数学学院(SchoolofMathematics)

第24卷第2期2021年3月高等数学研究STUDIES IN COLLEGE MATHEMATICSVol. 24 ,No. 2Mar. , 2021doi : 10. 3969/j. issn. 1008-1399. 2021. 02. 021二阶连续混合偏导数相等的一个简洁证明刘雁鸣(武汉工程大学数理学院!湖北武汉430205)摘 要 本文运用导数判别函数单调性的知识，通过构造函数给出了二阶连续混合偏导数相等的一个证明，比数 学分析中的证明方法简易”关键词二元函数；二阶混合偏导数中图分类号 O171 文献标识码 A 文章编号 1008 - 1399(2021)02 - 0064 - 02A Concise Proof of the Mixed Derivative TheoremLIU Yanming(School of Mathematics and Physics !Wuhan Institute of Technology !Wuhan 430205)Abstract By the construction of a function of two variables and the knowledge of the first derivative testforincreasinganddecreasingfunctions !thispaperpresentsaconciseproofofthe MixedDerivativeTheo- reminCalculusKeywords function of two variables , mixed sccond ―order partial derivatives在高等数学中有一个重要定理：二元函数Z = f(x,y )的两个二阶混合偏导数E 和E 如果连续， 则必相等.国内院校广泛使用同济版《高等数学》(下 册)1+对此不予证明，原因是一般教师所了解的数学 分析中的证明方法，学生不易接受和掌握.目前国内的数学分析教材对此定理的证明，均 采用了菲赫金哥尔兹的《微积分学教程(第一卷第8 版)》中的证明閃，如华罗庚的《高等数学引论(第二 册)》3+、华东师范大学《数学分析(下册)》4等.我们 尝试利用大一新生在高中就已经掌握的运用导数判 别函数的单调性知识来给出一个简单的证明.首先给出一个引理.引理 设函数h(.x,y)是定义在应上开集D 的 实值函数,如果在D 内的一个矩形区域J = La,*+X *9]上有混合偏导数x y>o 或者h x>o,那么有h ( ,d ) 一h(*, c) 一h(a 9) + h(a ,c)〉0 .收稿日期：2020 - 09 - 18修改日期2020 -11 -28作者简介：刘雁鸣0974 —)，女，内蒙赤峰，理学硕士，副教授,高等数学教 ，Email :liuyanminglaoshi@126 com7如果在J 上有混合偏导数E <0或者E <o,那么有h(*9) —h(*,c ) —h(a,9)+h(a ,c)<0.证明 首先，由假设h y>o,对于固定的变量 x 2[a,*+,函数h x作为定义在区间*9]上关于变量 y 的导数大于零，故由单调性判别法可知hl (x9) — h'x (x ,c)>0.函数h(x,d )—h(x,c )作为定义在区间[a,*+上 关于变量x 的一元函数的一阶导数为正，再次应用判别 知[h(*,d ) —h(*,c)] 一 [h(a,d ) — h(a,c )]>0. 此即 h(*,d )—h(*,c) 一h(a , d) + h(a , c)>0.对于 二元函数h(x,y ),变量x 与y 地位对等，故可以类 似的证明E x>0的情形.其次，由假设/4<0,对于固定的变量x 2 [a , *+,函数1作为定义在区间*9+上关于变量y 的 导数小于零，故由单调性判别法可知1 (x,9—h'x(x ,c )<0.而函数h(x 9) —h(x ,c )作为定义在区间[a,*] 上关于变量x 的函数的一阶导数为负，再次应用单第24卷第2期刘雁鸣：二阶连续混合偏导数相等的一个简洁证明65调性判别法可知, 9% 一 h(b ,c~) + 一,9) 一 X(a ,c) + <0.此即 h $ ,9)一h(b ,c)一h(a,9)+h(a,c )<0.对于二元函数h (xy ),变量x 与y 地位对等，故可以类 似的证明h ,x <0的情形.证毕•下面证明定理.令h(x,y ) =f(x,y )—(x ° ,0)+ E (x ° , ,0)](x —x 0)(y —,0).易知h(x , y )在D 内连续、具有二阶连续偏导数.对h(x , y)关于x 求偏导,得, y ) = f ：(x , y')—1* E , (x ° , ,0)+E (x 0, ,0)](y —,0),在点(x ° , ,0 )处，h xy (x ° , ,0 ) = 2* Ey (x ° , ,0 ) — f yx (x ° , ,0 ) ] •对h(x , y)关于y 求偏导,得h((x , y ) = f ,(x , y ) —2 * E, (x ° , ,0 )+ E (x ° , ,0)](x —x °),在 (xy 0 )处h yx (x ° , ,0 ) = 1* Ex (x ° , ,0 ) — Ey (x ° , ,0)+ •因此有 Ey (x ° , ,0 ) = —hyx (x ° , ,0 )•若E p (x ° , ,0).0 ,由于h (x , y )在D 内具有二 阶连续偏导数,故在点(x 0, ,0)的某邻域内均有E ,(x , y )〉0.在该邻域内取矩形区域[a , b ]X *c , 9+ ,有h(b , 9) 一h(b , c ) 一h(a , 9)+h(a , c )〉0.同时,又有h (x (x ° , ,0)<0 ,从而h (b 9 ) —h (b c ) —h (a 9 ) +h (a c ) <0 •这个矛盾表明,只有h,(x 0, ,0) = 0成立.因此fE xy (x 0 y 0 ) =fE yx (x 0 y 0 ) •证毕•这个证明比较简单,笔者在课堂教学中讲授过 数次,学生容易接受和掌握，也加深了对利用导数判别函数单调性的认识•参考文献*+同济大学数学系编.高等数学(下册)：M ].7版•匕京:高等教育出版社，2017：70.*+菲赫金哥尔茨著•微积分学教程(第一卷)M +杨張亮,叶彦谦译•版•匕京：高等教育出版社2006 = 347 - 348*+华罗庚,王元著.高等数学引论(第二册)[M ].1版•匕京：高等教育出版社2009:43 - 44.*+华东师范大学数学系编.数学分析(下册)[M+ 4版.北京：高等教育出版社，2016:139 - 140.(上接第63页)U 1)=1在yoz 面上的投①②影方程•解曲线%由两封闭球面相交而成,是一条封闭曲线，但由①一②消去x 得y +z =1 •这是变量y 和z 可在定任意取值的平面方程，说明方程组在消去x 的过程中也消去了对变量y 和变量z 的限制,因此曲线%在yoz 面上的投影曲线仅是方程组J ，+z =1所表示曲线的一部分,需要添加消去的限制l x =0条件使它成为所求投影曲线方程•由①知, —1 ,由②知y —1 —1 ,联立两不等式得0—y —1 ,因此曲线4 y +z = 1%在y o z 面上的投影方程是3(0—,—1).[x = 0通过以上讨论可看出，同济七版《高等数学》工教材对空间曲线C ：y z ) =0y z )=0① 在某一坐标面上②的投影曲线方程的定义更为恰当,空间曲线C 在坐标面上的投影曲线可能与它相应的投影柱面在坐标面上的准线吻合，也可能只是其投影柱面在相应坐标面上的准线的一部分•排除定理1的情形，如果曲线C非封闭,或曲线C 封闭,只要方程组消去某一变量如变量z 的同时并没有改变原方程组中另两个变量如x ( y ) 的曲 线 C 在 xoy 面 的 曲 线就是投影柱面在xoy 面的准线•空间曲线以参数方程形式给出时,类似讨论.参考文献*+同济大学数学系•高等数学(下册)[M+7版•匕京：高等教20147*+朱健民,李建平.高等数学(上册)[M ].2版•匕京：高等教2015.*+吴赣昌•高等数学(上册)[M ]. 3版•匕京：中国人民大学2009.*+水乃翔.从空间曲线投影问题的典型错误谈起*+•湖州师范学院学报：自然科学版1984 , ( S1)：55 -57.。

360教育集团介绍，新南威尔士大学现有10个学院，1个大学院，共75个系。

其中大学院是由澳大利亚国防部资助，是在1981年成立的，该处学员都是军校学生，在此学习艺术、科学、工程。

这个大学院也称为澳大利亚国防军事学院。

10个学院中，艺术学院是位于帕丁顿的前城市艺术学院并入新南威尔士大学而成的。

生命科学学院与科学技术学院皆建立于1997年。

其余的学院分别是：艺术与社会科学学院，建筑环境学院，贸易与经济学院，工程学院，法律学院，医药学院，管理学院。

文学与社会科学院文学与社会科学院(Faculty of Arts and Social Sciences) 位于Morven Brown、Robert Webster和Mathews等大楼。

隶属文学与社会科学院的共有12个学系。

教育系 (School of Education) 英文系 (School of English) 历史系 (School of History) 历史与科学哲学系 (School of History and Philosophy of Science) 媒体、电影和戏剧系 (School of Media,Film and Theatre) 现代语言系 (School of Modern Language Studies) 音乐与音乐教育系 (School of Music and Music Education) 哲学系 (School of Philosophy) 政治与国际关系学系 (School of Politics and International Relations) 社会科学与政策系 (School of Social Science and Policy) 社会工作系(School of Social Work) 社会学与人类学系 (School of Sociology and Anthropology)建筑环境学院建筑环境学院 (Faculty of the Built Environment) 提供的学士课包括建筑学、建筑物建筑管理、风景建筑、工业设计、内部建筑和市镇规划等。

教学机构土木工程学院 School of Civil Engineering机械工程学院 School of Mechanical Engineering电气工程学院 School of Electrical Engineering信息科学与技术学院 School of Information Science and Technology经济管理学院 School of Economics and Business Admini stration人文社会科学学院 School of Humanities and Social Sciences交通运输学院 School of Traffic and Transportation环境科学与工程学院 School of Environmental Science and Engineering外国语学院 School of Foreign Languages 物理科学与技术学院 School of Physical Science and Technology材料科学与工程学院 School of Material Science and Engineering建筑学院 School of Architecture艺术与传播学院 School of Arts and Communication数学学院 School of Mathematics物流学院 School of Logistics政治学院School of Politic al Science成人教育学院 School of Adult Education网络教育学院 School of Online Education软件学院 School of Software力学与工程学院 School of Mechanics and Engineering生命科学与工程学院 School of Life Science and Engineering公共管理学院 School of Public Administration 科技学院 School of Science and Technology体育工作部 Division of Physical Education现代工业技术培训中心 Modern Industrial Technology Training CenterCAD工程中心 CAD Engineering Center超导研究开发中心 Superconductivity Research and Development Cente r心理研究与咨询中心 Psychological Research and Counseling Center智能控制与仿真工程中心 Intelligent Control and Simulation Engineering Center 深圳研究院 Shenzhen Research Institute峨眉校区机构纪律检查委员会CPC Discipline Inspection Commission工会Trade Union Committee团委Communist Youth League Committee党委宣传统战部Publicity and United Front Work Department of CPC Committee 党委组织部Organization Department学校办公室Office of the President教务处Office of the Provost科学技术处Office of Science and Technology人事处Office of Human Resources计划财务处Office of Budget and Finance学生工作处（部）Office of Student Affairs武装保卫部Office of Public Safety校园规划建设处Office of Campus Planning and Construction国有资产及实验室管理处Office of State-owned Facilities and Labs Management后勤集团Logistics Group图书馆Library信息网络中心（计算中心）Information and Network Center (Computing Center)高等教育研究所Higher Education Research Institute成都办事处Liaison Office in Chengdu土木工程系Department of Civil Engineering机械工程系Department of Mechanical Engineering电气工程系Department of Electrical Engineering计算机与通信工程系Department of Computer and Telecommunication Engineering交通运输系Department of Traffic and Transportation外国语系Department of Foreign Languages人文社会科学系Department of Humanities and Social Sciences基础课部Division of Foundation Courses体育工作部Division of Physical Education三校区地址九里校区:四川省成都市二环路北一段111号西南交通大学邮政编码:610031 Jiuli Campus:Southwest Jiaotong UniversityNo.111, First Section, North of Second Ring RoadChengdu, Sichuan 610031P.R. China犀浦校区:四川省成都市高新区西部园区西南交通大学邮编:611756Xipu Campus:Southwest Jiaotong UniversityWest Section, High-tech ZoneChengdu, Sichuan 611756P.R. China峨眉校区: 四川省蛾眉山市西南交通大学邮编:614202Emei Campus:Southwest Jiaotong UniversityEmei, Sichuan 614202P.R. China以前办公自动化上发过的，作为一个有心人，应该把它保存在电脑上。

附件3中山大学各学院、直属系英文译名中国语言文学系Department of Chinese Language and Literature 历史学系Department of Histroy哲学系Department of Philosophy社会学与人类学学院School of Sociology and Anthropology人文高等研究院Institute for Advanced Studies in Humanities通识教育部University General Education Unit岭南学院Lingnan College外国语学院School of Foreign Languages法学院School of Law知识产权学院School of Intellectual Property政治与公共事务管理学院School of Government管理学院Business School创业学院Institute of Entrepreneurship马克思主义学院School of Marxism心理学系Department of Psychology传播与设计学院School of Communication and Design资讯管理学院School of Information Mangement数学学院School of Mathematics物理学院School of Physics—1—化学学院School of Chemistry地理科学与规划学院School of Geography and Planning生命科学学院School of Life Sciences工学院School of Engineering材料科学与工程学院School of Materials Science and Engineering 电子与信息工程学院School of Electronics and Information Technology中山大学—卡内基梅隆大学联合工程学院SYSU-CMU Joint Institute of Engineering微电子学院School of Microelectronics数据科学与计算机学院School of Data and Computer Science环境科学与工程学院School of Environmental Science and Engineering中山医学院Zhongshan School of Medicine中山大学光华口腔医学院•附属口腔医院Guanghua School of Stomatology, Hospital of Stomatology, Sun Yat-sen University公共卫生学院School of Public Health药学院School of Pharmaceutical Sciences护理学院School of Nursing逸仙学院Yat-sen Honor School体育部Department of Physical Education中国语言文学系（珠海）Department of Chinese Langauge and Literature (Zhuhai)—2—历史学系（珠海）Department of History (Zhuhai)哲学系（珠海）Department of Philosophy (Zhuhai)国际金融学院International School of Business & Finance国际翻译学院School of International Studies国际关系学院School of International Relations旅游学院School of Tourism Management数学学院（珠海）School of Mathematics (Zhuhai)物理与天文学院School of Physics and Astronomy大气科学学院School of Atmospheric Sciences海洋科学学院School of Marine Sciences地球科学与工程学院School of Earth Sciences and Engineering化学工程与技术学院School of Chemical Engineering and Technology海洋工程学院School of Ocean Engineering中法核工程与技术学院Sino-French Institute of Nuclear Engineering & Technology基础医学院（深圳）School of Basic Medicine (Shenzhen)临床医学院（深圳）School of Clinical Medicine (Shenzhen)药学院（深圳）School of Pharmaceutical Science (Shenzhen)—3—。

数学专业介绍英文Mathematics is a versatile and highly respected field of study that plays a crucial role in various aspects of our daily lives. A major in mathematics provides students with a strong foundation in logical reasoning, problem-solving skills, and critical thinking abilities. The curriculum typically includes courses in calculus, algebra, geometry, statistics, and differential equations, among others. Students also have the opportunity to explore more advanced topics such as number theory, mathematical analysis, and mathematical modeling.One of the key benefits of studying mathematics is the development of analytical skills that are highly sought after in a wide range of industries. Mathematics majors are equipped with the ability to analyze complex problems, formulate logical arguments, and devise innovative solutions. These skills are applicable in fields such as finance, engineering, computer science, and data analysis, making mathematics graduates highly marketable in today'scompetitive job market.Furthermore, a mathematics major opens up a diverse range of career opportunities for graduates. Many mathematics majors pursue careers in academia as professors or researchers, while others find employment in industries such as finance, technology, consulting, and government. The analytical and problem-solving skills gained through a mathematics degree are transferable to various professions, making mathematics graduates valuable assets in any workplace.In addition to its practical applications, mathematics is also a deeply theoretical and abstract field of study that appeals to those with a passion for pure mathematics. The beauty of mathematics lies in its elegant theories, complex structures, and profound connections to other disciplines such as physics, computer science, and economics. Students who are drawn to the theoretical aspects of mathematics can delve into topics such as topology, abstract algebra, and mathematical logic, exploring the fundamental principles that underpin themathematical universe.Moreover, studying mathematics can also lead to personal growth and intellectual fulfillment. The process of solving challenging mathematical problems, proving theorems, and exploring new mathematical concepts can be intellectually stimulating and rewarding. Mathematics majors develop a strong sense of discipline, perseverance, and creativity as they grapple with complex mathematical ideas and push the boundaries of their understanding. The pursuit of knowledge in mathematics is a journey of self-discovery that can lead to a deeper appreciation of the beauty and power of mathematics.Overall, a major in mathematics offers students a unique opportunity to cultivate their analytical skills, explore the theoretical foundations of mathematics, and pursue diverse career paths in academia, industry, and beyond. Whether you are interested in applying mathematics to real-world problems, delving into abstract mathematical theories, or simply seeking intellectual challenge and personal growth, a mathematics major can provide you withthe knowledge, skills, and opportunities to succeed in a variety of fields. Mathematics is not just a subject of study; it is a way of thinking, problem-solving, and understanding the world around us.。

外国语学院School of Foreign Languages and Cultures英语英语English 英语教育英语教育English Education日语日语Japanese阿拉伯语阿拉伯语Arabic政法学院School of Politics and Law法学法学Science of Law行政管理行政管理Administration Management思想政治管理思想政治管理Ideological and Political Management人文学院School of Humanities汉语言文学汉语言文学Chinese Literature新闻学新闻学Journalism历史学历史学History旅游管理旅游管理Tourism Management广告学广告学Science of Advertising物理电气信息学院School of Physics and Electrical Information Engineering物理学物理学Physics电气工程与自动化电气工程与自动化Electrical Engineering and Automation电子信息工程电子信息工程Electronic Information Engineering通信工程通信工程Communication Engineering网络工程网络工程Network Engineering资源环境学院School of Resources and EnvironmentResources and Environment and Urban and Rural 资源环境与城乡规划资源环境与城乡规划Planning地理信息系统Geographical Information System地理信息系统Environmental Science环境科学环境科学土木与水利工程学院School of Civil Engineering and Water ConservancyArchitecture建筑学建筑学Civil Engineering土木工程土木工程City Planning城市规划城市规划交通工程Transportation Engineering交通工程Agricultural Water Conservancy Engineering 农业水利工程农业水利工程工程管理Engineering Management工程管理农学院School of AgricultureFood Science and Engineering食品科学与工程食品科学与工程农学Agronomy农学Gardening园艺园艺植物保护Plant Protection植物保护Herbal Science草业科学草业科学林学Forestry林学Horticulture园林园林森林资源保护与游憩Forestry Resource Protection and Recreation森林资源保护与游憩Animal Medicine动物医学动物医学动物科学Zoology动物科学Agricultural Resources and Environment农业资源与环境农业资源与环境经济管理学院School of Economics and Management经济学Economics经济学Business Administration工商管理工商管理信息管理与信息系统Information Management and Information System 信息管理与信息系统Marketing市场营销市场营销Accounting会计学会计学E-Commerce电子商务电子商务农林经济管理Farming and Forestry Economy Management 农林经济管理教育科学学院School of Education ScienceEducational Technology教育技术学教育技术学应用心理学Applied Psychology应用心理学小学教育Elementary School Education小学教育数学计算机学院School of Mathematics and Computer ScienceMathematics and Applied Mathematics数学与应用数学数学与应用数学信息与计算机科学Information and Computer Science信息与计算机科学Software Engineering软件工程软件工程Computer Science and Technology计算机科学与技术计算机科学与技术生命科学学院School of Life SciencesBiological Science生物科学生物科学Biotechnology生物技术生物技术化学化工学院School of Chemistry and Chemical EngineeringChemistry化学化学应用化学应用化学Applied Chemistry化学工程与工艺化学工程与工艺Chemical Engineering and Technology制药工程制药工程Pharmaceutical Engineering材料化学材料化学Material Chemistry机械工程学院School of Mechanical Engineering过程装备与控制工程过程装备与控制工程Processing Equipment and Control Engineering机械工程及自动化机械工程及自动化Mechanical Engineering and Automation交通运输交通运输Traffic and Transportation农业机械化及自动化农业机械化及自动化Agricultural Mechanization and Automation音乐学院School of Music音乐学音乐学Musicology舞蹈学舞蹈学Dancing体育学院School of Physical Education体育教育体育教育 Physical Education美术学院School of Art美术学美术学Art艺术设计艺术设计Artistic Design国际教育学院国际教育学院School of International Education思想政治课教学研究部思想政治课教学研究部Department of Moral Education民族预科教育学院民族预科教育学院Preparatory School for Minority Nationalities继续教育学院继续教育学院School of Further Education远程教育学院远程教育学院School of Distance Learning高等职业技术学院高等职业技术学院School of Higher V ocational Technology新华学院新华学院Xinhua School船舶工程学院 C o l l e g e o f S h i p b u i l d i n g E n g i n e e r i n g建筑工程学院 C o l l e g e o f C i v i l E n g i n e e r i n g动力与能源工程学院 Co l l e g e o f P o w e r a n d E n e r g y E n g i n e e r i n g 自动化学院 C o l l e g e o f A u t o m a t i o n水声工程学院 Co l l e g e o f U n d e r w a t e r A c o u s t i c E n g i n e e r i n g 计算机科学与技术学院 Co l l e g e o f C o m p u t e r S c i e n c e a n d T e c h n o l o g y 机电工程学院 Co l l e g e o f M e c h a n i c a l a n d E l e c t r i c a l E n g i n e e r i n g 信息与通信工程学院 C o l l e g e o f I n f o r m a t i o n a n d C o m m u n i c a t i o n s E n g i n e e r i n g 经济管理学院 S c h o o lo f E c o n o m i c s a n d M a n a g e m e n t 材料科学与化学工程学院C o l l e g e o f M a t e r i a l S c i e n c e a n d C h e m i c a l E n g i n e e r i n g 理学院 C o l l e g e o f S c i e n c e人文社会科学学院 C o l l e g e o f H u m a n i t i e s a n d S o c i a l S c i e n c e s核科学与技术学院 C o l l e g e o f N u c l e a r S c i e n c e a n d T e c h n o l o g y外语系 F o r e i g n L a n g u a g e s D e p a r t m e n t体育军事训练部 P h y s i c a lE d u c a t i o n D e p a r t m e n t中文学科、专业名称 英文学科、专业名称哲学 Philosophy马克思主义哲学 Philosophy of Marxism中国哲学 Chinese Philosophy外国哲学 Foreign Philosophies逻辑学 Logic伦理学 Ethics美学 Aesthetics宗教学 Science of Religion科学技术哲学 Philosophy of Science and Technology经济学 Economics理论经济学 Theoretical Economics政治经济学 Political Economy经济思想史 History of Economic Thought经济史 History of Economic西方经济学 Western Economics世界经济 World Economics人口、资源与环境经济学 Population, Resources and Environmental Economics 应用经济学 Applied Economics国民经济学 National Economics区域经济学 Regional Economics财政学（含税收学） Public Finance (including Taxation)金融学（含保险学） Finance (including Insurance)产业经济学 Industrial Economics国际贸易学 International Trade劳动经济学 Labor Economics统计学 Statistics数量经济学 Quantitative Economics中文学科、专业名称 英文学科、专业名称国防经济学 National Defense Economics法学 Law法学 Science of Law法学理论 Jurisprudence法律史 Legal History宪法学与行政法学 Constitutional Law and Administrative Law刑法学 Criminal Jurisprudence民商法学(含劳动法学、社会保障法学) Civil Law and Commercial Law (including Science of Labour Law and Science of So cial Security Law )诉讼法学 Science of Procedure Laws经济法学 Science of Economic Law环境与资源保护法学 Science of Environment and Natural Resources Protection Law国际法学(含国际公法学、国际私法学、国际经济法学、) International law (including International Public law, International PrivateLaw and International Economic Law)军事法学 Science of Military Law政治学 Political Science政治学理论 Political Theory中外政治制度 Chinese and Foreign Political Institution科学社会主义与国际共产主义运动 Scientific Socialism and InternationalCommunist Movement中共党史(含党的学说与党的建设) History of the Communist Party of China(including the Doctrine of China Party and Party Building)马克思主义理论与思想政治教育 Education of Marxist Theory and Education in Ideology and Politics国际政治学 International Politics国际关系学 International Relations外交学 Diplomacy社会学 Sociology社会学 Sociology人口学 Demography人类学 Anthropology民俗学(含中国民间文学) Folklore (including Chinese Folk Literature) 民族学 Ethnology民族学 Ethnology马克思主义民族理论与政策 Marxist Ethnic Theory and Policy中国少数民族经济 Chinese Ethnic Economics中国少数民族史 Chinese Ethnic History中国少数民族艺术 Chinese Ethnic Art教育学 Education教育学 Education Science教育学原理 Educational Principle课程与教学论 Curriculum and Teaching Methodology教育史 History of Education比较教育学 Comparative Education学前教育学 Pre-school Education高等教育学 Higher Education成人教育学 Adult Education职业技术教育学 Vocational and Technical Education特殊教育学 Special Education教育技术学 Education Technology心理学 Psychology基础心理学 Basic Psychology发展与心理学 Developmental and Educational Psychology应用心理学 Applied Psychology体育学 Science of Physical Culture and Sports体育人文社会学 Humane and Sociological Science of Sports运动人体科学 Human Movement Science体育教育训练学 Theory of Sports Pedagogy and Training民族传统体育学 Science of Ethnic Traditional Sports文学 Literature中国语言文学 Chinese Literature文艺学 Theory of Literature and Art语言学及应用语言学 Linguistics and Applied Linguistics汉语言文字学 Chinese Philology中国古典文献学 Study of Chinese Classical Text中国古代文学 Ancient Chinese Literature中国现当代文学 Modern and Contemporary Chinese Literature中国少数民族语言文学 Chinese Ethnic Language andLiterature比较文学与世界文学 Comparative Literature and World Literature 外国语言文学 Foreign Languages and Literatures英语语言文学 English Language and Literature俄语语言文学 Russian Language and Literature法语语言文学 French Language and Literature德语语言文学 German Language and Literature日语语言文学 Japanese Language and Literature印度语言文学 Indian Language and Literature西班牙语语言文学 Spanish Language and Literature阿拉伯语语言文学 Arabic Language and Literature欧洲语言文学 European Language and Literature亚非语言文学 Asian-African Language and Literature外国语言学及应用语言学 Linguistics and Applied Linguistics in Foreign Languages新闻传播学 Journalism and Communication新闻学 Journalism传播学 Communication艺术学 Art艺术学 Art Theory音乐学 Music美术学 Fine Arts设计艺术学 Artistic Design戏剧戏曲学 Theater and Chinese Traditional Opera电影学 Film广播电视艺术学 Radio and television Art舞蹈学 Dance历史学 History历史学 History史学理论及史学史 Historical Theories and History of Historical Science 考古学及博物馆学 Archaeology and Museology历史地理学 Historical Geography历史文献学(含敦煌学、古文字学) Studies of Historical Literature (including Paleography and Studies of Dunhuang)专门史 History of Particular Subjects中国古代史 Ancient Chinese History中国近现代史 Modern and Contemporary Chinese History世界史 World History理学 Natural Science数学 Mathematics基础数学 Fundamental Mathematics计算数学 Computational Mathematics概率论与数理统计 Probability and Mathematical Statistics应用数学 Applied Mathematics运筹学与控制论 Operational Research and Cybernetics物理学 Physics理论物理 Theoretical Physics粒子物理与原子核物理 Particle Physics and Nuclear Physics原子与分子物理 Atomic and Molecular Physics等离子体物理 Plasma Physics凝聚态物理 Condensed Matter Physics声学 Acoustics光学 Optics无线电物理 Radio Physics化学 Chemistry无机化学 Inorganic Chemistry分析化学 Analytical Chemistry有机化学 Organic Chemistry物理化学（含化学物理） Physical Chemistry (including Chemical Physics) 高分子化学与物理 Chemistry and Physics of Polymers天文学 Astronomy天体物理 Astrophysics天体测量与天体力学 Astrometry and Celestial Mechanics地理学 Geography自然地理学 Physical Geography人文地理学 Human Geography地图学与地理信息系统 Cartography and Geography Information System大气科学 Atmospheric Sciences气象学 Meteorology大气物理学与大气环境 Atmospheric Physics and Atmospheric Environment 海洋科学 Marine Sciences物理海洋学 Physical Oceanography海洋化学 Marine Chemistry海洋生理学 Marine Biology海洋地质学 Marine Geology地球物理学 Geophysics固体地球物理学 Solid Earth Physics空间物理学 Space Physics地质学 Geology矿物学、岩石学、矿床学 Mineralogy, Petrology, Mineral Deposit Geology地球化学 Geochemistry古生物学与地层学（含古人类学） Paleontology and Stratigraphy (including Paleoanthropology)构造地质学 Structural Geology第四纪地质学 Quaternary Geology生物学 Biology植物学 Botany动物学 Zoology生理学 Physiology水生生物学 Hydrobiology微生物学 Microbiology神经生物学 Neurobiology遗传学 Genetics发育生物学 Developmental Biology细胞生物学 Cell Biology生物化学与分子生物学 Biochemistry and Molecular Biology生物物理学 Biophysics生态学 Ecology系统科学 Systems Science系统理论 Systems Theory系统分析与集成 Systems Analysis and Integration科学技术史 History of Science and Technology工学 Engineering力学 Mechanics一般力学与力学基础 General and Fundamental Mechanics固体力学 Solid Mechanics流体力学 Fluid Mechanics工程力学 Engineering Mechanics机械工程 Mechanical Engineering机械制造及其自动化 Mechanical Manufacture and Automation机械电子工程 Mechatronic Engineering机械设计与理论 Mechanical Design and Theory车辆工程 Vehicle Engineering光学工程 Optical Engineering仪器科学与技术 Instrument Science and Technology精密仪器及机械 Precision Instrument and Machinery测试计量技术及仪器 Measuring and Testing Technologies and Instruments 材料科学与工程 Materials Science and Engineering材料物理与化学 Materials Physics and Chemistry材料学 Materialogy材料加工工程 Materials Processing Engineering冶金工程 Metallurgical Engineering冶金物理化学 Physical Chemistry of Metallurgy钢铁冶金 Ferrous Metallurgy有色金属冶金 Non-ferrous Metallurgy动力工程及工程热物理 Power Engineering and Engineering Thermophysics 工程热物理 Engineering Thermophysics热能工程 Thermal Power Engineering动力机械及工程 Power Machinery and Engineering流体机械及工程 Fluid Machinery and Engineering制冷及低温工程 Refrigeration and Cryogenic Engineering化工过程机械 Chemical Process Equipment电气工程 Electrical Engineering电机与电器 Electric Machines and Electric Apparatus电力系统及其自动化 Power System and its Automation高电压与绝缘技术 High Voltage and Insulation Technology电力电子与电力传动 Power Electronics and Power Drives电工理论与新技术 Theory and New Technology of Electrical Engineering电子科学与技术 Electronics Science and Technology物理电子学 Physical Electronics电路与系统 Circuits and Systems微电子学与固体电子学 Microelectronics and Solid State Electronics电磁场与微波技术 Electromagnetic Field and Microwave Technology信息与通信工程 Information and Communication Engineering通信与信息系统 Communication and Information Systems信号与信息处理 Signal and Information Processing控制科学与工程 Control Science and Engineering控制理论与控制工程 Control Theory and Control Engineering检测技术与自动化装置 Detection Technology and Automatic Equipment系统工程 Systems Engineering模式识别与智能系统 Pattern Recognition and Intelligent Systems导航、制导与控制 Navigation, Guidance and Control计算机科学与技术 Computer Science and Technology计算机软件与理论 Computer Software and Theory计算机系统结构 Computer Systems Organization计算机应用技术 Computer Applied Technology建筑学 Architecture建筑历史与理论 Architectural History and Theory建筑设计及其理论 Architectural Design and Theory城市规划与设计（含风景园林规划与设计） Urban Planning and Design (including Landscape Planning and Design) 建筑技术科学 Building Technology Science土木工程 Civil Engineering岩土工程 Geotechnical Engineering结构工程 Structural Engineering市政工程 Municipal Engineering供热、供燃气、通风及空调工程 Heating, Gas Supply, Ventilating and Air Conditioning Engineering防灾减灾工程及防护工程 Disaster Prevention and Reduction Engineering and Protective Engineering 桥梁与隧道工程 Bridge and Tunnel Engineering水利工程 Hydraulic Engineering水文学及水资源 Hydrology and Water Resources水力学及河流动力学 Hydraulics and River Dynamics水工结构工程 Hydraulic Structure Engineering水利水电工程 Hydraulic and Hydro-Power Engineering港口、海岸及近海工程 Harbor, Coastal and Offshore Engineering测绘科学与技术 Surveying and Mapping大地测量学与测量工程 Geodesy and Survey Engineering摄影测量与遥感 Photogrammetry and Remote Sensing地图制图学与地理信息工程 Cartography and Geographic Information Engineering化学工程与技术 Chemical Engineering and Technology化学工程 Chemical Engineering化学工艺 Chemical Technology生物化工 Biochemical Engineering应用化学 Applied Chemistry工业催化 Industrial Catalysis地质资源与地质工程 Geological Resources and Geological Engineering矿产普查与勘探 Mineral Resource Prospecting and Exploration地球探测与信息技术 Geodetection and Information Technology地质工程 Geological Engineering矿业工程 Mineral Engineering采矿工程 Mining Engineering矿物加工工程 Mineral Processing Engineering安全技术及工程 Safety Technology and Engineering石油与天然气工程 Oil and Natural Gas Engineering油气井工程 Oil-Gas Well Engineering油气田开发工程 Oil-Gas Field Development Engineering油气储运工程 Oil-Gas Storage and Transportation Engineering纺织科学与工程 Textile Science and Engineering纺织工程 Textile Engineering纺织材料与纺织品设计 Textile Material and Textiles Design纺织化学与染整工程 Textile Chemistry and Dyeing and Finishing Engineering 服装设计与工程 Clothing Design and Engineering轻工技术与工程 The Light Industry Technology and Engineering制浆造纸工程 Pulp and Paper Engineering制糖工程 Sugar Engineering发酵工程 Fermentation Engineering皮革化学与工程 Leather Chemistry and Engineering交通运输工程 Communication and Transportation Engineering道路与铁道工程 Highway and Railway Engineering交通信息工程及控制 Traffic Information Engineering & Control交通运输规划与管理 Transportation Planning and Management载运工具运用工程 Vehicle Operation Engineering船舶与海洋工程 Naval Architecture and Ocean Engineering船舶与海洋结构物设计制造 Design and Construction of Naval Architecture and Ocean Structure轮机工程 Marine Engine Engineering水声工程 Underwater Acoustics Engineering航空宇航科学与技术 Aeronautical and Astronautical Science and Technology飞行器设计 Flight Vehicle Design航空宇航推进理论与工程 Aerospace Propulsion Theory and Engineering航空宇航器制造工程 Manufacturing Engineering of Aerospace Vehicle人机与环境工程 Man-Machine and Environmental Engineering兵器科学与技术 Armament Science and Technology武器系统与运用工程 Weapon Systems and Utilization Engineering兵器发射理论与技术 Armament Launch Theory and Technology火炮、自动武器与弹药工程 Artillery, Automatic Gun and Ammunition Engineering军事化学与烟火技术 Military Chemistry and Pyrotechnics核科学与技术 Nuclear Science and Technology核能科学与工程 Nuclear Energy Science and Engineering核燃料循环与材料 Nuclear Fuel Cycle and Materials核技术及应用 Nuclear Technology and Applications辐射防护及环境保护 Radiation and Environmental Protection农业工程 Agricultural Engineering农业机械化工程 Agricultural Mechanization Engineering农业水土工程 Agricultural Water-Soil Engineering农业生物环境与能源工程 Agricultural Biological Environmental and Energy Engineering农业电气化与自动化 Agricultural Electrification and Automation林业工程 Forestry Engineering森林工程 Forest Engineering木材科学与技术 Wood Science and Technology林产化学加工工程 Chemical Processing Engineering of Forest Products环境科学与工程 Environmental Science and Engineering环境科学 Environmental Science环境工程 Environmental Engineering生物医学工程 Biomedical Engineering食品科学与工程 Food Science and Engineering食品科学 Food Science粮食、油脂及植物蛋白工程 Cereals, Oils and Vegetable Protein Engineering 农产品加工及贮藏工程 Processing and Storage of Agriculture Products水产品加工及贮藏工程 Processing and Storage of Aquatic Products农学 Agriculture作物学 Crop Science作物栽培学与耕作学 Crop Cultivation and Farming System作物遗传育种学 Crop Genetics and Breeding园艺学 Horticulture果树学 Pomology蔬菜学 Olericulture茶学 Tea Science农业资源利用学 Utilization Science of Agricultural Resources土壤学 Soil Science植物营养学 Plant Nutrition植物保护学 Plant Protection植物病理学 Plant Pathology农业昆虫与害虫防治 Agricultural Entomology and Pest Control农药学 Pesticide Science畜牧学 Animal Science动物遗传育种与繁殖 Animal Genetics, Breeding and ReproductionScience动物营养与饲料科学 Animal Nutrition and Feed Science草业科学 Practaculture Science特种经济动物饲养学（含蚕、蜂等） The Rearing of Special-type Economic Animals (including Silkworm, Honeybees, etc.)兽医学 Veterinary Medicine基础兽医学 Basic Veterinary Medicine预防兽医学 Preventive Veterinary Medicine临床兽医学 Clinical Veterinary Medicine林学 Forestry林木遗传育种学 Forest Tree Genetics and Breeding森林培育学 Silviculture森林保护学 Forest Protection森林经理学 Forest Management野生动植物保护与利用 Wildlife Conservation and Utilization园林植物与观赏园艺 Ornamental Plants and Horticulture水土保持与荒漠化防治 Soil and Water Conservation and Desertification Combating水产学 Fisheries Science水产养殖学 Aquaculture Science捕捞学 Fishing Science渔业资源学 Science of Fisheries Resources医学 Medicine基础医学 Basic Medicine人体解剖与组织胚胎学 Human Anatomy, Histology and Embryology免疫学 Immunology病原生物学 Pathogenic Organisms病理学与病理生理学 Pathology and Pathophysiology法医学 Forensic Medicine放射医学 Radiation Medicine航空航天与航海医学 Aerospace and Nautical medicine临床医学 Clinical Medicine内科学（含心血管病学、血液病学、呼吸系病学、消化系病学、内分泌与代谢病学、肾脏病学、风湿病学、传染病学） Internal medicine (including Cardiology, Hematology, Respiratory, Gastroentero logy, Endocrinology and Metabolism, Nephrology, Rheuma-tology, Infectious Diseases) 儿科学 Pediatrics老年医学 Geriatrics神经病学 Neurology精神病与精神卫生学 Psychiatry and Mental Health皮肤病与性病学 Dermatology and Venereology影像医学与核医学 Imaging and Nuclear Medicine临床检验诊断学 Clinical Laboratory Diagnostics护理学 Nursing外科学（含普通外科学、骨外科学、泌尿外科学、胸心血管外科学、神经外科学、整形外科学、烧伤外科学、野战外科学） Surgery (General Surgery, Orthopedics, Urology, Cardiothoracic Surgery, Neuros urgery, Plastic Surgery, Burn Surgery, Field Surgery)妇产科学 Obstetrics and Gynecology眼科学 Ophthalmic Specialty耳鼻咽喉科学 Otolaryngology肿瘤学 Oncology康复医学与理疗学 Rehabilitation Medicine & Physical Therapy运动医学 Sports Medicine麻醉学 Anesthesiology急诊医学 Emergency Medicine口腔医学 Stomatology口腔基础医学 Basic Science of Stomatology口腔临床医学 Clinical Science of Stomatology公共卫生与预防医学 Public Health and Preventive Medicine流行病与卫生统计学 Epidemiology and Health Statistics劳动卫生与环境卫生学 Occupational and Environmental Health营养与食品卫生学 Nutrition and Food Hygiene儿少卫生与妇幼保健学 Maternal, Child and Adolescent Health卫生毒理学 Hygiene Toxicology军事预防医学 Military Preventive Medicine中医学 Chinese Medicine中医基础理论 Basic Theories of Chinese Medicine中医临床基础 Clinical Foundation of Chinese Medicine中医医史文献 History and Literature of Chinese Medicine方剂学 Formulas of Chinese Medicine中医诊断学 Diagnostics of Chinese Medicine中医内科学 Chinese Internal Medicine中医外科学 Surgery of Chinese Medicine中医骨伤科学 Orthopedics of Chinese Medicine中医妇科学 Gynecology of Chinese Medicine中医儿科学 Pediatrics of Chinese Medicine中医五官科学 Ophthalmology and Otolaryngoloy of Chinese Medicine针灸推拿学 Acupuncture and Moxibustion and Tuina of Chinese medicine民族医学 Ethnomedicine中西医结合医学 Chinese and Western Integrative Medicine中西医结合基础医学 Basic Discipline of Chinese and Western Integrative中西医结合临床医学 Clinical Discipline of Chinese and Western Integrative Medicine 药学 Pharmaceutical Science药物化学 Medicinal Chemistry药剂学 Pharmaceutics生药学 Pharmacognosy药物分析学 Pharmaceutical Analysis微生物与生化药学 Microbial and Biochemical Pharmacy药理学 Pharmacology中药学 Science of Chinese Pharmacology军事学 Military Science军事思想学及军事历史学 Military Thought and Military History军事思想学 Military Thought军事历史学 Military History战略学 Science of Strategy军事战略学 Military Strategy战争动员学 War Mobilization战役学 Science of Operations联合战役学 Joint Operation军种战役学（含第二炮兵战役学） Armed Service Operation (including Operation of Strategic Missile Force) 战术学 Science of Tactics合同战术学 Combined-Arms Tactics兵种战术学 Branch Tactics军队指挥学 Science of Command作战指挥学 Combat Command军事运筹学 Military Operation Research军事通信学 Military Communication军事情报学 Military Intelligence密码学 Cryptography军事教育训练学（含军事体育学） Military Education and Training (including Military Physical Training)军制学 Science of Military System军事组织编制学 Military Organizational System军队管理学 Military Management军队政治工作学 Science of Military Political Work军事后勤学与军事装备学 Science of Military Logistics and Military Equipment军事后勤学 Military Logistics后方专业勤务 Rear Special Service军事装备学 Military Equipment管理学 Management Science管理科学与工程 Management Science and Engineering工商管理学 Science of Business Administration会计学 Accounting企业管理学（含财务管理、市场营销学、人力资源管理学） Corporate Management (including Financial Management, Marketing, and Human Res ources Management)旅游管理学 Tourist Management技术经济及管理学 Technology Economy and Management农林经济管理学 Agricultural and Forestry Economics & Management农业经济管理学 Agricultural Economics & Management林业经济管理学 Forestry Economics & Management公共管理学 Science of Public Management行政管理学 Administration Management社会医学与卫生事业管理学 Social Medicine and Health Management教育经济与管理学 Educational Economy and Management社会保障学 Social Security土地资源管理学 Land Resource Management图书馆、情报与档案学 Science of Library, Information and Archival图书馆学 Library Science情报学 Information Science档案学 Archival Science————————————————————————————————工程硕士 Engineering Master领域名称 Domain Title……………………………………………………………………………………………工程硕士领域英文名称工程硕士领域机械工程 Mechanical Engineering光学工程 Optical Engineering仪器仪表工程 Instrument and Meter Engineering材料工程 Material Engineering冶金工程 Metallurgy Engineering动力工程 Power Engineering电气工程 Electrical Engineering电子与通信工程 Electrical and Communication Engineering控制工程 Control Engineering计算机技术 Computer Technology软件工程 Software Engineering建筑与土木工程 Architectural and Civil Engineering水利工程 Hydraulic Engineering测绘工程 Surveying and Mapping Engineering化学工程 Chemical Engineering地质工程 Geological Engineering矿业工程 Mining Engineering石油与天然气工程 Petroleum and Natural Gas Engineering纺织工程 Textile Engineering轻工技术与工程 Light Industry Technology and Engineering交通运输工程 Communication and Transportation Engineering 船舶与海洋工程 Shipbuilding and Oceanography Engineering兵器工程 Arms Engineering核能与核技术工程 Nuclear Energy and Nuclear Technology农业工程 Agricultural Engineering林业工程 Forestry Engineering环境工程 Environmental Engineering生物医学工程 Biomedical Engineering食品工程 Foodstuff Engineering航空工程 Aeronautical Engineering航天工程 Aerospace Engineering车辆工程 Vehicle Engineering制药工程 Pharmaceutical Engineering工业工程 Industrial Engineering工业设计工程 Industrial Design Engineering生物工程 Biotechnology Engineering项目管理 Project Management物流工程 Logistics Engineering-----------------------------------------------------------------------------------------------------------工商管理硕士 Master of Business Administration。