包郵微積分原理(上)

作者：崔建蓮

出版社：電子工業出版社出版時間：2023-07-01

開本：其他頁數： 320

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微積分原理(上) 版權信息

ISBN：9787121458392
條形碼：9787121458392 ; 978-7-121-45839-2
裝幀：一般膠版紙
冊數：暫無
重量：暫無
所屬分類：
自然科學
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數學

微積分原理(上) 內容簡介

微積分是理工科高等學校非數學類專業*基礎、重要的一門核心課程。許多后繼數學課程及物理和各種工程學課程都是在微積分課程的基礎上展開的，因此學好這門課程對每一位理工科學生來說都非常重要。本書在傳授微積分知識的同時，注重培養學生的數學思維、語言邏輯和創新能力，弘揚數學文化，培養科學精神。本套教材分上、下兩冊。上冊內容包括實數集與初等函數、數列極限、函數極限與連續、導數與微分、微分學基本定理及應用、不定積分、定積分、廣義積分和常微分方程。下冊內容包括多元函數的極限與連續、多元函數微分學及其應用、重積分、曲線積分、曲面積分、數項級數、函數項級數、傅里葉級數和含參積分。

微積分原理(上) 目錄

目錄第1 章實數集與初等函數··················1 1．1 實數集····································1 1．1．1 集合及其運算························1 1．1．2 映射···································3 1．1．3 可數集································3 1．1．4 實數集的性質························5 1．1．5 戴德金原理···························8 1．1．6 確界原理·····························8 習題1．1····································．10 1．2 初等函數······························．11 1．2．1 函數的概念························．11 1．2．2 函數的一些特性··················．12 1．2．3 函數的運算························．13 1．2．4 基本初等函數·····················．14 1．2．5 反函數及其存在條件·············．18 1．2．6 反三角函數························．19 *1．2．7 雙曲函數和反雙曲函數··········．22 *1．2．8 雙曲函數與三角函數之間的聯系·····························．24 習題1．2····································．24 第2 章數列極限····························．27 2．1 數列極限的概念·····················．27 習題2．1····································．30 2．2 數列極限的性質·····················．31 習題2．2····································．35 2．3 幾類特殊的數列·····················．36 2．3．1 無窮大數列與無窮小數列·······．36 2．3．2 無窮大數列與無界數列··········．36 2．3．3 Stolz 定理·························．38 習題2．3 ···································．40 2．4 實數連續性定理····················．41 2．4．1 單調有界定理·····················．41 2．4．2 閉區間套定理·····················．43 2．4．3 Bolzano-Weierstrass 定理·········．44 2．4．4 柯西收斂準則·····················．45 *2．4．5 有限覆蓋定理·····················．47 *2．4．6 聚點定理··························．48 習題2．4 ···································．48 *2．5 上極限與下極限···················．50 習題2．5 ···································．54 第3 章函數極限與連續··················．55 3．1 函數極限的概念····················．55 3．1．1 函數在一點的極限···············．55 3．1．2 函數在無窮遠處的極限··········．58 習題3．1 ···································．58 3．2 函數極限的性質及運算···········．59 3．2．1 函數極限的性質··················．59 3．2．2 函數極限的四則運算·············．60 3．2．3 復合函數的極限··················．62 習題3．2 ···································．62 3．3 函數極限的存在條件··············．63 3．3．1 函數極限與數列極限的關系·····．63 3．3．2 兩個重要極限·····················．65 3．3．3 無窮大量與無窮小量·············．67 3．3．4 等價無窮小量代換求極限········．69 習題3．3····································．71 3．4 函數的連續···························．73 3．4．1 函數連續的概念··················．73 3．4．2 間斷點及其分類··················．74 3．4．3 連續函數的局部性質·············．78 習題3．4····································．78 3．5 閉區間上連續函數的性質········．79 3．5．1 閉區間上連續函數的基本性質··．79 3．5．2 反函數的連續性··················．82 3．5．3 一致連續性························．83 習題3．5····································．86 第4 章導數與微分·························．89 4．1 導數的概念···························．89 4．1．1 導數概念的引出··················．89 4．1．2 函數可導的條件與性質··········．91 習題4．1····································．92 4．2 求導法則······························．94 4．2．1 導數的四則運算法則·············．94 4．2．2 反函數求導法則··················．96 4．2．3 復合函數的導數――鏈式法則···．97 4．2．4 隱函數求導法則··················．98 4．2．5 參數方程求導法則················．99 習題4．2····································102 4．3 函數的微分···························103 4．3．1 可微的概念························103 4．3．2 可微與可導的關系················104 4．3．3 微分在函數近似計算中的應用··105 4．3．4 微分的運算法則··················106 習題4．3····································106 4．4 高階導數與高階微分··············106 4．4．1 高階導數·························．107 4．4．2 高階微分·························．109 4．4．3 復合函數的微分·················．109 習題4．4 ··································．110 第5 章微分學基本定理及應用········．112 5．1 微分中值定理······················．112 5．1．1 極值的概念與費馬定理·········．112 5．1．2 微分中值定理····················．113 習題5．1 ··································．118 5．2 洛必達法則··························．120 5．2．1 0 0 型不定式極限·················．120 5．2．2 ∞ ∞ 型不定式極限················．123 5．2．3 其他類型不定式極限············．125 習題5．2 ··································．126 5．3 泰勒公式及應用···················．127 5．3．1 泰勒公式·························．128 5．3．2 基本初等函數的展開式·········．130 5．3．3 泰勒公式的應用·················．134 習題5．3 ··································．138 5．4 單調性與極值······················．140 5．4．1 函數的單調性····················．140 5．4．2 函數取極值的條件··············．142 習題5．4 ··································．145 5．5 函數的凸性與函數作圖··········．147 5．5．1 函數的凸性······················．147 5．5．2 曲線的漸近性····················．152 5．5．3 函數作圖·························．153 習題5．5 ··································．155 *5．6 方程求根的牛頓迭代公式·····．155 第6 章不定積分···························．160 6．1 原函數與不定積分················．160 6．1．1 原函數與不定積分的概念······．160 6．1．2 不定積分的線性運算·············162 6．1．3 常用的不定積分公式·············162 習題6．1····································163 6．2 不定積分計算························164 6．2．1 分部積分法························165 6．2．2 積分換元法························166 習題6．2····································174 6．3 有理函數的不定積分··············175 習題6．3····································178 6．4 可化為有理函數的不定積分·····179 6．4．1 三角有理函數的不定積分········179 6．4．2 某些無理函數的不定積分········182 習題6．4····································185 第7 章定積分·······························187 7．1 定積分的概念及可積條件········187 7．1．1 引例································187 7．1．2 定積分的概念·····················188 7．1．3 定積分的幾何意義················189 7．1．4 可積的必要條件··················190 7．1．5 可積準則··························191 習題7．1····································195 7．2 可積函數類及定積分的性質·····195 7．2．1 閉區間上的可積函數類··········195 *7．2．2 再論可積的充要條件·············196 7．2．3 定積分的性質·····················200 習題7．2····································203 7．3 定積分的計算························204 7．3．1 變上限積分························205 7．3．2 微積分基本定理··················208 7．3．3 積分換元法和分部積分法········210 習題7．3····································213 7．4 積分中值定理························216 習題7．4····································221 7．5 定積分的應用······················．221 *7．5．1 分析學應用······················．221 7．5．2 定積分的幾何應用··············．224 7．5．3 定積分的物理應用··············．232 習題7．5 ··································．236 第8 章廣義積分···························．239 8．1 無窮積分·····························．239 8．1．1 無窮積分的概念·················．239 8．1．2 無窮積分求值····················．240 8．1．3 無窮積分斂散性判別法·········．241 習題8．1 ··································．246 8．2 瑕積分································．248 8．2．1 瑕積分收斂的概念··············．248 8．2．2 無窮積分與瑕積分的關系······．249 8．2．3 瑕積分斂散性判別法············．250 習題8．2 ··································．254 第9 章常微分方程························．255 9．1 常微分方程的概念················．255 9．1．1 引例······························．255 9．1．2 常微分方程的概念··············．257 9．1．3 常微分方程的解·················．257 習題9．1 ··································．258 9．2 一階常微分方程的初等解法····．259 9．2．1 可分離變量的微分方程·········．259 9．2．2 齊次方程·························．261 9．2．3 可化為齊次方程類型的方程····．262 9．2．4 常數變易法······················．263 9．2．5 伯努利方程······················．265 習題9．2 ··································．267 9．3 一階微分方程初值問題的解····．268 9．3．1 初值問題解的存在專享性定理······························．268 *9．3．2 奇解······························．268 9．4 高階線性常微分方程··············269 9．4．1 可降階的高階微分方程··········269 9．4．2 高階線性常微分方程解的結構·····························273 9．4．3 高階非齊次方程的常數變易法·····························278 習題9．4····································280 9．5 常系數高階線性常微分方程·····281 9．5．1 常系數齊次線性常微分方程的特征值法··························281 9．5．2 常系數非齊次線性常微分方程的待定系數法·····················285 *9．5．3 常系數線性常微分方程的應用――質點的振動···············289 習題9．5····································291 9．6 歐拉方程······························292 習題9．6····································294 9．7 一階線性常微分方程組··········．294 9．7．1 解的疊加原理及解的存在專享性····························．294 9．7．2 一階線性常微分方程組解的結構······························．295 9．7．3 一階非齊次線性常微分方程組的常數變易法······················．298 9．7．4 從方程組的觀點看高階微分方程······························．299 9．8 常系數線性常微分方程組·······．301 9．8．1 矩陣A 可對角化的情形·········．301 9．8．2 矩陣A 不可對角化的情形······．302 9．8．3 矩陣A 有復特征根的情形······．305 *9．8．4 方程組初值問題解的一般形式·························．307 *9．8．5 非齊次方程的通解··············．309 習題9．8 ··································．310

展開全部

微積分原理(上) 作者簡介

崔建蓮，清華大學數學系副教授，主要研究方向為算子代數、算子理論及在量子信息中的應用，發表學術論文60多篇，SCI收錄50多篇。

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