Advanced Mathematics (Volume 1) cover

Advanced Mathematics (Volume 1)

Rating 3.3
Word count
90K words
Chapters
211
ISBN
9787115458742

Tags

Summary

This book is written for non-mathematics majors at regular higher education institutions, and can also be used by anyone seeking to improve their mathematical literacy and ability. The book consists of 7 chapters: Chapter 1 Functions and Limits; Chapter 2 Derivatives and Differentials; Chapter 3 The Mean Value Theorem and Applications of Derivatives; Chapter 4 Indefinite Integrals; Chapter 5 Definite Integrals; Chapter 6 Applications of Definite Integrals; Chapter 7 Differential Equations.

Editor's review

Edited by Yan Chao, Advanced Mathematics (Volume I) is a classic textbook published by Posts & Telecom Press for engineering and economics/management students. Guided by the Ministry of Education's Basic Requirements for Advanced Mathematics Courses, it systematically covers core topics: functions and limits, derivatives and differentials, the mean value theorem and applications of derivatives, indefinite integrals, definite integrals and their applications. Its strengths lie in its clear structure and rigorous logic, balancing theoretical completeness with practical application. The examples and exercises are well designed and appropriately graded, helping readers build knowledge step by step. As a foundational textbook, it does not pursue showy complex proofs but focuses on building a solid mathematical thinking framework, making it especially suitable as an introduction for first-year university students or for readers who need a systematic review of the basics. Its plain explanatory style and precise language make it a reliable choice for building a solid foundation in advanced mathematics.

Who it's for

A good fit for

  • First-year students majoring in engineering, science, economics, or management at colleges and universities
  • Self-learners who need a systematic review of the foundations of advanced mathematics
  • Exam candidates preparing for graduate entrance examinations who need to consolidate single-variable calculus
  • Professionals in engineering or technical fields who need to revisit mathematical tools

Not a good fit for

  • Upper-level mathematics majors seeking cutting-edge theory or deeper mathematical analysis
  • General-interest readers who want to learn mathematics through light storytelling or a completely beginner-friendly approach

Reading tips

  1. It is recommended to study the chapters in order, focusing on understanding the definition and geometric intuition of each concept.
  2. Complete the exercises after each section carefully; this is key to testing understanding and consolidating computational skills.
  3. Combine the ideas behind proofs of theorems (such as the mean value theorem) with their applications to appreciate their importance.

About the publisher

Posts & Telecom Press China · Founded in 1953

Posts & Telecom Press is known for its strengths in science and technology, computing, and information and communications titles, and has extended its publishing into photography, art, software learning, and inspirational growth, giving it a fairly diverse range. Its catalog includes computing and communications works such as Refactoring and Patterns (Revised Edition) and Huawei Switch Learning Guide (Second Edition), alongside bilingual collector's editions such as The Little Prince (Chinese-English-French illustrated collector's edition), The Great Gatsby, and Three Days to See. Beijing Ten Years: 1980-1990 and Ghost Blade: WLOP Personal Illustration Collection reflect its presence in photography and art. Its authors include Antoine de Saint-Exupéry, Helen Keller, Joshua Kerievsky, Wang Da, Zhang Zhaozeng, and WLOP.

Contents

  1. 版权信息
  2. 内容提要
  3. 前言
  4. 第1章 函数、极限与连续
  5. 1.1 集合与函数
  6. 1.1.1 集合、区间和邻域
  7. 1.1.2 函数的概念
  8. 1.1.3 函数的性质
  9. 1.1.4 反函数与复合函数
  10. 1.1.5 初等函数
  11. 习题1.1
  12. (A)
  13. (B)
  14. 1.2 数列的极限
  15. 1.2.1 数列的极限
  16. 1.2.2 收敛数列的性质
  17. 习题1.2
  18. 1.3 函数的极限
  19. 1.3.1 函数的极限
  20. 1.3.2 函数极限的性质
  21. 1.3.3 函数极限的四则运算法则
  22. 1.3.4 复合函数的极限运算法则
  23. 习题1.3
  24. (A)
  25. (B)
  26. 1.4 两个重要极限
  27. 1.4.1 极限存在的准则
  28. 1.4.2 两个重要极限
  29. 习题1.4
  30. (A)
  31. (B)
  32. 1.5 无穷大和无穷小
  33. 1.5.1 无穷小量与无穷大量
  34. 1.5.2 无穷小量的性质
  35. 1.5.3 无穷小量的比较
  36. 习题1.5
  37. (A)
  38. (B)
  39. 1.6 函数的连续与间断
  40. 1.6.1 函数的连续性
  41. 1.6.2 函数的间断点
  42. 1.6.3 连续函数的运算
  43. 1.6.4 闭区间上连续函数的性质
  44. 习题1.6
  45. (A)
  46. (B)
  47. 本章小结
  48. 复习题1
  49. 第2章 导数与微分
  50. 2.1 导数的概念
  51. 2.1.1 引例
  52. 2.1.2 导数的定义
  53. 2.1.3 导数的几何意义
  54. 2.1.4 函数的可导性与连续性的关系
  55. 习题2.1
  56. (A)
  57. (B)
  58. 2.2 函数的求导法则
  59. 2.2.1 导数的四则运算法则
  60. 2.2.2 反函数的求导法则
  61. 2.2.3 复合函数的求导法则
  62. 2.2.4 基本求导公式与求导法则
  63. 2.2.5 高阶导数
  64. 习题2.2
  65. (A)
  66. (B)
  67. 2.3 隐函数及由参数方程所确定的函数的导数
  68. 2.3.1 隐函数的求导法则
  69. 2.3.2 由参数方程所确定的函数的导数
  70. 习题2.3
  71. (A)
  72. (B)
  73. 2.4 函数的微分及其应用
  74. 2.4.1 微分的定义
  75. 2.4.2 微分的几何意义
  76. 2.4.3 基本微分公式与运算法则
  77. 2.4.4 微分在近似计算中的应用
  78. 习题2.4
  79. (A)
  80. (B)
  81. 本章小结
  82. 复习题2
  83. 第3章 微分中值定理与导数的应用
  84. 3.1 微分中值定理
  85. 3.1.1 罗尔定理
  86. 3.1.2 拉格朗日中值定理
  87. 3.1.3 ∗柯西中值定理
  88. 习题3.1
  89. (A)
  90. (B)
  91. 3.2 洛必达法则
  92. 3-21与型未定式
  93. 3.2.2 其他类型未定式
  94. 习题3.2
  95. (A)
  96. (B)
  97. 3.3 泰勒公式
  98. 3.3.1 泰勒中值定理的引入
  99. 3.3.2 泰勒中值定理
  100. 习题3.3
  101. (A)
  102. (B)
  103. 3.4 函数的单调性、极值与曲线的凹凸性
  104. 3.4.1 函数的单调性
  105. 3.4.2 函数的极值及其求法
  106. 3.4.3 曲线的凹凸性
  107. 习题3.4
  108. (A)
  109. (B)
  110. 3.5 最优化与导数的应用
  111. 3.5.1 函数的最大值与最小值
  112. 3.5.2 导数在物理上的应用
  113. 3.5.3 导数在经济上的应用
  114. 习题3.5
  115. (A)
  116. (B)
  117. 3.6 函数图形的描绘
  118. 3.6.1 渐近线
  119. 3.6.2 函数图形的描绘
  120. 习题3.6
  121. (A)
  122. (B)
  123. 3.7曲率
  124. 3.7.1 弧微分
  125. 3.7.2 曲率及其计算公式
  126. 3.7.3 曲率圆与曲率半径
  127. 习题3.7
  128. (A)
  129. (B)
  130. 本章小结
  131. 复习题3
  132. 第4章 不定积分
  133. 4.1 不定积分的概念与性质
  134. 4.1.1 原函数的概念
  135. 4.1.2 不定积分的概念
  136. 4.1.3 积分与导数(微分)的互逆运算性质
  137. 4.1.4 基本积分表
  138. 4.1.5 不定积分的性质
  139. 习题4.1
  140. (A)
  141. (B)
  142. 4.2 换元积分法
  143. 4.2.1 第一类换元法
  144. 4.2.2 第二类换元法
  145. 习题4.2
  146. (A)
  147. (B)
  148. 4.3 分部积分法
  149. 习题4.3
  150. (A)
  151. (B)
  152. 4.4 有理函数的积分
  153. 4.4.1 有理函数的积分
  154. 4.4.2 可化为有理函数的积分举例
  155. 习题4.4
  156. (A)
  157. (B)
  158. 本章小结
  159. 复习题4
  160. 第5章 定积分
  161. 5.1 定积分的概念与性质
  162. 5.1.1 引例
  163. 5.1.2 定积分的定义
  164. 5.1.3 定积分的几何意义
  165. 5.1.4 定积分的性质
  166. 习题5.1
  167. (A)
  168. (B)
  169. 5.2 微积分基本公式
  170. 5.2.1 引例:变速直线运动中位置函数与速度函数的关系
  171. 5.2.2 积分上限函数及其导数
  172. 5.2.3 牛顿-莱布尼茨公式
  173. 习题5.2
  174. (A)
  175. (B)
  176. 5.3 定积分的换元积分法与分部积分法
  177. 5.3.1 定积分的换元积分法
  178. 5.3.2 定积分的分部积分法
  179. 习题5.3
  180. (A)
  181. (B)
  182. 5.4 反常积分
  183. 5.4.1 无穷限的反常积分
  184. 5.4.2 无界函数的反常积分
  185. 习题5.4
  186. (A)
  187. (B)
  188. 本章小结
  189. 复习题5
  190. 第6章 定积分的应用
  191. 6.1 定积分的元素法
  192. 6.2 定积分的几何应用
  193. 6.2.1 平面图形的面积
  194. 6.2.2 体积
  195. 6.2.3 平面曲线的弧长
  196. 习题6.2
  197. (A)
  198. (B)
  199. 6.3 定积分的物理应用
  200. 6.3.1 变力做功
  201. 6.3.2 液体压力
  202. 6.3.3 引力
  203. 习题6.3
  204. 6.4 定积分在经济学中的应用
  205. 6.4.1 总量函数
  206. 6.4.2 资金的现值与将来值
  207. 习题6.4
  208. 本章小结
  209. 复习题6
  210. 附录 积分公式
  211. 参考文献

FAQ

How does this book compare with the Tongji University edition of Advanced Mathematics?
This book is equally rigorous, but may place more emphasis on connecting with subsequent engineering courses in the selection and arrangement of examples and exercises, and some explanations may be more concise and direct. Both are excellent textbooks; when choosing, refer to your school's course requirements or your personal study habits.
Is this book difficult to use for self-study?
As a standard textbook, it is self-contained and suitable for self-study. However, advanced mathematics is inherently abstract; when studying independently, it is advisable to use teaching videos or find channels for asking questions, and be sure to practice diligently.
Does this book include the content of Volume II?
No. This is Volume I, covering primarily single-variable calculus. Multivariable calculus, differential equations, infinite series, and other topics are usually taught in Volume II.

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About the author

颜超 Author

Yan Chao: public information is limited, and at present only one work is known, Higher Mathematics (Volume 1), commonly classified under examinations. Judging from this attribution and the work itself, Yan Chao appears to be one of the authors involved in compiling textbooks or exam-preparation materials for higher mathematics, aimed at the study and examination needs of basic university mathematics courses. Higher Mathematics (Volume 1) usually covers functions, limits, derivatives and integrals; it is an important introductory course for students in science, engineering and economics or management, and a required part of the mathematics entrance examination for postgraduate study. Books of this kind therefore tend to stress a complete knowledge system, well-targeted examples and exercises, and clear exposition. Given the limited material, Yan Chao's personal background, institutional affiliation and writing career cannot be confirmed, and no other works or academic activities are recorded. Readers choosing this work may judge it by edition, intended audience and examination requirements.

单娟 Author

Shan Juan is the author of Advanced Mathematics (Volume 1). Publicly available information about the author is limited, and details of her background are not documented. In the library, the book is categorized under examinations, suggesting that it is designed for exam review and self-study. Advanced mathematics is a foundational course for students in science, engineering, and some economics and management programs, typically covering functions, limits, continuity, derivatives and differentials, integrals and their applications. Judging from its positioning, the book likely emphasizes a systematic review of key concepts and training in problem-solving methods, helping readers master fundamental ideas and common techniques within a limited time. As no reliable records of the author's life, education, or professional positions are available, no further inferences are made here. For readers, Shan Juan's work can serve as supplementary material for coursework and pre-exam review, with its value lying mainly in the focused and practical organization of its content.

张超 Author

Zhang Chao is a Chinese book author whose works span a notably wide range of subjects. Public information about him is limited, and details of his life remain unclear. The collection includes 24 of his titles, covering categories such as photography, software learning, popular science, personal growth, programming and design, and law. Representative works include popular science and photography books such as A Cloud a Day and Starlight Collector: A Brief History of Small Astronomical Telescopes, technical hands-on titles such as Circuit Simulation with Multisim and Elasticsearch Source Code Analysis and Optimization in Practice, as well as growth-oriented themes in Networking Is Designed and legal practice content in Equity Structure Design. His writing is generally practical and accessible, focused on turning specialized knowledge into actionable introductory guidance.

周海青 Author

Zhou Haiqing: currently, public information is limited, and it is not possible to confirm nationality, dates of birth and death, or a complete career history. According to the library holdings, only one work appears under this name, Higher Mathematics (Volume 1), commonly classified under exam preparation. It can therefore be inferred that the book is intended for exam preparation or course study, covering the fundamentals of higher mathematics to support students' review and test-taking. In writing, the author may focus more on summarizing key points and matching them to question types than on extended theoretical discussion. As more verifiable information is unavailable, speculation about academic background, teaching experience, or the full range of writing would be inappropriate. Readers wishing to learn about this author can currently begin with the actual content of this textbook, considering its organization and explanatory style, and use that to judge whether it suits their stage of study and examination goals.

杜秀清 Author · Chinese

Du Xiuqing is a Chinese author about whom limited public information is available; at present, only one work by this author is known, Higher Mathematics (Volume 1). The book belongs to the exam-preparation category and is intended for the study of and examination in advanced mathematics. Its content centers on the foundational knowledge covered in the first volume of a higher mathematics course, serving students in science, engineering, and related fields at colleges and universities as they review for coursework and examinations. Judging from its common classification, the writing is closely aligned with examination requirements, emphasizing the organization of key concepts and practice so that readers can grasp the essentials within a limited time. Because details of the author's life, academic background, and research interests are not available in public sources, a complete picture of the author's work cannot be drawn; the existing book remains the main basis for understanding this author's editorial approach. For readers who need to review higher mathematics systematically, a supplementary teaching aid of this kind has clear practical value.

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