工科数学分析
课程编码：08N1120241-2 课程中文名称：工科数学分析
课程英文名称：Engineering Analysis of Mathematics 总学时：180 学分：11.0
先修课程：中学数学 课程简介：
本课程系统介绍数学分析中的基本概念和基本方法： 一、函数：函数及其性质，复合函数，初等函数，集合。
二、极限与连续：极限定义，极限的性质与运算法则，极限存在的判别法则，无穷小与无穷大，函数的连续性。
三、导数与微分：导数的基本公式与四则运算法则，反函数与复合函数求导法则，隐函数与参数式函数求导法则，高阶导数，微分。
四、微分中值定理：微分中值定理，洛必达法则，泰勒公式。 五、不定积分：原函数与不定积分，换元积分法，分步积分法。
六、定积分：定积分的定义与性质，定积分存在的判别法则,微积分基本定理，定积分的换元积分法和分步积分法，反常积分。
七、导数与积分应用：函数的极值与最大（小）值，函数的分析作图法，曲线的弧长与弧微分，曲率，定积分的应用。
八、微分方程：一阶微分方程，几种可积的高阶微分方程，线性微分方程。
九、多元函数微分学：偏导数与高阶偏导数，全微分，复合函数求导法，隐函数求导法，偏导数的几何应用，多元函数的泰勒公式与极值，方向导数与梯度。
十、多元函数积分学：黎曼积分定义，二重积分，三重积分，第一型曲线积分，第一型曲面积分，黎曼积分的应用。
十一、第二型曲线积分与第二型曲面积分：向量场，第二型曲线积分，格林公式，第二型曲面积分，高斯公式，通量与散度，斯托克斯公式，环量与旋度。
十二、无穷级数：无穷级数及其性质，正项级数，任意项级数，函数项级数，幂级数，傅里叶级数。
十三、复变函数：复变函数的极限与连续，解析函数，复变函数的积分，解析函数的级数表示。
十四、微分几何：曲线论的基本知识，曲面论的第一基本形式，曲面论的第二基本形式。
Course Description:
This course is intended to introduce those methods and concepts in analysis of mathematics： 1. Functions: Functions and their properties, composite functions, elementary functions, set. 2. Limits and continuity: Definition of limit, limit properties and laws, criteria of existence of limits, infinities and infinitesimals, continuity.
3. Derivatives and differentials: Differentiation formulas and rules, chain rule, implicit differentiation, higher derivatives, differentials.
4. Mean value theorem: Mean value theorem of differentials, L`hospital rule, Taylor formula. 5. Infinite integrals: Antiderivatives and infinite integrals, substitution rule, integration by parts.
6. Definite integrals: Definition and properties of definite integral, criteria of existence of definite integrals, fundamental theorem of calculus, substitution rule, integration by parts, improper integrals.
7. Applications of differentiation and integration: Maximum and minimum values, curve sketching, arc length and differential of arc length, curvature, applications of integration.
8. Differential equations: First-order differential equations, some integrable higher-order differential equations, linear differential equations.
9. Multivariable differential calculus: Partial derivatives and higher partial derivatives, total differentials, chain rule, implicit differentiation, applications of partial derivatives to geometry, Taylor formula and extreme values, directional derivatives and gradients.
10. Multivariable integral calculus: Definition of Riemann integrals, double integrals, triple integrals, line integrals, surface integrals, applications of Riemann integrals.
11. Line and surface integrals of vector field: Vector fields, line integrals, Green’s theorem, surface integrals, Gauss’ theorem, flux and divergence, Stokes’ theorem, circulation and curl.
12. Infinite series: Infinite series and their properties, series with positive terms, numerical series, series of functions, power series, Fourier series.
13. Functions of complex variable: Limits and continuity, analytic functions, integrals, series expression of analytic function.
14. Differential geometry: Fundamental knowledge of curve theory, first fundamental form of a surface, second fundamental form of a surface.
2工科数学分析
