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)好這門課程對每一位理工科學(xué)生來說都非常重要。本套教材在傳授微積分知識的同時(shí)，注重培養(yǎng)學(xué)生的數(shù)學(xué)思維、語言邏輯和創(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。上冊內(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)級數(shù)、函數(shù)項(xiàng)級數(shù)、傅里葉級數(shù)和含參積分。
崔建蓮，清華大學(xué)數(shù)學(xué)系副教授。2002年7月獲得中科院數(shù)學(xué)研究所博士學(xué)位，2004年4月北京大學(xué)博士后出站，香港大學(xué)訪問學(xué)者，韓國首爾大學(xué)訪問學(xué)者，美國威廉瑪麗學(xué)院訪問學(xué)者。2004年4月入職清華大學(xué)數(shù)學(xué)系，現(xiàn)為數(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10 章 多元函數(shù)的極限與連續(xù)··········1
10.1 n ? 中的點(diǎn)集拓?fù)浜忘c(diǎn)列··········.1
10.1.1 n ? 中的點(diǎn)集拓?fù)洹ぁぁぁぁぁぁぁぁぁぁぁぁぁぁぁぁぁぁ?
10.1.2 n ? 中的點(diǎn)列·························6
10.1.3 n ? 的完備性·························7
*10.1.4 n ? 中的等價(jià)范數(shù)···················8
習(xí)題10.1 ··································.10
10.2 多元函數(shù)與多元向量值函數(shù)····.11
10.2.1 多元函數(shù)的概念··················.11
10.2.2 二元函數(shù)的圖像··················.12
10.2.3 多元向量值函數(shù)··················.16
習(xí)題10.2 ··································.17
10.3 多元函數(shù)的極限···················.18
10.3.1 多元函數(shù)的重極限···············.18
10.3.2 多元函數(shù)的累次極限············.19
10.3.3 向量值函數(shù)的極限···············.21
習(xí)題10.3 ··································.23
10.4 多元函數(shù)和向量值函數(shù)的
連續(xù)性·······························.24
10.4.1 多元函數(shù)連續(xù)的概念············.24
10.4.2 多元函數(shù)對各個(gè)變量的分別
連續(xù)·······························.26
10.4.3 多元連續(xù)函數(shù)的性質(zhì)············.27
習(xí)題10.4 ··································.28
第11 章 多元函數(shù)微分學(xué)················.30
11.1 多元函數(shù)的偏導(dǎo)數(shù)與全微分····.30
11.1.1 多元函數(shù)的偏導(dǎo)數(shù)···············.30
11.1.2 多元函數(shù)的全微分···············.32
11.1.3 函數(shù)可微的條件··················.34
11.1.4 全微分在函數(shù)近似計(jì)算中的
應(yīng)用······························.37
習(xí)題11.1 ··································.38
11.2 高階偏導(dǎo)數(shù)與復(fù)合函數(shù)的
微分··································.39
11.2.1 高階偏導(dǎo)數(shù)·······················.39
11.2.2 復(fù)合函數(shù)的微分··················.41
11.2.3 一階全微分的形式不變性·······.43
習(xí)題11.2 ··································.44
11.3 方向?qū)?shù)與梯度···················.46
11.3.1 方向?qū)?shù)·························.46
11.3.2 梯度······························.48
習(xí)題11.3 ··································.50
11.4 向量值函數(shù)的微分················.51
11.4.1 向量值函數(shù)的微分···············.51
11.4.2 復(fù)合映射的微分··················.54
習(xí)題11.4 ··································.55
11.5 隱函數(shù)微分法與逆映射微分法··.56
11.5.1 隱函數(shù)的微分····················.56
11.5.2 逆映射的微分····················.64
習(xí)題11.5 ··································.64
第12 章 多元函數(shù)微分學(xué)應(yīng)用··········.67
12.1 多元函數(shù)微分學(xué)的幾何應(yīng)用····.67
12.1.1 空間曲線·························.67
12.1.2 空間曲面的切平面與法線·······.69
12.1.3 空間曲線的切線與法平面·······.72
習(xí)題12.1 ··································.76
12.2 高階全微分與泰勒公式··········.77
12.2.1 高階全微分·······················.77
12.2.2 泰勒公式·························.79
習(xí)題12.2 ··································.82
12.3 多元函數(shù)的極值···················.82
12.3.1 無條件極值·······················.83
12.3.2 條件極值·························.87
習(xí)題12.3 ··································.95
第13 章 重積分····························.98
13.1 二重積分的概念及性質(zhì)··········.98
13.1.1 二重積分的概念··················.98
13.1.2 可積的條件·······················100
13.1.3 二重積分的性質(zhì)··················101
習(xí)題13.1 ··································103
13.2 二重積分的計(jì)算···················104
13.2.1 直角坐標(biāo)系·······················104
13.2.2 二重積分的坐標(biāo)變換············108
習(xí)題13.2 ·································.114
13.3 三重積分···························.116
13.3.1 直角坐標(biāo)系······················.117
13.3.2 一般坐標(biāo)變換···················.119
13.3.3 柱坐標(biāo)變換·······················120
13.3.4 球坐標(biāo)變換·······················122
習(xí)題13.3 ··································124
13.4 重積分在幾何和物理中的
應(yīng)用··································125
13.4.1 空間曲面的面積··················126
13.4.2 重積分在物理中的應(yīng)用··········128
習(xí)題13.4 ··································131
*13.5 n 重積分····························132
13.5.1 若當(dāng)測度的定義··················132
13.5.2 若當(dāng)可測的等價(jià)條件············134
13.5.3 若當(dāng)測度的運(yùn)算性質(zhì)············135
13.5.4 n 重積分··························138
13.5.5 n 維球坐標(biāo)變換··················139
第14 章 曲線積分·························143
14.1 第一型曲線積分——關(guān)于弧長
的曲線積分·························143
14.1.1 第一型曲線積分的概念··········143
14.1.2 第一型曲線積分的性質(zhì)·········.145
14.1.3 第一型曲線積分的計(jì)算·········.146
14.1.4 柱面?zhèn)让娣e的計(jì)算··············.148
習(xí)題14.1 ·································.149
14.2 第二型曲線積分——關(guān)于坐標(biāo)
的曲線積分························.150
14.2.1 第二型曲線積分的概念·········.150
14.2.2 兩類曲線積分之間的關(guān)系······.151
14.2.3 第二型曲線積分的計(jì)算·········.151
習(xí)題14.2 ·································.155
14.3 格林公式···························.157
14.3.1 格林公式························.157
14.3.2 曲線積分與積分路徑無關(guān)的
條件·····························.160
14.3.3 求微分式的原函數(shù)··············.161
14.3.4 全微分方程······················.164
習(xí)題14.3 ·································.166
第15 章 曲面積分························.170
15.1 第一型曲面積分——關(guān)于面積
的曲面積分························.170
15.1.1 第一型曲面積分的概念·········.170
15.1.2 第一型曲面積分的計(jì)算·········.171
習(xí)題15.1 ·································.174
15.2 第二型曲面積分——關(guān)于坐標(biāo)
的曲面積分························.175
15.2.1 第二型曲面積分的概念·········.175
15.2.2 第二型曲面積分的計(jì)算·········.178
習(xí)題15.2 ·································.181
15.3 高斯公式和斯托克斯公式······.182
15.3.1 高斯公式························.182
15.3.2 斯托克斯公式···················.185
15.3.3 空間曲線積分與積分路徑無關(guān)
的條件···························.189
習(xí)題15.3 ·································.190
15.4 場論初步···························.192
15.4.1 梯度場···························.192
15.4.2 散度場···························.193
15.4.3 旋度場···························.195
15.4.4 三種運(yùn)算的聯(lián)合運(yùn)用············196
15.4.5 平面向量場·······················196
*15.4.6 曲線坐標(biāo)系·······················198
15.4.7 正交曲線坐標(biāo)系下的梯度、旋度、
散度和拉普拉斯算子············200
習(xí)題15.4 ··································204
第16 章 數(shù)項(xiàng)級數(shù)·························206
16.1 級數(shù)的斂散性······················207
16.1.1 級數(shù)收斂與發(fā)散的概念··········207
16.1.2 收斂級數(shù)的性質(zhì)··················208
習(xí)題16.1 ··································210
16.2 正項(xiàng)級數(shù)···························.211
習(xí)題16.2 ··································220
16.3 任意項(xiàng)級數(shù)·························221
16.3.1 萊布尼茨（Leibniz）判別法····221
16.3.2 絕對收斂級數(shù)的性質(zhì)············222
16.3.3 條件收斂級數(shù)的兩個(gè)判別法·····226
*16.3.4 無窮乘積·························229
習(xí)題16.3 ··································229
第17 章 函數(shù)項(xiàng)級數(shù)······················232
17.1 函數(shù)列·······························232
17.1.1 函數(shù)列的一致收斂···············232
17.1.2 函數(shù)列極限函數(shù)的分析性·······237
習(xí)題17.1 ··································238
17.2 函數(shù)項(xiàng)級數(shù)·························239
17.2.1 函數(shù)項(xiàng)級數(shù)的收斂域············239
17.2.2 函數(shù)項(xiàng)級數(shù)的一致收斂性·······240
17.2.3 和函數(shù)的分析性··················243
*17.2.4 兩個(gè)例子·························247
習(xí)題17.2 ··································251
17.3 冪級數(shù)·······························252
17.3.1 冪級數(shù)的收斂域與收斂半徑·····252
17.3.2 冪級數(shù)和函數(shù)的分析性··········255
習(xí)題17.3 ··································261
17.4 函數(shù)的冪級數(shù)展開················262
17.4.1 泰勒級數(shù)、麥克勞林級數(shù)·······263
17.4.2 函數(shù)可展開為泰勒級數(shù)的條件····264
17.4.3 基本初等函數(shù)的麥克勞林級數(shù)··.265
17.4.4 利用冪級數(shù)求數(shù)的近似值······.268
習(xí)題17.4 ·································.270
第18 章 傅里葉級數(shù)·····················.271
18.1 函數(shù)的傅里葉級數(shù)···············.272
18.1.1 以2π 為周期函數(shù)的傅里葉級數(shù)··.272
18.1.2 以2l 為周期函數(shù)的傅里葉級數(shù)··.278
習(xí)題18.1 ·································.280
18.2 傅里葉級數(shù)的逐點(diǎn)收斂性······.281
18.2.1 傅里葉級數(shù)的性質(zhì)··············.281
18.2.2 傅里葉級數(shù)的逐點(diǎn)收斂·········.284
習(xí)題18.2 ·································.291
18.3 傅里葉級數(shù)的平方平均收斂···.292
18.3.1 正交投影及Bessel 不等式······.292
18.3.2 三角多項(xiàng)式······················.295
18.3.3 Fejér 核與一致逼近·············.296
18.3.4 均方收斂························.299
習(xí)題18.3 ·································.306
18.4 傅里葉積分簡介··················.308
18.4.1 傅里葉級數(shù)的復(fù)數(shù)形式·········.308
18.4.2 傅里葉積分：啟發(fā)式介紹······.309
18.4.3 傅里葉積分：嚴(yán)格理論·········.312
習(xí)題18.4 ·································.318
18.5 函數(shù)逼近定理·····················.319
18.5.1 魏爾斯特拉斯第一逼近定理····.319
18.5.2 魏爾斯特拉斯第二逼近定理····.325
習(xí)題18.5 ·································.327
第19 章 含參積分························.328
19.1 含參定積分························.328
習(xí)題19.1 ·································.332
19.2 含參廣義積分·····················.333
19.2.1 含參廣義積分的一致收斂性····.333
19.2.2 含參廣義積分的分析性·········.336
19.2.3 歐拉積分：伽馬函數(shù)與貝塔
函數(shù)·····························.342
習(xí)題19.2 ·································.346
參考文獻(xiàn)······································.348
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