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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/vector-calculus-engineers
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课程名称：工程师的向量微积分 课程概述：该课程涵盖了向量微积分的基本理论及其应用。第一周我们介绍标量场和向量场，第二周学习场的微分，第三周探讨多维积分和曲线坐标系。第四周重点讲解线积分和面积分，最后在第五周学习向量微积分的基本定理，包括梯度定理、散度定理和斯托克斯定理。这些定理在电磁学和流体力学等核心工程学科中至关重要。 一些大学可能将该课程称为多变量微积分或三重微积分。修习此课程的先决条件是两学期的单变量微积分（微分与积分）。 课程由53段短视频组成，每个视频后会有一些解题练习，并在每个重要主题后设有短小的练习测验。每周结束时会有一个评估测验，问题及测验解答可以在教师提供的讲义中找到。 课程内容大纲： 1. 向量：定义向量并学习加减法和点积、叉积的乘法，研究直线和平面的解析几何，并定义克罗内克δ和李维-西维塔符号以证明向量恒等式。 2. 微分：定义标量场和向量场的偏导数，衍生最小二乘法，使用链式法则对多个变量的函数进行微分，定义梯度、散度、旋度和拉普拉斯算子，推导电磁波方程。 3. 积分与曲线坐标：学习双重和三重积分，定义极坐标、圆柱坐标和球坐标，以简化具有对称性的数学问题，并学习如何在曲线坐标系中使用微分算子和变换雅可比矩阵。 4. 线积分与面积分：学习如何计算标量场和向量场在曲线或表面上的积分，推导与力场相关的功-能定理，以及流体通过表面的质量通量。 5. 基本定理：学习微积分基本定理及向量微积分的基本定理，包括梯度定理、散度定理和斯托克斯定理，并展示它们在导出连续方程和能量守恒定律中的应用。 欢迎下载讲义：[讲义下载链接](http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf) 观看推广视频：[视频链接](https://youtu.be/qUseabHb6Vk)
Name:Vectors
Description:Vectors are mathematical constructs that have both length and direction. We define vectors and show how to add and subtract them, and how to multiply them using the dot and cross products. We apply vectors to study the analytical geometry of lines and planes, and define the Kronecker delta and the Levi-Civita symbol to prove vector identities. Finally, we define the important concepts of scalar and vector fields.
Name:Differentiation
Description:Scalar and vector fields can be differentiated. We define the partial derivative and derive the method of least squares as a minimization problem. We learn how to use the chain rule for a function of several variables, and derive the triple product rule used in chemical engineering. We define the gradient, divergence, curl, and Laplacian. We learn some useful vector calculus identities and derive them using the Kronecker delta and Levi-Civita symbol. We use vector identities to derive the electromagnetic wave equation from Maxwell's equation in free space. Electromagnetic waves form the basis of all modern communication technologies.
Name:Integration and Curvilinear Coordinates
Description:Integration can be extended to functions of several variables. We learn how to perform double and triple integrals. We define curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, and use them to simplify problems with circular, cylindrical or spherical symmetry. We learn how to write differential operators in curvilinear coordinates and how to change variables in multidimensional integrals using the Jacobian of the transformation.
Name:Line and Surface Integrals
Description:Scalar or vector fields can be integrated over curves or surfaces. We learn how to take the line integral of a scalar field and use the line integral to compute arc lengths. We then learn how to take line integrals of vector fields by taking the dot product of the vector field with tangent unit vectors to the curve. Consideration of the line integral of a force field results in the work-energy theorem. Next, we learn how to take the surface integral of a scalar field and use the surface integral to compute surface areas. We then learn how to take the surface integral of a vector field by taking the dot product of the vector field with the normal unit vector to the surface. The surface integral of a velocity field is used to define the mass flux of a fluid through a surface.
Name:Fundamental Theorems
Description:The fundamental theorem of calculus links integration with differentiation. Here, we learn the related fundamental theorems of vector calculus. These include the gradient theorem, the divergence theorem, and Stokes' theorem. We show how these theorems are used to derive continuity equations and the law of conservation of energy. We show how to define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell's equations into differential form.
Vector Calculus for Engineers covers both basic theory and applications. In the first week we learn about scalar and vector fields, in the second week about differentiating fields, in the third week about multidimensional integration and curvilinear coordinate systems. The fourth week covers line and surface integrals, and the fifth week covers the fundamental theorems of vector calculus, including the gradient theorem, the divergence theorem and Stokes’ theorem. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics. Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. Two semesters of single variable calculus (differentiation and integration) are a prerequisite. The course is organized into 53 short lecture videos, with a few problems to solve following each video. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of five weeks to the course, and at the end of each week there is an assessed quiz. Download the lecture notes: http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Watch the promotional video: https://youtu.be/qUseabHb6Vk
