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所在平台: Udemy |
课程主页: https://www.udemy.com/course/calculus-3-multivariable-calculus-part-1-of-2/
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课程名称：微积分3（多变量微积分），第1部分 课程概述： 本课程是微积分3（多变量微积分）的第一部分，基于Robert A. Adams和Christopher Essex的《微积分, 完整课程》第8或9版。本部分内容主要围绕向量场展开，包含以下主题： 1. 课程介绍及准备知识概述（第10章） 2. R^n中的解析几何（2维和3维）：点、位置向量、直线和平面、点之间的距离 3. 圆锥曲线（圆、椭圆、抛物线、双曲线） 4. 二次曲面（球体、圆柱体、圆锥体、椭球体、抛物面等） 5. R^n中的拓扑学：距离、开球、邻域、开集和闭集、内点、外点、边界点 6. 坐标系：笛卡尔坐标系、极坐标、柱坐标和球坐标 学习目标：理解在R^2和R^3中简单方程和不等式所表示的几何对象，判断集合的开闭性，确定内点、外点和边界点，描述不同坐标系下的点及几何对象。 随后，课程内容将介绍向量值函数和参数曲线，以及多变量函数的可微性，具体内容包括： - 向量值函数和参数化曲线 - 多变量函数的极限、连续性、偏导数、梯度及雅可比矩阵 - 泰勒公式与多变量函数的优化（使用拉格朗日乘子法解决约束优化问题） 每部分都有具体的学习目标，旨在帮助学生在多变量微积分的各种概念上打下坚实的基础。 课程还提供了详细的学习资源，包括255个视频和216个解决问题的文本，首部分内容和相关教材可在“001 Outline_Calculus3.pdf”中找到。确保与教授确认您需要掌握的课程部分，以便进行期中考试的准备。
Calculus 3 (multivariable calculus), part 1 of 2Towards and through the vector fields, part 1 of 2(Chapter numbers in Robert A. Adams, Christopher Essex: Calculus, a complete course. 8th or 9th edition.)C0: Introduction to the course; preliminaries (Chapter 10: very briefly; most of the chapter belongs to prerequisites) S1. About the courseS2. Analytical geometry in R^n (n = 2 and n = 3): points, position vectors, lines and planes, distance between points (Ch.10.1)S3. Conic sections (circle, ellipse, parabola, hyperbola)S4. Quadric surfaces (spheres, cylinders, cones, ellipsoids, paraboloids etc) (Ch.10.5)S5. Topology in R^n: distance, open ball, neighbourhood, open and closed set, inner and outer point, boundary point (Ch.10.1)S6. Coordinates: Cartesian, polar, cylindrical, spherical coordinates (Ch.10.6)You will learn: to understand which geometrical objects are represented by simpler equations and inequalities in R^2 and R^3, determine whether a set is open or closed, if a point is an inner, outer or boundary point, determine the boundary points, describe points and other geometrical objects in the different coordinate systems.C1: Vector-valued functions, parametric curves (Chapter 11: 11.1, 11.3)S7. Introduction to vector-valued functionsS8. Some examples of parametrisationS9. Vector-valued calculus; curve: continuous, differentiable and smoothS10. Arc lengthS11. Arc length parametrisationYou will learn: Parametrise some curves (straight lines, circles, ellipses, graphs of functions of one variable);if r(t) = (x(t), y(t), z(t)) is a function describing a particle's position in R^3 with respect to time t, describe position, velocity, speed and acceleration; compute arc length of parametric curves, arc length parametrisation.C2: Functions of several variables; differentiability (Chapter 12) S12. Real-valued functions in multiple variables, domain, range, graph surface, level curves, level surfacesYou will learn: describe the domain and range of a function, Illustrate a function f(x,y) with a surface graph or with level curves.S13. Limit, continuityYou will learn: calculate limit values, determine if a function has limit value or is continuous at one point, use common sum-, product-,.rules for limits.S14. Partial derivative, tangent plane, normal line, gradient, JacobianYou will learn: calculate first-order partial derivatives, compute scalar products (two formulas) and cross pro- duct, give formulas for normals and tangent planes; understand functions from R^n to R^m, gradients and Jacobians.S15. Higher partial derivatesYou will learn: compute higher order partial derivatives, use Schwarz' theorem. Solve and verify some simple PDE's.S16. Chain rule: different versionsYou will learn: calculate the chain rule using dependency diagrams and matrix multiplication.S17. Linear approximation, linearisation, differentiability, differentialYou will learn: determine if a function is differentiable in a point, linearisation of a real-valued function, use linearisation to derive an approximate value of a function, use the test for differentiability (continuous partial derivatives), and properties of differentiable functions.S18. Gradient, directional derivativesYou will learn: calculate the gradient, find the direction derivative in a certain direction, properties of gradients, understand the geometric interpretation of the directional derivative, give a formula for the tangent and normal lines to a level curve.S19. Implicit functionsYou will learn: calculate the Jacobian determinant, derive partial derivatives with dependent and free variables of implicit functions.S20. Taylor's formula, Taylor's polynomialYou will learn: derive Taylor's polynomials and Taylor's formula. Understand quadratic forms and learn how to determine if they are positive definite, negative definite, or indefinite.C3: Optimisation of functions of several variables (Chapter 13: 13.1-3)S21. Optimisation on open domains (critical points)S22. Optimisation on compact domainsS23. Lagrange multipliers (optimisation with constraints)You will learn: classify critical points: local max and min, saddle points; find max and min values for a given function and region; use Lagrange multipliers with one or more conditions.Make sure that you check with your professor what parts of the course you will need for your midterms. 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 255 videos and their titles, and with the texts of all the 216 problems solved during this course, is presented in the resource file "001 Outline_Calculus3.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
