《微積分（下）》是2023年電子工業出版社出版的圖書，作者是尤超、尹遜波、靳水林。
基本介紹
- 中文名：微積分（下）
- 作者：尤超、尹遜波、靳水林
- 出版社：電子工業出版社
- 出版時間：2023年2月
- 頁數：232 頁
- 定價：52 元
- 開本：16 開
- ISBN：9787121450051
內容簡介,圖書目錄,
內容簡介
本書主要針對拔尖創新人才培養而編寫，分上、下兩冊. 上冊內容包括極限與連續、導數與微分、微分 中值定理及導數套用、不定積分、定積分及其套用、微分方程；下冊內容包括多元函式微分學、多元數量值函式積分學、多元向量值函式積分學、無窮級數. 本書可作為高等學校理工科專業微積分課程的教材，也適合準備考研的學生參考.
圖書目錄
第 7 章 多元函式微分學 ·······································································································.1
7.1 多元函式的基本概念及其極限 ·····················································································.1
7.1.1 n 維歐式空間 ········································································································.1
7.1.2 多元函式 ···············································································································.2
7.1.3 多元函式的重極限與連續 ························································································.3
7.1.4 多元函式的累次極限 ·······························································································.7
習題 7.1 ···························································································································.8
7.2 偏導數與高階偏導數 ····································································································.9
7.2.1 偏導數的定義 ·········································································································.9
7.2.2 高階偏導數 ··········································································································.11
習題 7.2 ·························································································································.14
7.3 全微分 ······················································································································.14
習題 7.3 ·························································································································.18
7.4 複合函式求導法 ··········································································································.19
習題 7.4 ·························································································································.25
7.5 隱函式求導法 ··············································································································.26
習題 7.5 ·························································································································.30
7.6 偏導數的幾何套用 ······································································································.31
7.6.1 空間曲線的切線與法平面 ······················································································.31
7.6.2 曲面的切平面與法線 ·····························································································.33
7.6.3 二元函式全微分的幾何意義 ···················································································.35
習題 7.6 ·························································································································.35
7.7 多元函式的極值 ··········································································································.36
7.7.1 多元函式的無條件極值 ··························································································.36
7.7.2 條件極值與拉格朗日乘數法 ···················································································.42
習題 7.7 ·························································································································.46
7.8 方嚮導數與梯度 ··········································································································.46
7.8.1 場的基本概念 ·······································································································.46
7.8.2 方嚮導數 ·············································································································.46
7.8.3 梯度 ····················································································································.49
習題 7.8 ·························································································································.51
綜合題 ···························································································································.52
第 8 章 多元數量值函式積分學 ······························································································.53
8.1 黎曼積分 ······················································································································.53
8.1.1 二重積分的概念 ···································································································.53
8.1.2 黎曼積分的概念 ···································································································.54
8.1.3 黎曼積分的分類 ···································································································.54
8.1.4 黎曼積分的性質 ···································································································.55
習題 8.1 ·························································································································.57
8.2 二重積分的計算 ··········································································································.58
8.2.1 直角坐標系下二重積分的計算 ················································································.58
8.2.2 極坐標系下二重積分的計算 ···················································································.62
8.2.3 二重積分的對換對稱性 ··························································································.66
8.2.4 二重積分的變數代換 ·····························································································.67
習題 8.2 ·························································································································.70
8.3 三重積分的計算 ··········································································································.72
8.3.1 直角坐標系下三重積分的計算 ················································································.72
8.3.2 柱坐標系下三重積分的計算 ···················································································.75
8.3.3 球坐標系下三重積分的計算 ···················································································.77
8.3.4 三重積分一般的變數代換 ······················································································.79
習題 8.3 ·························································································································.81
8.4 第一型曲線積分 ··········································································································.83
習題 8.4 ·························································································································.86
8.5 第一型曲面積分的計算 ·······························································································.87
習題 8.5 ·························································································································.93
8.6 黎曼積分的套用實例 ··································································································.93
8.6.1 物體質心 ·············································································································.94
8.6.2 轉動慣量 ·············································································································.96
習題 8.6 ·························································································································.98
8.7 含參變數的積分 ··········································································································.99
習題 8.7 ·······················································································································.101
綜合題 ······················································································································.101
第 9 章 多元向量值函式積分學 ····························································································.104
9.1 向量場 ························································································································.104
9.1.1 向量場與向量線 ·································································································.104
9.1.2 向徑的導數 ········································································································.105
9.2 第二型曲線積分 ········································································································.106
9.2.1 第二型曲線積分的定義 ························································································.106
9.2.2 第二型曲線積分的計算 ························································································.108
9.2.3 兩種類型曲線積分的關係 ····················································································.111
習題 9.2 ·······················································································································.112
9.3 格林公式及平面流速場的環量與旋度 ·····································································.113
9.3.1 格林公式 ···········································································································.113
9.3.2 平面流速場的環量與旋度 ····················································································.117
習題 9.3 ·······················································································································.117
9.4 平面曲線積分與路徑無關的條件及保守場 ······························································.118
9.4.1 平面曲線積分與路徑無關的條件 ··········································································.118
9.4.2 保守場、原函式、全微分方程 ··············································································.123
習題 9.4 ·······················································································································.126
9.5 第二型曲面積分 ········································································································.127
9.5.1 第二型曲面積分的定義 ························································································.127
9.5.2 第二型曲面積分的計算 ························································································.130
習題 9.5 ·······················································································································.135
9.6 高斯公式、通量與散度 ·····························································································.136
9.6.1 高斯公式 ···········································································································.136
9.6.2 向量場的通量與散度 ···························································································.139
9.6.3 第二型曲面積分的三個等價條件 ··········································································.140
習題 9.6 ·······················································································································.142
9.7 斯托克斯公式、環量與旋度 ·····················································································.144
9.7.1 斯托克斯公式 ·····································································································.144
9.7.2 向量場的環量與旋度 ···························································································.146
習題 9.7 ·······················································································································.150
9.8 外微分形式 ················································································································.151
9.8.1 外積和外微分形式 ······························································································.151
9.8.2 外微分運算 ········································································································.151
9.8.3 梯度、旋度與散度的數學意義 ··············································································.152
9.8.4 多變數微積分的基本定理（一般的斯托克斯公式） ················································.154
綜合題 ···························································································································.15
第 10 章 無窮級數 ···········································································································.156
10.1 數項級數的斂散性 ··································································································.156
10.1.1 斂散性的概念 ···································································································.156
10.1.2 數項級數的幾個基本性質 ···················································································.158
10.1.3 柯西收斂原理 ···································································································.160
習題 10.1 ·····················································································································.161
10.2 正項級數的斂散性及其判別法 ···············································································.162
習題 10.2 ·····················································································································.172
10.3 任意項級數的斂散性及其判別法 ···········································································.174
習題 10.3 ·····················································································································.181
10.4 函式項級數 ··············································································································.182
10.4.1 函式項級數的收斂域與和函式 ············································································.182
10.4.2 函式項級數的一致收斂性 ···················································································.183
習題 10.4 ·····················································································································.189
10.5 冪級數 ············································································································.189
10.5.1 冪級數的收斂半徑和收斂域 ···············································································.190
10.5.2 冪級數的運算 ···································································································.194
習題 10.5 ·····················································································································.196
10.6 函式的冪級數展開 ··································································································.197
10.6.1 泰勒級數及直接展開法 ······················································································.197
10.6.2 間接展開法 ······································································································.201
10.6.3 冪級數和函式 ···································································································.204
習題 10.6 ·····················································································································.206
10.7 冪級數的套用舉例 ··································································································.208
10.7.1 函式值的近似計算 ····························································································.208
10.7.2 在積分計算中的套用 ·························································································.209
10.7.3 方程的冪級數解法 ····························································································.210
習題 10.7 ·····················································································································.212
10.8 傅立葉級數 ··············································································································.213
10.8.1 三角函式系的正交性 ·························································································.213
10.8.2 傅立葉級數 ······································································································.214
10.8.3 正弦級數和餘弦級數 ·························································································.216
10.8.4 以 2l 為周期的函式的傅立葉級數 ········································································.217
10.8.5 有限區間上的函式的傅立葉展開 ·········································································.219
習題 10.8 ·····················································································································.220
綜合題 ························································································································.222
