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所在平台: Udemy |
课程主页: https://www.udemy.com/course/calculus-3-multivariable-calculus-part-2-of-2/
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课程名称：微积分 3（多变量微积分），第二部分，共二部分 课程概述：本课程为微积分 3（多变量微积分）的第二部分，主要探讨向量场及其相关的积分与向量微积分理论。课程内容基于Robert A. Adams与Christopher Essex的《Calculus, a complete course》第8或第9版。 课程内容： 1. **多重积分** - 学习计算双重积分，涵盖Riemann积分、平面中的集合和曲线等基础知识。 - 通过单积分迭代计算双重积分，理解x-简单和y-简单区域的概念，应用Fubini定理进行双重积分的迭代。 2. **变量变化** - 学习通过变量替换（主要是极坐标）来计算双重积分。 3. **不定积分** - 学习如何判断不定积分的收敛性与发散性，使用双重积分的均值定理计算二维函数的平均值。 4. **三重积分** - 学习计算三重积分，应用Fubini定理及变量替换（球坐标或圆柱坐标），计算各种变量变化下的Jacobian。 5. **多重积分的应用** - 探索多重积分在质量、表面积、质心等方面的实际应用。 6. **向量场** - 了解平面及空间中的向量场，保守向量场的相关概念，计算保守向量场的势函数。 7. **曲线积分** - 学习函数与向量场的曲线积分，及其在质量、弧长、功等计算中的应用，掌握三种计算向量场曲线积分的方法。 8. **曲面与曲面积分** - 理解作为二元函数图像描述的曲面，以及参数化曲面的定义，学会计算曲面积分，应用于质量与面积的计算。 9. **流量积分与向量微积分** - 学习流量积分的计算，了解表面的方向性、法向量场的选择及其与边界的关系，运用Green定理、高斯定理及Stokes定理。 课程总结： 在课程结束时，学生将能够定义并计算向量场的旋度和散度，并掌握多种基本公式；熟练运用Green定理、高斯定理和Stokes定理，并有效评估何时能够（以及方便地）应用这些定理。 注意：请根据您的教授的要求，确认您需掌握课程的哪些部分，以备最终考试之用。课程的详细大纲及每个章节的视频内容和问题解答可在资源文件 "001 Outline_Calculus3_part2.pdf" 中找到。
Calculus 3 (multivariable calculus), part 2 of 2Towards and through the vector fields, part 2 of 2: Integrals and vector calculus(Chapter numbers in Robert A. Adams, Christopher Essex: Calculus, a complete course. 8th or 9th edition.)C4: Multiple integrals (Chapter 14)S1. Introduction to the courseS2. Repetition (Riemann integrals, sets in the plane, curves)S3. Double integralsYou will learn: compute double integrals on APR (axis-parallel rectangles) by iteration of single integrals; x-simple and y-simple domains; iteration of double integrals (Fubini's theorem).S4. Change of variables in double integralsYou will learn: compute double integrals via variable substitution (mainly to polar coordinates).S5. Improper integralsYou will learn: motivate if an improper integral is convergent or divergent; use the mean-value theorem for double integrals in order to compute the mean value for a two-variable function on a compact connected set.S6. Triple integralsS7. Change of variables in triple integralsYou will learn: compute triple integrals by Fubini's theorem or by variable substitution to spherical or cylindrical coordinates; compute the Jacobian for various kinds of change of variables.S8. Applications of multiple integrals such as mass, surface area, mass centre.You will learn: apply multiple integrals for various aims.C5: Vector fields (Chapter15)S9. Vector fieldsS10. Conservative vector fieldsYou will learn: about vector fields in the plane and in the space; conservative vector fields; use the necessary condition for a vector field to be conservative; compute potential functions for conservative vector fields.S11. Line integrals of functionsS12. Line integral of vector fieldsYou will learn: calculate both kinds of line integrals (the ones of functions, and the ones of vector fields) and use them for computations of mass, arc length, work; three methods for computation of line integrals of vector fields.S13. SurfacesYou will learn: understand surfaces described as graphs to two-variable functions f:R^2->R and as parametric surfaces, being graphs of r:R^2->R^3; determine whether a surface is closed and determine surfaces' boundary; determine normal vector to surfaces.S14. Surface integralsYou will learn: calculate surface integrals of scalar functions and use them for computation of mass and area.S15. Oriented surfaces and flux integralsYou will learn: determine orientation of a surface; determine normal vector field; choose orientation of a surface which agrees with orientation of the surface's boundary; calculate flux integrals and use them for computation of the flux of a vector field across a surface.C6: Vector calculus (Chapter16: 16.1-16.5)S16. Gradient, divergence and curl, and some identities involving them; irrotational and solenoidal vector fields (Ch. 16.1-2)S17. Green's theorem in the plane (Ch. 16.3)S18. Gauss' theorem (Divergence Theorem) in 3-space (Ch. 16.4)S19. Stokes' theorem (Ch. 16.5)S20. Wrap-up Multivariable calculus / Calculus 3, part 2 of 2.You will learn: define and compute curl and divergence of (two- and three-dimensional) vector fields and proof some basic formulas involving gradient, divergence and curl; apply Green's, Gauss's and Stokes's theorems, estimate when it is possible (and convenient) to apply these theorems.Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.A detailed description of the content of the course, with all the 200 videos and their titles, and with the texts of all the 152 problems solved during this course, is presented in the resource file "001 Outline_Calculus3_part2.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
