微積分是理工科高等學校非數(shù)學類專業(yè)最基礎、重要的一門核心課程。許多后繼數(shù)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(yǎng)學生的數(shù)學思維、語言邏輯和創(chuàng)新能力，弘揚數(shù)學文化，培養(yǎng)科學精神。本套教材分上、下兩冊。上冊內(nèi)容包括實數(shù)集與初等函數(shù)、數(shù)列極限、函數(shù)極限與連續(xù)、導數(shù)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(nèi)容包括多元函數(shù)的極限與連續(xù)、多元函數(shù)微分學及其應用、重積分、曲線積分、曲面積分、數(shù)項級數(shù)、函數(shù)項級數(shù)、傅里葉級數(shù)和含參積分。
崔建蓮，清華大學數(shù)學系副教授，主要研究方向為算子代數(shù)、算子理論及在量子信息中的應用，發(fā)表學術論文60多篇，SCI收錄50多篇。
目錄
第1 章 實數(shù)集與初等函數(shù)··················1
1.1 實數(shù)集····································1
1.1.1 集合及其運算························1
1.1.2 映射···································3
1.1.3 可數(shù)集································3
1.1.4 實數(shù)集的性質(zhì)························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(shù)······························.11
1.2.1 函數(shù)的概念························.11
1.2.2 函數(shù)的一些特性··················.12
1.2.3 函數(shù)的運算························.13
1.2.4 基本初等函數(shù)·····················.14
1.2.5 反函數(shù)及其存在條件·············.18
1.2.6 反三角函數(shù)························.19
*1.2.7 雙曲函數(shù)和反雙曲函數(shù)··········.22
*1.2.8 雙曲函數(shù)與三角函數(shù)之間
的聯(lián)系·····························.24
習題1.2····································.24
第2 章 數(shù)列極限····························.27
2.1 數(shù)列極限的概念·····················.27
習題2.1····································.30
2.2 數(shù)列極限的性質(zhì)·····················.31
習題2.2····································.35
2.3 幾類特殊的數(shù)列·····················.36
2.3.1 無窮大數(shù)列與無窮小數(shù)列·······.36
2.3.2 無窮大數(shù)列與無界數(shù)列··········.36
2.3.3 Stolz 定理·························.38
習題2.3 ···································.40
2.4 實數(shù)連續(xù)性定理····················.41
2.4.1 單調(diào)有界定理·····················.41
2.4.2 閉區(qū)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(shù)極限與連續(xù)··················.55
3.1 函數(shù)極限的概念····················.55
3.1.1 函數(shù)在一點的極限···············.55
3.1.2 函數(shù)在無窮遠處的極限··········.58
習題3.1 ···································.58
3.2 函數(shù)極限的性質(zhì)及運算···········.59
3.2.1 函數(shù)極限的性質(zhì)··················.59
3.2.2 函數(shù)極限的四則運算·············.60
3.2.3 復合函數(shù)的極限··················.62
習題3.2 ···································.62
3.3 函數(shù)極限的存在條件··············.63
3.3.1 函數(shù)極限與數(shù)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(shù)的連續(xù)···························.73
3.4.1 函數(shù)連續(xù)的概念··················.73
3.4.2 間斷點及其分類··················.74
3.4.3 連續(xù)函數(shù)的局部性質(zhì)·············.78
習題3.4····································.78
3.5 閉區(qū)間上連續(xù)函數(shù)的性質(zhì)········.79
3.5.1 閉區(qū)間上連續(xù)函數(shù)的基本性質(zhì)··.79
3.5.2 反函數(shù)的連續(xù)性··················.82
3.5.3 一致連續(xù)性························.83
習題3.5····································.86
第4 章 導數(shù)與微分·························.89
4.1 導數(shù)的概念···························.89
4.1.1 導數(shù)概念的引出··················.89
4.1.2 函數(shù)可導的條件與性質(zhì)··········.91
習題4.1····································.92
4.2 求導法則······························.94
4.2.1 導數(shù)的四則運算法則·············.94
4.2.2 反函數(shù)求導法則··················.96
4.2.3 復合函數(shù)的導數(shù)——鏈式法則···.97
4.2.4 隱函數(shù)求導法則··················.98
4.2.5 參數(shù)方程求導法則················.99
習題4.2····································102
4.3 函數(shù)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(shù)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(shù)與高階微分··············106
4.4.1 高階導數(shù)·························.107
4.4.2 高階微分·························.109
4.4.3 復合函數(shù)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(shù)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(diào)性與極值······················.140
5.4.1 函數(shù)的單調(diào)性····················.140
5.4.2 函數(shù)取極值的條件··············.142
習題5.4 ··································.145
5.5 函數(shù)的凸性與函數(shù)作圖··········.147
5.5.1 函數(shù)的凸性······················.147
5.5.2 曲線的漸近性····················.152
5.5.3 函數(shù)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(shù)與不定積分················.160
6.1.1 原函數(shù)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(shù)的不定積分··············175
習題6.3····································178
6.4 可化為有理函數(shù)的不定積分·····179
6.4.1 三角有理函數(shù)的不定積分········179
6.4.2 某些無理函數(shù)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(shù)類及定積分的性質(zhì)·····195
7.2.1 閉區(qū)間上的可積函數(shù)類··········195
*7.2.2 再論可積的充要條件·············196
7.2.3 定積分的性質(zhì)·····················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(shù)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 初值問題解的存在唯一性
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(jié)構(gòu)·····························273
9.4.3 高階非齊次方程的常數(shù)
變易法·····························278
習題9.4····································280
9.5 常系數(shù)高階線性常微分方程·····281
9.5.1 常系數(shù)齊次線性常微分方程的
特征值法··························281
9.5.2 常系數(shù)非齊次線性常微分方程
的待定系數(shù)法·····················285
*9.5.3 常系數(shù)線性常微分方程的
應用——質(zhì)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
唯一性····························.294
9.7.2 一階線性常微分方程組解的
結(jié)構(gòu)······························.295
9.7.3 一階非齊次線性常微分方程組的
常數(shù)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(shù)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
書單推薦
