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Jingdong book

微积分与解析几何(影印版 原书第2版)

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Content Description

《微积分与解析几何(影印版 原书第2版)》除具有标准微积分教材的内容外,书中例子偏重实际,侧重于微积分的应用。同时补充了三角函数、极坐标等理论知识,使学生从高中到大学平稳过渡。文中穿插数学史与数学文化的相关内容,同时附录中提供了大量的补充内容以及严格的理论证明,适合不同层次的学生按需要学习。附加问题生动有趣,多是相关内容的经典结论

Author Description

乔治·西蒙斯(George F.Simmons) ,博士毕业于耶鲁大学。他通常能用简练精妙的语言表达深刻的数学思想,他在科罗拉多学院任教的日子,以其鲜明的个性以及引入入胜的教学方式,赢得学生的喜爱。

Catalogue

致教师
致学生
第一部分
第1章 数、函数与图形
1.1 引言
1.2 数轴与坐标平面 毕达哥拉斯
1.3 直线的斜率和方程
1.4 圆与抛物线 笛卡儿和费马
1.5 函数的概念
1.6 函数的图形
1.7 三角函数的引入:函数sinθ和cosθ
复习小结:定义、概念及方法
附加问题
第2章 函数的导数
2.1 什么是微积分 切线问题
2.2 如何计算切线的斜率
2.3 导数的定义
2.4 速度与变化率 牛顿和莱布尼茨
2.5 极限的概念 两个三角函数的极限
2.6 连续函数 中值定理和其他定理
复习小结:定义、概念及方法
附加问题
第3章 导数的运算
3.1 多项式函数的导数
3.2 函数积、商的求导法则
3.3 复合函数求导和链式法则
3.4 一些三角函数的导数
3.5 隐函数和分数指数函数的求导
3.6 高阶导数
复习小结:概念、公式及方法
附加问题
第4章 导数的应用
4.1 递增函数与递减函数 最大值与最小值
4.2 凹性与拐点
4.3 最大值和最小值问题的应用
4.4 更多最大/最小值问题 光的反射与折射
4.5 复合函数的变化率
4.6 牛顿法解方程
4.7 (选学)经济学上的应用 边际分析法
复习小结:概念及方法
附加问题
第5章 不定积分和微分方程
5.1 引言
5.2 微分与切线逼近
5.3 不定积分 换元积分法
5.4 微分方程 分离变量法
5.5 重力作用下的运动 逃逸速度和黑洞
复习小结:概念及方法
附加问题
第6章 定积分
6.1 引言
6.2 面积问题
6.3 “∑”符号与某些特殊求和
6.4 曲线下的面积 定积分 黎曼
6.5 极限思想下的面积计算
6.6 微积分基本定理
6.7 定积分的性质
复习小结:概念及方法
附加问题
附录:希波克拉底拱形
第7章 定积分的应用
7.1 引言:定积分的直观含义
7.2 两条曲线之间的面积
7.3 体积计算1:圆盘法
7.4 体积计算2:圆柱壳法
7.5 弧长
7.6 旋转曲面的面积
7.7 功和能
7.8 流体静力学
复习小结:概念与方法
附加问题
附录:阿基米德与球体体积
第二部分
第8章 指数函数与对数函数
8.1 引言
8.2 指数与对数的回顾
8.3 数e和函数y=e^x
8.4 自然对数和函数y=lnx 欧拉
8.5 应用 人口增长和放射性衰变
8.6 更多应用--控制人口增长
复习小结:概念及公式
附加问题
第9章 三角函数
9.1 三角函数的回顾
9.2 正弦和余弦函数的导数
9.3 正弦和余弦函数的积分 蒲丰投针问题
9.4 其他四个三角函数的导数
9.5 反三角函数
9.6 简谐运动:钟摆问题
9.7 (选学) 双曲函数
复习小结:定义及公式
附加问题
第10章 积分法
10.1 简介 基本公式
10.2 换元法
10.3 三角函数的积分
10.4 三角换元法
10.5 完全平方法
10.6 部分分式法
10.7 分部积分法
10.8 综合法 处理复杂类型的积分策略
10.9 数值积分 辛普森法则
复习小结:公式及方法
附加问题
附录1:悬链线或悬挂链曲线
附录2:沃利斯乘积:pi/2=2/1*2/3*4/3*4/5*6/5*6/7...
附录3:莱布尼茨如何发现公式:pi/4=1-1/3+1/5-1/7+...
第11章 积分的进一步应用
11.1 离散系统的质心
11.2 形心
11.3 帕普斯定理
11.4 惯性矩
复习小结:定义及概念
附加问题
第12章 不定式和反常积分
12.1 简介 中值定理的回顾
12.2 "0/0"不定式:洛必达法则
12.3 其他类型的不定式
12.4 反常积分
12.5 正态分布:高斯
复习小结:定义及概念
附加问题
第13章 常数项无穷级数
13.1 什么是无穷级数
13.2 收敛数列
13.3 收敛和发散级数
13.4 收敛级数的一般性质
13.5 正项级数 比较判别法
13.6 积分判别法 欧拉常数
13.7 比值判别法和根值判别法
13.8 交错级数的判别
复习小结:定义、概念及判别方法
附加问题
附录1:欧拉发现公式∑1/n^2=pi^2/6
附录2:更多关于无理数的问题:证明pi为无理数
附录3:关于级数∑1/Pn,其中Pn为素数
第14章 幂级数
14.1 引言
14.2 收敛区间
14.3 幂级数的微分与积分
14.4 泰勒级数和泰勒公式
14.5 应用泰勒公式的计算
14.6 微分方程的应用
14.7 (选学)幂级数的运算
14.8 (选学)复数和欧拉公式
复习小结:定义、公式及方法
附加问题
附录:伯努利数和欧拉的众多美妙的发现
第三部分
第15章 圆锥曲线
15.1 引言 圆锥截面
15.2 重新审视圆与抛物线
15.3 椭圆
15.4 双曲线
15.5 焦点-准线-偏心的定义
15.6 (可选)二次方程 绕坐标轴旋转
复习小结:定义及性质
附加问题
第16章 极坐标
16.1 极坐标系
16.2 极坐标方程的更多图像
16.3 圆、圆锥曲线和螺旋线的极坐标方程
16.4 弧长和切线
16.5 极坐标中的面积
复习小结:定义及公式
附加问题
第17章 参数方程及平面内的向量
17.1 曲线的参数方程
17.2 摆线和其他类似曲线
17.3 向量代数 单位向量i和j
17.4 向量函数的导数 速度和加速度
17.5 曲率和单位法向量
17.6 加速度的切分量和法分量
17.7 开普勒定理和牛顿的万有引力定律
复习小结:定义及公式
附加问题
附录1:最速降线问题的伯努利解法
第18章 三维空间的向量与曲面
18.1 三维空间的坐标和向量
18.2 两个向量的标量积
18.3 两个向量的向量积
18.4 直线和平面
18.5 圆柱坐标和旋转曲面
18.6 二次曲面
18.7 圆柱坐标和球面坐标
复习小结:定义及方程
第19章 偏导数
19.1 多元函数
19.2 偏导数
19.3 曲面的切平面
19.4 增量和微分 基本引理
19.5 方向导数和梯度
19.6 偏导数的链式法则
19.7 最大值和最小值问题
19.8 条件极值 拉格朗日乘数法
19.9(选学)拉普拉斯方程、热传导方程和波动方程 拉普拉斯和傅里叶
19.10 (选学)隐函数
复习小结:定义及方法
第20章 重积分
20.1 累次积分-体积
20.2 二重积分和累次积分
20.3 二重积分的物理应用
20.4 极坐标下的二重积分
20.5 三重积分
20.6 圆柱坐标
20.7 球面坐标 万有引力定律
20.8 曲面面积 勒让德公式
复习小结:方法和公式
附录:欧拉公式∑1/n^2=pi^2/6的二重积分证明
第21章 曲线积分和曲面积分 格林公式高斯公式和斯托克斯公式
21.1平面上的曲线积分
21.2 与路径无关:保守场
21.3 格林公式
21.4 曲面积分和高斯公式
21.5 斯托克斯公式
21.6 麦克斯韦方程组 终极思考
复习小结:概念及定理

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