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微積分原理(下) 版權信息
- ISBN：9787121464249
- 條形碼：9787121464249 ; 978-7-121-46424-9
- 裝幀：平塑
- 冊數：暫無
- 重量：暫無
- 所屬分類：>>
微積分原理(下) 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。本套教材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(下) 目錄
目錄
第10 章 多元函數的極限與連續··········1
10.1 n ? 中的點集拓撲和點列··········.1
10.1.1 n ? 中的點集拓撲···················1
10.1.2 n ? 中的點列·························6
10.1.3 n ? 的完備性·························7
*10.1.4 n ? 中的等價范數···················8
習題10.1 ··································.10
10.2 多元函數與多元向量值函數····.11
10.2.1 多元函數的概念··················.11
10.2.2 二元函數的圖像··················.12
10.2.3 多元向量值函數··················.16
習題10.2 ··································.17
10.3 多元函數的極限···················.18
10.3.1 多元函數的重極限···············.18
10.3.2 多元函數的累次極限············.19
10.3.3 向量值函數的極限···············.21
習題10.3 ··································.23
10.4 多元函數和向量值函數的
連續性·······························.24
10.4.1 多元函數連續的概念············.24
10.4.2 多元函數對各個變量的分別
連續·······························.26
10.4.3 多元連續函數的性質············.27
習題10.4 ··································.28
第11 章 多元函數微分學················.30
11.1 多元函數的偏導數與全微分····.30
11.1.1 多元函數的偏導數···············.30
11.1.2 多元函數的全微分···············.32
11.1.3 函數可微的條件··················.34
11.1.4 全微分在函數近似計算中的
應用······························.37
習題11.1 ··································.38
11.2 高階偏導數與復合函數的
微分··································.39
11.2.1 高階偏導數·······················.39
11.2.2 復合函數的微分··················.41
11.2.3 一階全微分的形式不變性·······.43
習題11.2 ··································.44
11.3 方向導數與梯度···················.46
11.3.1 方向導數·························.46
11.3.2 梯度······························.48
習題11.3 ··································.50
11.4 向量值函數的微分················.51
11.4.1 向量值函數的微分···············.51
11.4.2 復合映射的微分··················.54
習題11.4 ··································.55
11.5 隱函數微分法與逆映射微分法··.56
11.5.1 隱函數的微分····················.56
11.5.2 逆映射的微分····················.64
習題11.5 ··································.64
第12 章 多元函數微分學應用··········.67
12.1 多元函數微分學的幾何應用····.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
習題12.2 ··································.82
12.3 多元函數的極值···················.82
12.3.1 無條件極值·······················.83
12.3.2 條件極值·························.87
習題12.3 ··································.95
第13 章 重積分····························.98
13.1 二重積分的概念及性質··········.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··················101
習題13.1 ··································103
13.2 二重積分的計算···················104
13.2.1 直角坐標系·······················104
13.2.2 二重積分的坐標變換············108
習題13.2 ·································.114
13.3 三重積分···························.116
13.3.1 直角坐標系······················.117
13.3.2 一般坐標變換···················.119
13.3.3 柱坐標變換·······················120
13.3.4 球坐標變換·······················122
習題13.3 ··································124
13.4 重積分在幾何和物理中的
應用··································125
13.4.1 空間曲面的面積··················126
13.4.2 重積分在物理中的應用··········128
習題13.4 ··································131
*13.5 n 重積分····························132
13.5.1 若當測度的定義··················132
13.5.2 若當可測的等價條件············134
13.5.3 若當測度的運算性質············135
13.5.4 n 重積分··························138
13.5.5 n 維球坐標變換··················139
第14 章 曲線積分·························143
14.1 **型曲線積分——關于弧長
的曲線積分·························143
14.1.1 **型曲線積分的概念··········143
14.1.2 **型曲線積分的性質·········.145
14.1.3 **型曲線積分的計算·········.146
14.1.4 柱面側面積的計算··············.148
習題14.1 ·································.149
14.2 第二型曲線積分——關于坐標
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系······.151
14.2.3 第二型曲線積分的計算·········.151
習題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的
條件·····························.160
14.3.3 求微分式的原函數··············.161
14.3.4 全微分方程······················.164
習題14.3 ·································.166
第15 章 曲面積分························.170
15.1 **型曲面積分——關于面積
的曲面積分························.170
15.1.1 **型曲面積分的概念·········.170
15.1.2 **型曲面積分的計算·········.171
習題15.1 ·································.174
15.2 第二型曲面積分——關于坐標
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算·········.178
習題15.2 ·································
第10 章 多元函數的極限與連續··········1
10.1 n ? 中的點集拓撲和點列··········.1
10.1.1 n ? 中的點集拓撲···················1
10.1.2 n ? 中的點列·························6
10.1.3 n ? 的完備性·························7
*10.1.4 n ? 中的等價范數···················8
習題10.1 ··································.10
10.2 多元函數與多元向量值函數····.11
10.2.1 多元函數的概念··················.11
10.2.2 二元函數的圖像··················.12
10.2.3 多元向量值函數··················.16
習題10.2 ··································.17
10.3 多元函數的極限···················.18
10.3.1 多元函數的重極限···············.18
10.3.2 多元函數的累次極限············.19
10.3.3 向量值函數的極限···············.21
習題10.3 ··································.23
10.4 多元函數和向量值函數的
連續性·······························.24
10.4.1 多元函數連續的概念············.24
10.4.2 多元函數對各個變量的分別
連續·······························.26
10.4.3 多元連續函數的性質············.27
習題10.4 ··································.28
第11 章 多元函數微分學················.30
11.1 多元函數的偏導數與全微分····.30
11.1.1 多元函數的偏導數···············.30
11.1.2 多元函數的全微分···············.32
11.1.3 函數可微的條件··················.34
11.1.4 全微分在函數近似計算中的
應用······························.37
習題11.1 ··································.38
11.2 高階偏導數與復合函數的
微分··································.39
11.2.1 高階偏導數·······················.39
11.2.2 復合函數的微分··················.41
11.2.3 一階全微分的形式不變性·······.43
習題11.2 ··································.44
11.3 方向導數與梯度···················.46
11.3.1 方向導數·························.46
11.3.2 梯度······························.48
習題11.3 ··································.50
11.4 向量值函數的微分················.51
11.4.1 向量值函數的微分···············.51
11.4.2 復合映射的微分··················.54
習題11.4 ··································.55
11.5 隱函數微分法與逆映射微分法··.56
11.5.1 隱函數的微分····················.56
11.5.2 逆映射的微分····················.64
習題11.5 ··································.64
第12 章 多元函數微分學應用··········.67
12.1 多元函數微分學的幾何應用····.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
習題12.2 ··································.82
12.3 多元函數的極值···················.82
12.3.1 無條件極值·······················.83
12.3.2 條件極值·························.87
習題12.3 ··································.95
第13 章 重積分····························.98
13.1 二重積分的概念及性質··········.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··················101
習題13.1 ··································103
13.2 二重積分的計算···················104
13.2.1 直角坐標系·······················104
13.2.2 二重積分的坐標變換············108
習題13.2 ·································.114
13.3 三重積分···························.116
13.3.1 直角坐標系······················.117
13.3.2 一般坐標變換···················.119
13.3.3 柱坐標變換·······················120
13.3.4 球坐標變換·······················122
習題13.3 ··································124
13.4 重積分在幾何和物理中的
應用··································125
13.4.1 空間曲面的面積··················126
13.4.2 重積分在物理中的應用··········128
習題13.4 ··································131
*13.5 n 重積分····························132
13.5.1 若當測度的定義··················132
13.5.2 若當可測的等價條件············134
13.5.3 若當測度的運算性質············135
13.5.4 n 重積分··························138
13.5.5 n 維球坐標變換··················139
第14 章 曲線積分·························143
14.1 **型曲線積分——關于弧長
的曲線積分·························143
14.1.1 **型曲線積分的概念··········143
14.1.2 **型曲線積分的性質·········.145
14.1.3 **型曲線積分的計算·········.146
14.1.4 柱面側面積的計算··············.148
習題14.1 ·································.149
14.2 第二型曲線積分——關于坐標
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系······.151
14.2.3 第二型曲線積分的計算·········.151
習題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的
條件·····························.160
14.3.3 求微分式的原函數··············.161
14.3.4 全微分方程······················.164
習題14.3 ·································.166
第15 章 曲面積分························.170
15.1 **型曲面積分——關于面積
的曲面積分························.170
15.1.1 **型曲面積分的概念·········.170
15.1.2 **型曲面積分的計算·········.171
習題15.1 ·································.174
15.2 第二型曲面積分——關于坐標
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算·········.178
習題15.2 ·································
展開全部
微積分原理(下) 作者簡介
崔建蓮，清華大學數學系副教授。2002年7月獲得中科院數學研究所博士學位，2004年4月北京大學博士后出站，香港大學訪問學者，韓國首爾大學訪問學者，美國威廉瑪麗學院訪問學者。2004年4月入職清華大學數學系，現為數學系副教授，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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