毕业论文-对称性在简化积分运算中的应用.doc

毕业设计（论文） 题 目：对称性在简化积分运算中的应用学生姓名：学 号：所在学院：金融与数学学院专业班级：应数1001班届 别：指导教师：目 录前言11.对称性在定积分中的应用21.1相关定理及其应用22.对称性在重积分计算中的应用32.1对称性在二重积分中的应用32.2对称性在三重积分计算中的应用43.对称性在曲线积分计算中的应用53.1对称性在第一类曲线积分中的应用53.2 对称性在第二类曲线积分计算中的应用64.对称性在曲面积分计算中的应用94.1对称性在第一类曲面积分计算中的应用94.2 对称性在第二类曲面积分运算中的应用105.化积分区域为对称区域的几种方法11结束语12参考文献:12皖西学院2014届本科毕业设计（论文）对称性在简化积分运算中的应用摘 要：在计算积分中，恰当的使用轮换对成性和对称性，以及奇偶性都可以简化计算。本文主要结合实例阐述对称性在化简几类积分计算中的妙用。具体总结平移变换和区域划分方法来构造对称性。关键字：对称性；奇偶性；定积分；曲线积分；曲面积分Symmetry In The Integral CalculationAbstract ：In calculating of the calculus , proper use translatable symmetry、symmetry and parity can simplify the calculation . This paper mainly use example to elaborated symmetry in the simplify the calcalation .Specific summarize parallel moving transformation and divide the area to construct symmetry.Keywords:Symmetry; parity; definite integral;curve points; surface integrals前言微积分是高等数学中的难点和重点。在很多复杂的微积分证明和计算过程中，尤其是涉及多元微积分问题,常规的方法很难解决，恰当的利用积分区域的对称性和被积函数的奇偶性，可以大大简化积分计算。积分计算中,有很多积分区域存在对称性的问题.合理、恰当的利用其所具有的对称性的性质，则可以使其计算过程得到简化,甚至有些问题直接可以判断出其结果.当积分形式不具有对称性时，有时可以通过变换积分的区域形成对称。本文具体总结平移变换和区域划分等方法来构造对称性，从而达到简化积分计算的效果。本文主要结合实例阐述对称性在简化定积分、二重积分、三重积分、曲线积分、曲面积分计算中的妙用。1.对称性在定积分中的应用1.1相关定理及其应用 定理1.1【1】设在区间可积：（1） 若为奇函数，则；（2） 若为偶函数，则例1.1：计算积分其中为偶函数，则令，则在定积分的计算中，当积分区间为任意有限区间时，该积分区间一定关于直线对称，由此我们可得以下出定理。 定理1.24 设f(x)在a,b上连续，则例1.2 计算定积分解： 令，则由定理1.2知以上是对称性在定积分计算中的应用，可以得出对称性可以大大的简化定积分的运算。2.对称性在重积分计算中的应用2.1对称性在二重积分中的应用相关定理及应用定理2.1.135 若D关于x轴对称，D1位于x轴上半部分，当函数是关于y的奇函数，即时，；当函数是关于y的偶函数，即时，定理2.1.25 若D关于y轴对称，D2位于y轴右半部份，当函数f(x,y)是关于x的奇函数，即时：；当函数f(x,y)是关于x的偶函数，即时：。定理2.1.36 若区域D为关于原点对称，其中D3为D中关于原点对称的右侧。 当为奇函数即时，有 当为偶函数即时，有 推论1 设D是有界平面区域，函数在平面内连续，且D关于x轴、y轴对称，则 （1）若函数关于变量x、y都是偶函数，则， （2）若函数关于其中一个变量x或者变量y为奇函数，则为方便叙述，以下为轮换对称性的定理和定义：定理2.1.57 设函数在xoy平面上的有界区域D上连续，且D关于x,y存在轮换对称性，则定义2.1.47 设D为一有界可度量平面区域（或光滑平面曲线段），若，则称区域D（或光滑平面曲线段）关于x,y具有轮换对称性。例2.1.2 设区域D由x=0,y=0,x+y=1围城，求解 由题意得，变量x,y互换，积分区域区域D不变则所以2.2对称性在三重积分计算中的应用 三重积分应用对称性定理如下：定理2.2.18 设函数是定义在空间有界闭区间区域上的连续函数，且关于坐标平面对称，则 （1） 若是关于变量的奇函数，则；（2） 若是关于变量的偶函数，则。其中是的前半部分，同理可写出关于坐标平面对称时的情形相似于二重积分，得出结论定理2.2.2 设函数为定义在空间有界区域的连续函数，且关于原点对称，则（1） 若，则（2） 若，则 ，类似于二重积分，为方便叙述，三重积分轮换对称性的定理与定义如下：定理2.2.27 设函数f（x,y,z)是定义在空间有界区域上的连续函数，且关于变量x,y,z具有轮换对称性，则定义2.2.17 设是一个有界可度量的集合体（可为空间区域、空间曲线或曲面块），且它的边界光滑，若，则称关于变量具有轮换对称性。例2.2.2计算三重积分，解：有题意知，关于为奇函数，由上述定理知3.对称性在曲线积分计算中的应用3.1对称性在第一类曲线积分中的应用平面上第一类曲线积分的对称性定理：定理3.1.110 设平滑分段光滑曲线关于轴（或轴）对称，且在上有定义、可积，则（1） 若为关于（或）的奇函数，则；（2） 若为关于（或）的偶函数，则， 例3.1.1 设L是圆周，求解12 解：，因为关于轴、轴对称，且关于变量和是偶函数，由上述推论可得 为位于第一象限的部分。又因为故当曲线关于原点对称时，相关定理如下：定理3.1.211 设平面分段光滑曲线关于原点对称，且在L上有定义、可积，则（1） 若是的上半平面或右半平面部分。曲线的轮换对称性定理如下:定理3.1.3 设平面分段光滑曲线关于存在轮换对称性，在上有定义且可积，则3.2 对称性在第二类曲线积分计算中的应用由第二类曲线积分的物理背景为变力做功可知，它与曲线的方向相关，与上述积分对称性的几种结论不同，与第二类曲线积分相关结论如下。定理3.2.112 设为平面上分段光滑的定向曲线，为定义在上的连续函数；（1） 当关于轴对称时： 若是关于的偶函数，则； 若是关于的奇函数，则 若是关于的偶函数，则； 若是关于的奇函数，则 （2） 当关于轴对称时： 若是关于的偶函数，则； 若是关于的奇函数，则 若是关于的偶函数，则； 若是关于的奇函数，则 。（3）当关于原点对称时： 若，关于为偶函数，则 若，关于为奇函数，则，是的上半或右半平面部分。（1） （3）证明如下，（2）证明方法类似于（1），此处不做重复。证明 （1）若关于轴对称，设，且令,则若为的偶函数，则若为的奇函数，则L1为L在x轴上方部分。若Q(x,y)为y的奇函数，则若Q(x,y)为y的偶函数，则L1为L在x轴上方部分。（3） 若关于原点对称时，令,分别为关于原点对称的两部分之一，则， 令的参数方程为，则的参数方程为，；则如果P(x,y),Q(x,y)关于（x,y)是偶函数，则，得到；如果P(x,y),Q(x,y)关于（x,y)是奇函数，则，得到。L1是L的右半或上半平面部分。综上所述，定理得证。例 3.2.1 求解第二类曲线积分，L是椭圆沿顺时针方向。解 因为于原点对称，已知，都是的偶函数，由上述定理知相应轮换对称性定理如下定理3.2.212 设是平面上分段光滑的定向曲线，是定义在上的连续函数。若曲线关于具有轮换对称性，则例 3.2.2 求解第二类曲线积分，沿逆时针方向。解 ，已知关于轴对称,关于为偶函数由上述定理易知，有题意知关于存在轮换对称性，由上述定理已知，则4.对称性在曲面积分计算中的应用4.1对称性在第一类曲面积分中的应用在第一类曲面积分中，与上述类似，可以利用积分区域的对称性（关于坐标面对称、原点对称、轮换对称）以及被积函数奇偶性，简化计算第一类曲面积分，相关定理如下。定理4.1.113 设分块光滑曲面关于坐标面对称，且在上有定义、可积，则(1) 若是关于的奇函数，则，(2) 若是关于的偶函数， 则 其中同理得出曲面关于坐标面对称的相应结论。例4.1.1 求解第一类曲面积分，是球面上部分解 因为曲面关于坐标面对称，且是关于的奇函数，由上定理知因为曲面关于坐标面对称，且是关于的奇函数，由上定理知又因为，则第一类曲面积分轮换对称性的定理如下定理4.1.212 设分片光滑曲面关于存在轮换对称性，并且在上有定义且可积，即例4.1.2 求第一类曲面积分，为。解 由题意知 关于有轮换对称性，则 则4.2 对称性在第二类曲面积分运算中的应用第二类曲面积分类似于第二类曲线积分，根据其定义及物理背景，推导出对称性在第二类曲面积分中的结论。定理4.2.114 设积分曲面光滑或分段光滑，且，曲面和的法线方向相反，若曲面、关于平面对称，则（1） 若，则；（2） 若,则，为的的部分。轮换对称性定理在第二类曲面积分中如下：定理4.2.213 设积分曲面光滑或分段光滑，函数在曲面上有定义和可积，若积分曲面关于有轮换对称性，则小结：通过上述相关定理、定义的陈述和证明，我们可以把与积分相关的对称性统一成一个形式。现将被积函数用表示，积分区域记为，在区域上在的积分记为，相关定理如下。定理15 设是定义为上的连续函数，且具有某种对称性，记的对称点为，则（1） 若；（2） 若；为的一半。（3） 。（注：上述定理在第二类曲线积分、第二类曲面积分领域中无效）5.化积分区域为对称区域的几种方法以下是当积分区域不具有对称性时，可以用下面方法转化为具有对称性的区域。方法一 重新划分区域，构造对称性当积分区域不对称时，可以将积分局域重新进行分割划分，使得每个小区域都具有对称性，从而在划分的每个小区域使用上述的方法进行运算求和。例5.1 计算，其中D是由y=2x,y=-2,x=1围城的平面区域。解： 为使用对称性简化计算，对整个区域D重新划分为和,是上方的部分，是直线下方的部分，易知关于轴对称，关于轴对称，关于和为奇函数。则方法二 平移变换构造对称性当积分区域关于某条坐标轴平行时，可以通过平移坐标轴或积分区域，使得积分区域变为对称性区域，可以使得计算简化。例5.2 求解，其中D：解：已知积分区域不存在对称性，平移变换，的方程变为，则，关于轴对称，是关于的奇函数，由定理2.1.1得，结束语 求解积分过程中，可以通过使用轮换对称性、被积函数的奇偶性、积分区域的对称性来简化问题。在应用对称性简化积分计算时，被积表达式必须同时满足被积函数与积分区域两个方面的条件。本文总结了几类积分领域的的对称性定理，以及在这几个方面的应用形式。对于不能直接运用对称性简化积分运算的，也介绍了几种将非对称性变换为对称性求解的方法。参考文献:1 华东师范大学数学系编。数学分析上册M 北京：高等教育出版社。19912 程希旺 对称心在曲面积分和曲线积分计算中的应用J 2007(10）3华东师范大学数学系编。数学分析下册M 北京：高等教育出版社。19914华东六省工科数学系列编委会 高等数学学习指导书M 沈阳科学技术出版社 19915孙钦福 二重积分的对称性定理及应用 曲阜师范大学学报J 2008:29:9-106孙仁华 二重积分计算中的若干技巧J 湖南冶炼职业技术学院学报 2008:8（2）：102-1047陈云新 轮换对称性在积分中的应用J 高等数学研究 2001（4）：29-318王宪杰 对称区域上二重积分和三重积分的计算J 牡丹江师范学院学报 2007（4）：65-669梁应仙等 对称性在三重积分计算中的应用J 沈阳大学学报 2003.15（4）100-10110刘渭川 利用对称性计算曲线积分与曲面积分J 河南科学 2006.24（6）：810-81211王慧等 对称性在两类积分曲线积分中的应用J 淮北师范大学学报 2011.（32）：412刘富贵等 利用对称性计算第二类曲线积分与曲面积分的方法J 武汉理工大学学报 2006.30（6）13张霞 关于曲面积分对称性的研究J 安庆师范学院学报（自然科学版）2007.13（2）83-8614严水传 关于对称性在积分计算中的应用补遗J 高等数学研究 2002（5）：28-3115李久平 广义对称性在积分计算中的应用 工科数学 2001.17（3）：97-9913致谢我的该篇毕业论文是在邵毅书记的细心指导下完成的，邵书记和蔼的态度和博学的知识给了我很大的鼓舞和帮助，他认证严谨的态度和爱岗敬业的精神对我产生了深远的影响，在此我向他表示衷心的谢意！感谢所有的老师，没有你们授予的知识的积累，我完成这篇论文的过程就不会有那么大的信心和动力。最后，我向所有评阅老师表达我衷心的感谢！ 请删除以下内容，O(_)O谢谢！The origin of taxation in the United States can be traced to the time when the colonists were heavily taxed by Great Britain on everything from tea to legal and business documents that were required by the Stamp Tax. The colonists disdain for this taxation without representation (so-called because the colonies had no voice in the establishment of the taxes) gave rise to revolts such as the Boston Tea Party. However, even after the Revolutionary War and the adoption of the U.S. Constitution, the main source of revenue for the newly created states was money received from customs and excise taxes on items such as carriages, sugar, whiskey, and snuff. Income tax first appeared in the United States in 1862, during the Civil War. At that time only about one percent of the population was required to pay the tax. A flat-rate income tax was imposed in 1867. The income tax was repealed in its entirety in 1872. Income tax was a rallying point for the Populist party in 1892, and had enough support two years later that Congress passed the Income Tax Act of 1894. The tax at that time was two percent on individual incomes in excess of $4,000, which meant that it reached only the wealthiest members of the population. The Supreme Court struck down the tax, holding that it violated the constitutional requirement that direct taxes be apportioned among the states by population (pollock v. farmers loan & trust, 158 U.S. 601, 15 S. Ct. 912, 39 L. Ed. 1108 1895). After many years of debate and compromise, the sixteenth amendment to the Constitution was ratified in 1913, providing Congress with the power to lay and collect taxes on income without apportionment among the states. The objectives of the income tax were the equitable distribution of the tax burden and the raising of revenue. Since 1913 the U.S. income tax system has become very complex. In 1913 the income tax laws were contained in eighteen pages of legislation; the explanation of the tax reform act of 1986 was more than thirteen hundred pages long (Pub. L. 99-514, Oct. 22, 1986, 100 Stat. 2085). Commerce Clearing House, a publisher of tax information, released a version of the Internal Revenue Code in the early 1990s that was four times thicker than its version in 1953. Changes to the tax laws often reflect the times. The flat tax of 1913 was later replaced with a graduated tax. After the United States entered world war i, the War Revenue Act of 1917 imposed a maximum tax rate for individuals of 67 percent, compared with a rate of 13 percent in 1916. In 1924 Secretary of the Treasury Andrew W. Mellon, speaking to Congress about the high level of taxation, stated, The present system is a failure. It was an emergency measure, adopted under the pressure of war necessity and not to be counted upon as a permanent part of our revenue structure. The high rates put pressure on taxpayers to reduce their taxable income, tend to destroy individual initiative and enterprise, and seriously impede the development of productive business. Ways will always be found to avoid taxes so destructive in their nature, and the only way to save the situation is to put the taxes on a reasonable basis that will permit business to go on and industry to develop. Consequently, the Revenue Act of 1924 reduced the maximum individual tax rate to 43 percent (Revenue Acts, June 2, 1924, ch. 234, 43 Stat. 253). In 1926 the rate was further reduced to 25 percent. The Revenue Act of 1932 was the first tax law passed during the Great Depression (Revenue Acts, June 6, 1932, ch. 209, 47 Stat. 169). It increased the individual maximum rate from 25 to 63 percent, and reduced personal exemptions from $1,500 to $1,000 for single persons, and from $3,500 to $2,500 for married couples. The national industrial recovery act of 1933 (NIRA), part of President franklin d. roosevelts new deal, imposed a five percent excise tax on dividend receipts, imposed a capital stock tax and an excess profits tax, and suspended all deductions for losses (June 16, 1933, ch. 90, 48 Stat. 195). The repeal in 1933 of the eighteenth amendment, which had prohibited the manufacture and sale of alcohol, brought in an estimated $90 million in new liquor taxes in 1934. The social security act of 1935 provided for a wage tax, half to be paid by the employee and half by the employer, to establish a federal retirement fund (Old Age Pension Act, Aug. 14, 1935, ch. 531, 49 Stat. 620). The Wealth Tax Act, also known as the Revenue Act of 1935, increased the maximum tax rate to 79 percent, the Revenue Acts of 1940 and 1941 increased it to 81 percent, the Revenue Act of 1942 raised it to 88 percent, and the Individual Income Tax Act of 1944 raised the individual maximum rate to 94 percent. The post-World War II Revenue Act of 1945 reduced the individual maximum tax from 94 percent to 91 percent. The Revenue Act of 1950, during the korean war, reduced it to 84.4 percent, but it was raised the next year to 92 percent (Revenue Act of 1950, Sept. 23, 1950, ch. 994, Stat. 906). It remained at this level until 1964, when it was reduced to 70 percent. The Revenue Act of 1954 revised the Internal Revenue Code of 1939, making major changes that were beneficial to the taxpayer, including providing for child care deductions (later changed to credits), an increase in the charitable contribution limit, a tax credit against taxable retirement income, employee deductions for business expenses, and liberalized depreciation deductions. From 1954 to 1962, the Internal Revenue Code was amended by 183 separate acts. In 1974 the employee retirement income security act (ERISA) created protections for employees whose employers promised specified pensions or other retirement contributions (Pub. L. No. 93-406, Sept. 2, 1974, 88 Stat. 829). ERISA required that to be tax deductible, the employers plan contribution must meet certain minimum standards as to employee participation and vesting and employer funding. ERISA also approved the use of individual retirement accounts (IRAs) to encourage tax-deferred retirement savings by individuals. The Economic Recovery Tax Act of 1981 (ERTA) provided the largest tax cut up to that time, reducing the maximum individual rate from 70 percent to 50 percent (Pub. L. No. 97-34, Aug. 13, 1981, 95 Stat. 172). The most sweeping tax changes since world war ii were enacted in the Tax Reform Act of 1986. This bill was signed into law by President ronald reagan and was designed to equalize the tax treatment of various assets, eliminate tax shelters, and lower marginal rates. Conservatives wanted the act to provide a single, low tax rate that could be applied to everyone. Although this single, flat rate was not included in the final bill, tax rates were reduced to 15 percent on the first $17,850 of income for singles and $29,750 for married couples, and set at 28 to 33 percent on remaining income. Many deductions were repealed, such as a deduction available to two-income married couples that had been used to avoid the marriage penalty (a greater tax liability incurred when two persons filed their income tax return as a married couple rather than as individuals). Although the personal exemption exclusion was increased, an exemption for elderly and blind persons who itemize deductions was repealed. In addition, a special capital gains rate was repealed, as was an investment tax credit that had been introduced in 1962 by President john f. kennedy. The Omnibus Budget Reconciliation Act of 1993, the first budget and tax act enacted during the Clinton administration, was vigorously debated, and passed with only the minimum number of necessary votes. This law provided for income tax rates of 15, 28, 31, 36, and 39.6 percent on varying levels of income and for the taxation of social security income if the taxpayer receives other income over a certain level. In 2001 Congress enacted a major income tax cut at the urging of President george w. bush. Over the course of 11 years the law reduces marginal income tax rates across all levels of income. The 36 percent rate will be lowered to 33 percent, the 31 percent rate to 28 percent, the 28 percent rate to 25 percent. In addition, a new bottom 10 percent rate was created. Since the early 1980s, a flat-rate tax system rather than the graduated bracketed method has been proposed. (The graduated bracketed method is the one that has been used since graduated taxes were introduced: the percentage of tax differs based on the amount of taxable income.) The flat-rate system would impose one rate, such as 20 percent, on all income and would eliminate special deductions, credits, and exclusions. Despite firm support by some, the flat-rate tax has not been adopted in the United States. Regardless of the changes made by legislators since 1913, the basic formula for computing the amount of tax owed has remained basically the same. To determine the amount of income tax owed, certain deductions are taken from an individuals gross income to arrive at an adjusted gross income, from which add