Calculus II
微积分II是微积分I的延续，主要内容包括积分技巧、反常积分、无穷级数、幂级数、多元函数微分学、重积分、曲线积分与曲面积分。学生将深入学习各种积分方法，理解级数收敛性的判别法则，掌握多元函数微积分的基本概念和计算技巧。课程为微分方程、复分析、概率论等后续课程提供必要的数学基础，培养学生处理高维问题的能力。
Calculus II continues from Calculus I, covering integration techniques, improper integrals, infinite series, power series, multivariable differential calculus, multiple integrals, and line and surface integrals. Students will master various integration methods, convergence tests for series, and fundamental concepts of multivariable calculus. The course provides essential mathematical foundations for differential equations, complex analysis, and probability theory.
选择一个特殊函数（如指数函数、三角函数或贝塞尔函数），研究其泰勒级数展开与收敛性，使用部分和进行数值逼近并分析误差。同时运用数值积分方法（梯形法、辛普森法）计算相关定积分，比较不同方法的精度与收敛速度。要求包含理论推导、数值计算实现、误差分析和结果可视化。
Choose a special function (exponential, trigonometric, or Bessel function), study its Taylor series expansion and convergence, use partial sums for numerical approximation and analyze errors. Apply numerical integration methods (trapezoidal rule, Simpson's rule) to compute related definite integrals, comparing accuracy and convergence speed of different methods.