确定是强化我们的洞察力的过程，所以我们在周围任何地方都可以发现那样的情景模式，我们数学教育的目标是为了飞跃到现实的脚步之前并分享数学运算带来的理智愉悦的体验。

微积分的发展史是数学的重要做成部分。极限、函数、导数、积分和无穷级数等内容在微积分中所体现。微积分这门学科依然在现代数学教育史中占据一席之地。微分学和积分学统称为微积分学。这两大部分间起桥梁作用的是著名的微积分学基本定理。微积分研究的主要内容是丰富、变化的。微积分课程是先进思想的传播课程，后来人们也将微积分称为数学分析。微积分作为工具广泛用于在科学，经济学，工程等领域，用于解决许多问题，这是代数这门学科独自解决是不能满足的。

一、微积分

希腊数学家阿基米德是第一个找到切线方向的曲线，除了一个圆圈，在一个方法类似于微积分。当研究螺旋时，他一个点的运动分开成两个部分，一个径向运动部件和一个圆周运动部件，然后继续增加双组分在一起从而找到切线运动的曲线。

印度数学家及天文学家阿雅巴塔在499年为解决无穷小天文问题，采用了新的观念和表达方式，创造性的利用了一种基本微分方程形式。Manjula，在世纪十周年开个玩会，详细阐述了该微分方程一个评论。该方程skara最终导致Bha 二世时12世纪发展一种衍生为代表无穷小观念的改变，而他描述早期版本的“罗尔定理”。在15世纪，一个早期版本中值定理parameshvara是在天文学的喀拉拉学校里他的评定中和数学登顶，巴卡拉II被首次描述（1370-1460）。

在17世纪，欧洲数学家撒向后拉线，皮埃尔、德、费，布莱斯帕斯科，约翰沃利斯和其他学者讨论了概念的衍生体系，特别是，在Methodus广告disquirendam maximan最小风险，在tangentibus等linearum curvarum，开发出一种方法测定费最大值、最小值，并对各种曲线的切线相当于分化。艾萨克、牛顿后来写他自己的想法微分早期直接来自“费马研究极值的问题”。

二、积分学

计算面积和体积是微积分的基本功能。这可以追溯到莫斯科纸莎的手写稿（约1820年），其中一名埃及数学家成功的运用微积分知识计算金字塔形锥体的体积。希腊geometers被人认为是利用无穷小解决问题的显著体现者。德谟克利特是称自己是第一个经过慎密，严肃考虑后，将对象划分成无数的横切面。但他不能把其合理化，如果想把光滑的斜坡构思成数学中的一个离散的锥形截面，这个问题是他走入到了思想紧闭区，所以他必须创造出新的理论。大约在相同的时间点上，季诺也急于这方面的思考，使得他焦头烂额后，给出了无穷小理论的悖论。

安提和欧多克斯把它们进行分割成若干的部分，能过计算出地区和固体的面积和体积，这个过程穷尽了他们所熟悉的一般方法，阿基米德进一步创造了这一方法，利用启发式思想，最后得出结论是有点类似于现代的概念。（阿基米德将方形上的抛物线的研究方法应用于球体和圆柱体。）到了牛顿时代，这些方法都

过时了，它不应该被认为是无穷理论的基础，这一说法在此期间流传，然而，希腊数学家说恰是它的这种方法被应用于几何证明是可以被确定为一个正确的理论。

11世纪积分学走上了一个新的层次，一位在埃及工作的伊拉克籍数学家实在欧洲挺有盛誉的，他提出的问题推动了对四次方程求根问题的思考。在他所学的书中，有效解决了上述问题。其中，采用了一种方法很容易的确定出整数求和的问题。他在抛物面上求体积，而且能够将其进行推广，最终得到多项式的积分，并以此获得新的荣誉，他的这种方式接近于一般多项式的积分，但是还有限制因素就是四次多项式。

三、现代微积分

詹姆斯、格雷戈里能证明17世纪中叶微积分一个版本第二基本定理是受限制的。牛顿和莱布尼兹判别法通常被人为是现代的发明，微积分理论日渐成熟与17世纪后期。他们最重要的贡献是发展发展微积分基本定理。同时，大量利用莱布尼兹判别法做出一致的工作和有用的符号以及概念，牛顿是第一个在该领域组织成为一个一致的主题，并提供了一些方法，也是最重要的应用，尤其是积分学的理论基础。巴德费惠更斯，沃利斯以及其他许多人也做出了重要贡献。

四、应用

微积分作为工具解决了物理学以及天文学的问题，这是当代科学的起源。18世纪这些应用不断增多，直接接近拉普拉斯和拉格朗日整体研究的分析领域，拉格朗日（1773年）称我们应该引入动态的潜力理论，虽然名称是“潜在功能”，但科学基本的回忆录必须是绿色的（1827年，1828年印）。进入分析物理问题的其他应用程序种类繁多，在这个地方是不可能的，赫姆霍兹用自己的劳动特别声明，因为在他的动力，电力等理论所做出的贡献，并带来了他伟大的分析能力，并用来承担力学的基本公理，以及对于那些纯粹的数学。此外，微积分被引入到社会科学，新古典经济学。今天它成为主流经济学的有价值得工具。

博耶，卡尔、数学史、纽约：约.威利父子，1991年。

威廉.瑟斯顿，通告阿米尔数学SOC.1990年

Introduction to the calculus

Mathematics is like a flight of fancy,but one in which the fanciful turns out to be real and to have been present all long.Doing mathematics has the feel of fanciful invention,but it is really a process for sharpening for sharpening our perception so that we discover patterns that are everywhere around.To share in the delight and the intellectual experience of mathematics-to fly where before we walked-that is the goal mathematical education.

History of Calculus is part of the history of mathematics focused on limits,functions,derivatives,integrals,and infinte series.the subject,known historicall as infinitesimal calculus,constitutes a major part of modern mathematics education.It has two major-branches,differential calculus and integral calculus,which are related by the fundamental theorem of calculus.calculus is the study of change,in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.A course in calculus is a gateway to other,more advanced courses in mathematics devoted to the study of functions and functions and limits,broadly called mathematical analysis.Calculus has widespread applications in science,economics,and engineering and can solve many problems for which algebra alone is insufficient..

Differential calculus

The Greek mathematician Archimedes was the first to find the tangent to a curve,other than a circle,in a method akin to differential calculus.while studying the spiral,he separated a point's motion into two components,one radial motion component and one circular motion component,and then continued to add the two component motions together thereby finding the tangent to the curve.

The Indian mathematician astronomer Aryabhata in499used notion of infinitesimals and expressed an astronomical problem in the from of a basic differential equation.Manjula,in the10th century,elaborated on this differential equation in a commentary.this equation eventually led Bh a skara II in the12th century to develop the concept of a derivative representing infinitesimal change,and he described an early from of“Rolle’s theorem”.In the15th century,an early version of the mean value theorem was first described by parameshvara(1370-1460)from the Kerala school of astronomy and mathematics in his commentaries on.

Govindasv miand and Bhaskara II.

In the17th century,European mathematicians Isaac Barrow,Pierre de Fermat,Blaise pascal,John Wallis and others discussed the idea of a derivative.In particular,in Methodused disquirendam maximamet minima and in De tangentibus linearum curvarum,Fermat developed a method for detemining maxima,minima and tangents to various curves that was equivalent to differentiation..Isaac Newton would later write that his own early ideas about calculus came directly from“Fermat’s way of drawing tangents”.

Integral calculus

Calculating volumes and areas,the basic function of integral calculus,can be

traced back to the Moscow papyrus(c.1820BC),in which an Egyptian mathematician successfully calculated the volume of a pyramidal frustum.Greek geometers are credited with a significant use of infinitesimals.Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross sections,but his inability to rationalize discrete cross sections with a cone’s smooth prevented him from accepting the idea.At approximately the same time,zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create.

The next major step in integral calculus came in the11th century,when I bal-haytham(known as Alhacen in Europe),an Iraqi mathematician working in Egypt,devised what is now known as“Alhazen’s problem”,which leads to an equation of the fourth degree,in his Book of Optics.while solving this problem,he his the first mathematician to derive the formula for the sum of the fourth powers,using a method that is readily generalizable for determining the general formula for the sum of any integral powers.He performed integration in order to find the volume of a parabolic,and was able to generalize his result for the integrals of polynomials up to the fourth degree.He thus came close to finding a general formula for the integrals of polynomials,but he was not concerned with any polynomials higher than the fourth degree.

Antiphon and later Emulous are generally credited with implementing the method of exhaustion,which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes.Archimedes developed this meth further,while also inventing heuristic methods which resemble modern day concepts somewhat.(See Archimedes’Quadrate of the Parabla,The Method,Archimedes on spheres Cylinders.)It was not until the time of Newton that these methods were made obsolete.It should not be thought that infinitesimals were put on rigorous footing during this time,however.only when it was supplemented by a proper geomeyric proof would Greek mathematicians accept a proposition as true.

Modern calculus

James Gregory was able to prove a restricted version of the second fundamental theorem of calculus in the mid-17thcentury.Newton and Leibniz are usually credited with the invention of modern infinitesimal calculus in the late17th century.their most important contributions were the development of the fundamental theorem of calculus.Also,Leibniz did a great deal of work with developing consistent and useful notation and concepts.Newton was the first to organize the field into one consistent subject,and also provided some of the first and most important applications,especially of integral calculus.Important contributions were also made by barrow,de fermat,Huygens,Wallis and many others.

Applications

The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science.All throught the eighteenth century these applications were multiplied until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of

analysis.To Lagrange(1773)we owe the introduction of the theory of the potential into dynamics,although the name“potential function”and the fundamental memoir of the subject are due to Green(1872,prited in1828).The labors of Helmholtz should be especially mentioned,since he contributed to the theories of dynamics,electricity,etc,and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics.Furthermore,infinitesimal calculus was introduced into the social sciences,starting with Neo classical economics.Today,iit is a valuable tool in mainstream economics.

Boyer,car.a History of Mathematies.New York:John Wiley Sons,1991.

William Thurston,Notices Amir.Math.Soc.1990.