2021五邑大學數(shù)學分析研究生考試大綱及參考書目 正文

五邑大學2021年碩士學位研究生招生
《數(shù)學分析》考試大綱
                                    
一、課程性質(zhì)、目的和任務
數(shù)學分析是本科數(shù)學學科各專業(yè)的基礎(chǔ)課程，通過本課程的學習，培養(yǎng)學生具備比較扎實的函數(shù)理論、嚴謹邏輯思維能力、鍛煉學生的空間想象力、掌握應用函數(shù)理論解決相關(guān)實際問題的能力，為最終使學生具有較好的數(shù)學素質(zhì)打下堅實的基礎(chǔ)。
 
二、基本要求
掌握實數(shù)的完備性理論、極限理論、函數(shù)的連續(xù)性理論、微積分理論、級數(shù)理論。能應用所學的函數(shù)理論分析、解決實際問題。
 
三、考試范圍
 
   （一） 實數(shù)與函數(shù)                                      
1. 實數(shù)的分類與主要性質(zhì), 絕對值與不等式                                (A)
        不足近似和過剩近似及其應用                                          (B)
     2. 區(qū)間、鄰域、確界的概念                                              (A)
        確界原理                                                            (A)
     3. 函數(shù)的相關(guān)概念、表示法                                              (A)
        函數(shù)的四則運算、復合、反函數(shù)                                        (B)
        函數(shù)的圖象                                                          (C)
        初等函數(shù)                                                            (C)
     4. 四類具有特殊性質(zhì)的函數(shù)                                              (B)
   （二） 數(shù)列極限                                    
     1. 極限思想                                                            (B)
        數(shù)列極限概念                                                        (A)
     2. 收斂數(shù)列的性質(zhì)                                                      (A)
        收斂數(shù)列的四則運算法則                                              (B)
        一些常見的極限                                                      (A)
        子列及其性質(zhì)                                                        (A)
     3. 單調(diào)有界定理、柯西準則及其應用                                      (A)
  （三） 函數(shù)極限                             
     1.各種類型的函數(shù)極限的概念                                             (A)
     2.函數(shù)極限的性質(zhì)及其應用                                               (A)
     3.歸結(jié)原理、柯西準則及其應用                                           (A)
     4.兩個重要極限                                                         (A)
     5.無窮小與無窮大的概念、相互關(guān)系                                       (B)
       無窮小的比較                                                         (C)
       等價無窮小及其應用                                                   (A)
       函數(shù)的漸近線及其求法                                                 (A)
   （四） 函數(shù)的連續(xù)性                                 
     1.連續(xù)的概念                                                           (A)
       間斷點及其分類                                                       (B)
     2.連續(xù)函數(shù)的局部性質(zhì)和整體性質(zhì)                                         (A)
       反函數(shù)與復合函數(shù)的連續(xù)性                                             (A)
     3.初等函數(shù)的連續(xù)性                                                     (B)
   （五）導數(shù)和微分                               
     1.導數(shù)的概念、幾何意義                                                 (A)
     2.求導法則                                                             (A)
     3.參變量函數(shù)的求導法則                                                 (A)
     4.微分概念、微分的運算法則                                             (A)
       微分在近似計算的應用                                                 (B)
     5.高階導數(shù)與高階微分的概念、求法                                       (A)
       Leibniz公式                                                          (B)
       高階微分                                                             (B)
   （六）  微分中值定理及其應用                          
    1.羅爾定理、拉格朗日定理與函數(shù)的單調(diào)性                                  (A)
    2.柯西中值定理                                                          (A)
    3.泰勒公式及其應用                                                      (A)
      常用的幾個函數(shù)的馬克勞林展式                                          (A)
    4.洛比達法則及其應用                                                    (A)
    5.函數(shù)極值的存在性及求法、最值及其應用                                  (A)
    6.函數(shù)的凸性和拐點                                                      (B)
    7.函數(shù)的圖形討論                                                        (B)
  （七） 實數(shù)的完備性                                
    1.區(qū)間套定理、柯西準則、聚點定理、有限覆蓋定理                          (A)
      完備性定理的等價性                                                    (B)
    2.區(qū)間上連續(xù)函數(shù)的性質(zhì)的證明                                            (B)
                                        
   （八） 不定積分                                   
    1.原函數(shù)與不定積公的概念、性質(zhì)                                          (A)
      基本積分公式                                                          (A)
    2.分部積公法與換元積分法                                                (A)
    3.有理函數(shù)的不定積分                                                    (A)
      簡單無理函數(shù)與三角函數(shù)的不定積分                                      (B)
   （九） 定積分                                  
    1. 定積分的定義                                                         (B)
    2. 牛頓－萊布尼茨公式                                                   (A)
    3. 小和與大和的概念                                                     (B)
       定積分存在的條件                                                     (B)
       可積函數(shù)的分類                                                       (A)
    4. 定積分的性質(zhì)與積分中值定理                                           (A)
    5. 變限積分及其性質(zhì)                                                     (A)
       第二積分中值定理                                                     (C)
       定積分的換元法與分部積分法及其應用                                   (A)
       泰勒公式的積分型余項                                                 (B)
    6. 上和與下和的性質(zhì)、積分存在的充分必要條件                             (B)
  （十） 定積分的應用                                   
    1.求平面圖形的面積                                              (A)
    2.求截面面積已知的立體圖形的體積、旋轉(zhuǎn)體的體積                  (A)
    3.平面曲線的弧長                                                (A)
       曲率                                                         （C)
    4. 微元法、求旋轉(zhuǎn)曲面的面積                                      (A)
    5. 利用定積分求液體的靜壓力、引力、變力做功                      (A)
  （十）反常積分                                 
    1. 反常積分及其收斂性的概念                                       (B)
    2. 無窮積分的性質(zhì)及其收斂判別法                                   (A)
    3. 瑕積分的性質(zhì)及其斂散性判別法                                   (A)
  （十二） 數(shù)項級數(shù)                                   
    1.數(shù)項級數(shù), 部分和, 收斂與發(fā)散, 余項等概念                       (B)
      柯西收斂準則, 收斂級數(shù)的性質(zhì)                                   (A)
    2. 正項級數(shù)及其收斂判別法                                         (A)
    3. 一般項級數(shù)的收斂判別法                                         (A)
  （十三） 函數(shù)列與函數(shù)項級數(shù)                              
    1.函數(shù)列與函數(shù)項級數(shù)的概念                                       (B)
      收斂與一致收斂的概念, 函數(shù)級數(shù)的收斂域                         (A)
      函數(shù)列與函數(shù)項級數(shù)一致收斂的判別法                             (A)
    2.一致收斂函數(shù)列和函數(shù)項級數(shù)的性質(zhì)                               (A)
  （十四）冪級數(shù)                                        
    1.冪級數(shù)的收斂區(qū)間, 收斂半徑              (B)
      冪級數(shù)的性質(zhì) (A)
    2.冪級數(shù)的泰勒展開和麥克勞林展開式                               (A)
      基本初等函數(shù)的冪級數(shù)展開                                       (A)
    3.復變量的指數(shù)函數(shù), 歐拉公式                                     (C)
  （十五） 傅立葉級數(shù)                                 
    1. 三角級數(shù),傅立葉級數(shù)的概念                                      (C)
       以2p為周期的函數(shù)的傅立葉級數(shù)的展開式                          (A)
    2. 以2l為周期的函數(shù)的傅立葉級數(shù)展開                              (A)
  （十六） 多元函數(shù)的極限于連續(xù)                           
    1. 多元函數(shù)
       平面點集的相關(guān)概念                                                   (B)
       柯西準則, 區(qū)域套定理,聚點定理                                        (B)
       多元函數(shù)的概念                                                       (B)
    2. 二元函數(shù)的極限                                                       (A)
    3. 二元函數(shù)的連續(xù)性及其性質(zhì)                                             (A)
  （十七） 多元函數(shù)的微分學                            
    1. 多元函數(shù)的偏導數(shù)和全微分的概念, 聯(lián)系; 可微的條件; 偏導數(shù)的應用 (A)
       全微分的幾何意義                                               (B)
    2. 多元復合函數(shù)的偏導數(shù)與全微分                                   (A)
    3.方向?qū)?shù)與梯度的概念, 計算方法                                 (B)
    4.高階偏導數(shù), 中值定理及泰勒公式                                 (A)
          二元函數(shù)的極值                                                 (A)
 （十八） 隱函數(shù)                                           
   1. 隱函數(shù)的概念                                                          (B)
      隱函數(shù)偏導數(shù)和高階偏導數(shù)                                       (A)
   2.隱函數(shù)組的概念,存在性                                          (A)
   3.隱函數(shù)的幾何應用                                               (A)
   4.用拉格朗日乘數(shù)求條件極值                                       (A)
  （十九） 含參量積分                                
   1.含參量積分的概念                                               (B)
     含參量積分的連續(xù)性和可導性                                     (A)
   2.含參量反常積分的性質(zhì), 收斂判別法                               (A)
   3.Г函數(shù)和В函數(shù)的定義，性質(zhì)及其應用                             (B)
  （二十）曲線積分                                         
   1.第一型曲線積分的概念與求法                                     (A)
   2.第二型曲線積分的概念與計算                                     (A)
  （二十一）重積分                                      
   1.平面圖形的內(nèi),外面積; 二重積分的定義、可積條件、性質(zhì)            (A)
   2.化二重積分為累次積分                                           (A)
     用二重積分計算曲面的面積                                       (B)
   3.格林公式,曲線積分與路徑的無關(guān)性                                (A)
   4.二重積分的變量變換, 用極坐標計算二重積分                       (A)
   5.三重積分的定義                                                 (B)
     三重積分的計算                                                 (A)
   6.三重積分的簡單應用                                             (B)
 （二十二） 曲面積分                                      
   1.第一型曲線積分的概念、計算                                     (A)
   2.曲面的側(cè)                                                      （B)
      第二型曲面積分的定義、性質(zhì)、計算                               (A)
      兩類曲面積分之間的關(guān)系                                         (B)
   3.高斯公式與斯托克斯公式及其應用                                 (A)
 
四、主要教材及參考書
   1. 教材:
     華東師范大學數(shù)學系. 數(shù)學分析（第三版）[M]. 北京: 高等教育出版社, 2001.
   2. 主要參考書
    [1] 陳傳璋. 數(shù)學分析[M]. 北京: 人民教育出版社, 1992.
    [2]  Б.П.吉米多維奇. 數(shù)學分析習題集[M]. 北京:  人民教育出版社, 1997.
[3] 裴禮文. 數(shù)學分析中的典型問題與方法[M].  北京: 高等教育出版社, 1993.
 
五、說明
    對知識層次的要求含義是，A：掌握；B：理解；C：了解。

五邑大學

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