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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/calculus-through-data-and-modelling-techniques-of-integration
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课程名称：通过数据与建模的微积分：积分技巧 课程概述：本课程在之前对单变量函数在区间上的积分概念的基础上，扩展了我们对积分的理解，涵盖了多变量函数的积分技巧。首先，我们将学习如何在平面中对实值多变量函数进行积分。接着，我们将介绍向量函数，这些函数将点与向量相对应。课程的最终部分将为学习向量微积分奠定基础。最后，我们将引入在处理离散数据时近似定积分的技术，并通过同行评审的项目，将这些技术应用于实际问题中。 课程大纲： 模块1：迭代积分 描述：在本模块中，我们将定积分的概念扩展到对两变量或三变量函数的双重和三重积分。这些思想将用于计算更一般区域的面积、体积和质量。双重积分还可用于在涉及两个随机变量时计算概率。这一单变量微积分的扩展是后续学习向量微积分定理等重要工具的第一步。 模块2：平面区域上的双重积分 描述：对于函数 f(x) 的积分，积分区域始终是实数线的一个区间。而对于双重积分，我们希望扩展对多变量函数 f(x,y) 的积分能力，不仅限于矩形，也涵盖平面中的更一般区域。在本模块中，我们将开发相关工具和技术以实现这一目标。 模块3：向量函数 描述：向量值函数，也称为向量函数，是一种数学函数，定义在一个或多个变量上，其值域为多维向量或无限维向量的集合。向量值函数的输入可以是标量或向量，但其输出是向量。本模块将研究这些新类型的函数，并开发这些新数学对象的示例和应用。这些将成为未来模块中向量微积分发展的关键部分。 模块4：数据与积分 描述：尽管我们学会了广泛的代数工具来寻找反导数和使用微积分基本定理评估定积分，但在某些情况下，使用反导数并不可行。这可能是因为函数过于复杂，无法找到光滑的反导数，或者我们在处理离散数据而非连续函数。在本模块中，我们引入数值积分的概念和算法，它们使我们能够估算定积分的值。我们要解决的基本问题是：计算给定精度下定积分的近似解。本模块将介绍多种近似积分的方法。
Name:Module 1: Iterated Integrals
Description:In this module, we extend the idea of a definite integral to double and even triple integrals of functions of two or three variables. These ideas are then used to compute areas, volumes, and masses of more general regions. Double integrals are also used to calculate probabilities when two random variables are involved. This extension of single variable calculus is the first step towards major tools that arise later in this specialization involving theorems of vector calculus.
Name:Module 2: Double Integrals Over Plane Regions
Description:For integrals of a function f(x), the region over which we integrate is always an interval of the real line. But for double integrals, we want to expand our abilities to integrate a multivariable function f(x,y) not only over rectangles, but also over more general regions in the plane. In this module, we develop the tools and techniques to do that.
Name:Vector Functions
Description:A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector, but the output of this function is a vector. In this way, points are assigned to vectors. In this module, we will study these new types of functions and develop examples and applications of these new mathematical objects. They will play a key part in the development of vector calculus in future modules.
Name:Integration with Data
Description:Despite the broad algebraic tools we have learned to find antiderivatives and evaluate definite integrals using the Fundamental Theorem of Calculus, there are times when using antiderivatives is not possible. This could be because the function is too complicated in a way where no nice antiderivative exists, or that we are working with discrete data instead of a continuous function. In this module we introduce the notions and algorithms of numerical integration, which allow us to estimate the values of definite integrals. This is the basic problem we seek to solve: compute an approximate solution to a definite integral to a given degree of accuracy. There are many methods for approximating the integral to the desired precision, and we introduce a few here.
In this course, we build on previously defined notions of the integral of a single-variable function over an interval. Now, we will extend our understanding of integrals to work with functions of more than one variable. First, we will learn how to integrate a real-valued multivariable function over different regions in the plane. Then, we will introduce vector functions, which assigns a point to a vector. This will prepare us for our final course in the specialization on vector calculus. Finally, we will introduce techniques to approximate definite integrals when working with discrete data and through a peer reviewed project on, apply these techniques real world problems.
