微積分是理工科高等學(xué)校非數(shù)學(xué)類專業(yè)最基礎(chǔ)、重要的一門核心課程。許多后繼數(shù)學(xué)課程及物理和各種工程學(xué)課程都是在微積分課程的基礎(chǔ)上展開的，因此學(xué)好這門課程對(duì)每一位理工科學(xué)生來(lái)說(shuō)都非常重要。本書在傳授微積分知識(shí)的同時(shí)，注重培養(yǎng)學(xué)生的數(shù)學(xué)思維、語(yǔ)言邏輯和創(chuàng)新能力，弘揚(yáng)數(shù)學(xué)文化，培養(yǎng)科學(xué)精神。本套教材分上、下兩冊(cè)。上冊(cè)內(nèi)容包括實(shí)數(shù)集與初等函數(shù)、數(shù)列極限、函數(shù)極限與連續(xù)、導(dǎo)數(shù)與微分、微分學(xué)基本定理及應(yīng)用、不定積分、定積分、廣義積分和常微分方程。下冊(cè)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(xué)及其應(yīng)用、重積分、曲線積分、曲面積分、數(shù)項(xiàng)級(jí)數(shù)、函數(shù)項(xiàng)級(jí)數(shù)、傅里葉級(jí)數(shù)和含參積分。
崔建蓮，清華大學(xué)數(shù)學(xué)系副教授，主要研究方向?yàn)樗阕哟鷶?shù)、算子理論及在量子信息中的應(yīng)用，發(fā)表學(xué)術(shù)論文60多篇，SCI收錄50多篇。
目錄
第1 章 實(shí)數(shù)集與初等函數(shù)··················1
1.1 實(shí)數(shù)集····································1
1.1.1 集合及其運(yùn)算························1
1.1.2 映射···································3
1.1.3 可數(shù)集································3
1.1.4 實(shí)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(xí)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(yùn)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(xí)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(xí)題2.1····································.30
2.2 數(shù)列極限的性質(zhì)·····················.31
習(xí)題2.2····································.35
2.3 幾類特殊的數(shù)列·····················.36
2.3.1 無(wú)窮大數(shù)列與無(wú)窮小數(shù)列·······.36
2.3.2 無(wú)窮大數(shù)列與無(wú)界數(shù)列··········.36
2.3.3 Stolz 定理·························.38
習(xí)題2.3 ···································.40
2.4 實(shí)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(zhǔn)則·····················.45
*2.4.5 有限覆蓋定理·····················.47
*2.4.6 聚點(diǎn)定理··························.48
習(xí)題2.4 ···································.48
*2.5 上極限與下極限···················.50
習(xí)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(diǎn)的極限···············.55
3.1.2 函數(shù)在無(wú)窮遠(yuǎn)處的極限··········.58
習(xí)題3.1 ···································.58
3.2 函數(shù)極限的性質(zhì)及運(yùn)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(yùn)算·············.60
3.2.3 復(fù)合函數(shù)的極限··················.62
習(xí)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(guān)系·····.63
3.3.2 兩個(gè)重要極限·····················.65
3.3.3 無(wú)窮大量與無(wú)窮小量·············.67
3.3.4 等價(jià)無(wú)窮小量代換求極限········.69
習(xí)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(diǎn)及其分類··················.74
3.4.3 連續(xù)函數(shù)的局部性質(zhì)·············.78
習(xí)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(xí)題3.5····································.86
第4 章 導(dǎo)數(shù)與微分·························.89
4.1 導(dǎo)數(shù)的概念···························.89
4.1.1 導(dǎo)數(shù)概念的引出··················.89
4.1.2 函數(shù)可導(dǎo)的條件與性質(zhì)··········.91
習(xí)題4.1····································.92
4.2 求導(dǎo)法則······························.94
4.2.1 導(dǎo)數(shù)的四則運(yùn)算法則·············.94
4.2.2 反函數(shù)求導(dǎo)法則··················.96
4.2.3 復(fù)合函數(shù)的導(dǎo)數(shù)——鏈?zhǔn)椒▌t···.97
4.2.4 隱函數(shù)求導(dǎo)法則··················.98
4.2.5 參數(shù)方程求導(dǎo)法則················.99
習(xí)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(dǎo)的關(guān)系················104
4.3.3 微分在函數(shù)近似計(jì)算中的應(yīng)用··105
4.3.4 微分的運(yùn)算法則··················106
習(xí)題4.3····································106
4.4 高階導(dǎo)數(shù)與高階微分··············106
4.4.1 高階導(dǎo)數(shù)·························.107
4.4.2 高階微分·························.109
4.4.3 復(fù)合函數(shù)的微分·················.109
習(xí)題4.4 ··································.110
第5 章 微分學(xué)基本定理及應(yīng)用········.112
5.1 微分中值定理······················.112
5.1.1 極值的概念與費(fèi)馬定理·········.112
5.1.2 微分中值定理····················.113
習(xí)題5.1 ··································.118
5.2 洛必達(dá)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(xí)題5.2 ··································.126
5.3 泰勒公式及應(yīng)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(yīng)用·················.134
習(xí)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(xí)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(xí)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(yùn)算·············162
6.1.3 常用的不定積分公式·············162
習(xí)題6.1····································163
6.2 不定積分計(jì)算························164
6.2.1 分部積分法························165
6.2.2 積分換元法························166
習(xí)題6.2····································174
6.3 有理函數(shù)的不定積分··············175
習(xí)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(wú)理函數(shù)的不定積分········182
習(xí)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(zhǔn)則··························191
習(xí)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(xí)題7.2····································203
7.3 定積分的計(jì)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(xí)題7.3····································213
7.4 積分中值定理························216
習(xí)題7.4····································221
7.5 定積分的應(yīng)用······················.221
*7.5.1 分析學(xué)應(yīng)用······················.221
7.5.2 定積分的幾何應(yīng)用··············.224
7.5.3 定積分的物理應(yīng)用··············.232
習(xí)題7.5 ··································.236
第8 章 廣義積分···························.239
8.1 無(wú)窮積分·····························.239
8.1.1 無(wú)窮積分的概念·················.239
8.1.2 無(wú)窮積分求值····················.240
8.1.3 無(wú)窮積分?jǐn)可⑿耘袆e法·········.241
習(xí)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(wú)窮積分與瑕積分的關(guān)系······.249
8.2.3 瑕積分?jǐn)可⑿耘袆e法············.250
習(xí)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(xí)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(xí)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(xí)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(yīng)用——質(zhì)點(diǎn)的振動(dòng)···············289
習(xí)題9.5····································291
9.6 歐拉方程······························292
習(xí)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(diǎn)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(duì)角化的情形·········.301
9.8.2 矩陣A 不可對(duì)角化的情形······.302
9.8.3 矩陣A 有復(fù)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(xí)題9.8 ··································.310
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