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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/calculus-through-data-and-modelling-vector-calculus
课程评论：没有评论
课程名称：通过数据与建模学习微积分：向量微积分 概述：本课程继续您对微积分的学习，重点关注积分在向量值函数或向量场中的应用。这些函数将向量分配给空间中的点，从而使我们能够发展高级理论，并将其应用于现实世界问题。我们定义了线积分，以计算向量场所做的功。课程的最后我们将学习格林定理，该定理描述了在闭合路径上的某些类型的线积分与双重积分之间的关系。在离散情况下，该定理称为鞋带定理，允许我们测量多边形的面积。我们使用该定理的这一版本，开发更多的数据分析工具，通过同行评审项目来进行实践学习。完成本课程后，您将掌握任何以单变量或多变量微积分为基础的高级数学、计算机科学或数据科学所需的所有工具。 课程大纲： 模块1：向量场和线积分 描述：在本模块中，我们定义了向量场的概念，即将向量应用于给定点的函数。然后，我们发展这些新函数在平面和空间沿一般曲线的积分概念。线积分最早于19世纪初被开发，最初用于解决流体流动、力、电和磁问题。今天，它们仍然是高级数学理论和向量微积分的核心。 模块2：线积分的基本定理 描述：在本模块中，我们介绍保守向量场的概念。在向量微积分中，保守向量场是某函数f的梯度，称为势函数。保守向量场具有路径无关的特点，即在两个点之间选择的任何路径都不会改变线积分的值。我们将阐述一个关于保守向量场线积分的重要定理，称为线积分的基本定理。这将使我们能够证明，对于保守系统，沿着配置空间路径移动所做的功仅依赖于路径的端点。 模块3：格林定理 描述：在本模块中，我们阐述并应用向量微积分的一个主要工具：格林定理。格林定理给出了一个两维向量场在平面上闭合路径的线积分与其所包围区域的双重积分之间的关系。二维保守场在闭合路径上的积分为零是格林定理的一个特例。
Name:Module 1: Vector Fields and Line Integrals
Description:In this module, we define the notion of a Vector Field, which is a function that applies a vector to a given point. We then develop the notion of integration of these new functions along general curves in the plane and in space. Line integrals were developed in the early19th century initially to solve problems involving fluid flow, forces, electricity, and magnetism. Today they remain at the core of advanced mathematical theory and vector calculus.
Name:Module 2: The Fundamental Theorem for Line Integrals
Description:In this module, we introduce the notion of a Conservative Vector Field. In vector calculus, a conservative vector field is a vector field that is the gradient of some function f, called the potential function. Conservative vector fields have the property that the line integral is path independent, which means the choice of any path between two points does not change the value of the line integral. Conversely, path independence of the line integral is equivalent to the vector field being conservative. We then state and formalize an important theorem about line integrals of conservative vector fields, called the Fundamental Theorem for Line Integrals. This will allow us to show that for a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path.
Name:Module 3: Green's Theorem
Description:In this module we state and apply a main tool of vector calculus: Green's Theorem. Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses. The fact that the integral of a two-dimensional conservative field over a closed path is zero is a special case of Green's theorem.
This course continues your study of calculus by focusing on the applications of integration to vector valued functions, or vector fields. These are functions that assign vectors to points in space, allowing us to develop advanced theories to then apply to real-world problems. We define line integrals, which can be used to fund the work done by a vector field. We culminate this course with Green's Theorem, which describes the relationship between certain kinds of line integrals on closed paths and double integrals. In the discrete case, this theorem is called the Shoelace Theorem and allows us to measure the areas of polygons. We use this version of the theorem to develop more tools of data analysis through a peer reviewed project. Upon successful completion of this course, you have all the tools needed to master any advanced mathematics, computer science, or data science that builds off of the foundations of single or multivariable calculus.
