Vector Calculus for Engineers
About This Course
We cover 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 integrating fields. The fourth 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 42 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 four weeks to the course, and at the end of each week there is an assessed quiz. Lecture notes can be downloaded from http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf
Course Syllabus
Vectors
-A vector is a mathematical construct that has both length and direction. We will define vectors and learn how to add and subtract them, and how to multiply them using the scalar and vector products (dot and cross products). We will use vectors to learn some analytical geometry of lines and planes, and learn about the Kronecker delta and the Levi-Civita symbol to prove vector identities. The important concepts of scalar and vector fields will be introduced.
Differentiation
-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. From the del differential operator, we define the gradient, divergence, curl and Laplacian. We learn some useful vector calculus identities and how to derive them using the Kronecker delta and Levi-Civita symbol. Vector identities are then used to derive the electromagnetic wave equation from Maxwell’s equation in free space. Electromagnetic waves form the basis for all modern communication technologies.
Integration and Curvilinear Coordinates
-Scalar and vector fields can be integrated. We learn about double and triple integrals, and line integrals and surface integrals. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used to simplify problems with cylindrical or spherical symmetry. We learn how to change variables in multidimensional integrals using the Jacobian of the transformation.
Fundamental Theorems
-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, define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell’s equations into their more famous differential form.
Course presenter
Jeffrey R. Chasnov
Professor
Bio
Jeffrey R. Chasnov is a Professor of Mathematics at the Hong Kong University of Science and Technology, where he has been teaching since 1993. He is an expatriate American from New York and California, and earned his BA from UC Berkeley and PhD from Columbia University, with post-doctoral fellowships at NASA, Stanford, and Grenoble, France, and a sabbatical semester at Harvey Mudd College. He is the author of numerous research articles in fluid turbulence and mathematical biology, and has authored online textbooks and videos for his courses on differential equations, matrix algebra, mathematical biology and scientific computation. Before and after work, he enjoys family life, swimming and tennis, and takes great pleasure in his family’s annual camping and skiing vacations.
FAQ
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Coursera is the most popular MOOC provider in the world based on the number of students (over 45 million learners) and has an active catalog of 3,800+ online courses.
As well as these individual courses and 16 online degrees, Coursera offers 400 groups of courses known as Specializations, MasterTracks, and Professional Certificates.
How do Coursera courses work?
Coursera is an online education provider that offers online courses, popularly known as MOOCs or Massive Open Online Courses, from top universities around the world. Currently, Coursera boasts an active catalog of more than 3800 online courses created by these partner institutions.
Coursera courses consist of pre-recorded video lectures that you can watch on a weekly schedule or when it’s convenient for you. They also have student discussion forums, homework/assignments, and online quizzes or exams.
Generally speaking, Coursera courses are free to audit (i.e. watch videos) but if you want to earn a course Certificate, you will need to pay. An investigation by Class Central discovered 1150+ Coursera courses that include assessments for free although other courses have the assessments behind a paywall.
Many of the Coursera courses are part of Coursera Specializations, a microcredential offered by Coursera. Specializations consist of a sequence of courses and for some Specializations the last course is a Capstone project. If learners earn a certificate for each course in a Specialization, they will receive a Specialization certificate. Specializations are usually geared towards in-demand skills in business and technology. You can take single courses or the whole specialization.
Other courses are grouped into MasterTracks and Professional Certificates. Coursera has launched over 400 microcredentials.
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