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Online Course

Vector Calculus for Engineers

The Hong Kong University of Science and Technology via Coursera


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.

Lecture notes can be downloaded from


-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.

-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 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 of all modern communication technologies.

Integration and Curvilinear Coordinates
-Integration can be extended to functions of several variables. We learn how to perform double and triple integrals. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used 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.

Line and Surface Integrals
-Scalar or vector fields can be integrated on curves or surfaces. We learn how to take the line integral of a scalar field and use line integrals 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 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 the surface.

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, derive the law of conservation of energy, define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell's equations into their more aesthetically pleasing differential form.

Taught by

Jeffrey R. Chasnov

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4.8 rating, based on 170 reviews

Start your review of Vector Calculus for Engineers

  • Anonymous

    Anonymous completed this course.

    I can only deliver a mixed review. The course presents a generous amount of material, and all the basics are covered, but the presentation, especially in the final week, is perfunctory at best, grinding through derivations and leaving many steps for the...
  • Syed Murtaza Jaffar

    Syed Murtaza Jaffar completed this course.

    This indeed is one of the BEST courses in Vector Calculus with the BEST instructor teaching it. Professor Chasnov is highly organized and presents the contents in a clear manner. I have become fond of his excellent teaching style. Over and above, all engineers must take this course. I hope he teaches courses in PDEs, Integral Transforms, Complex Variables, ... in times to come to benefit the motivated mathematics learners all around the globe! This is terrific effort from him. I wish the best comes his way as a reward for his dedication. God bless.
  • Anonymous

    Anonymous completed this course.

    Week three is the pivotal week for learning that I struggled with. Line and Surface integrals just did not come easy to me. A tutorial on the line and surface integrals in greater depth would have helped me since it is difficult to visualize what these always mean. The instruction was excellent, but I feel I needed extra help. Would love to take a course in just line and surface integrals.

    An extremely valuable course for anyone in physics or engineering. Take it as soon as you can.
  • Anonymous
    our professors explanation and command on the subject is very high. sir, has helped me in understanding physics concepts in a much simpler way.this course is very useful for both mathematics and physics students.In short duration it could cover all areas of vector calculus I request sir to include more no. of problems and solve them so that students can be confident applying the concepts of vector calculus in solving problems related to electromagnetic waves and transmission lines
  • Anonymous

    Anonymous completed this course.

    Finished all the course in about 2 weeks. It is very good if you want to refresh your memory on vector calculus(in my case). If you want a solid foundation, then you should supplement it with lots of more examples from some textbook(s). Otherwise, things are explained very well, and the examples are not too difficult to scare you away! Great course.
  • Anonymous
    A great refresher course if you already know vector calculus and would like to take a cursory glance to brush up the concepts. I didn't have the in-depth knowledge of the topic but tackling it on your own can at first seem daunting. It had been something...
  • Dale K Garman
    My review here isn't so much about this particular course. Instead, it is about the instructor Jeff Chasnov. I have already taken 4 courses through him on the Coursera platform: Differential Equations for Engineers, Matrix Algebra for Engineers, Vector...
  • Anonymous
    This Course is helpful for more knowledge about Vector.

    This course easy to understand and always helps us☺️
  • Anonymous

    Anonymous completed this course.

    Prof. Jeff has one of the best MOOCs out there for Vector Calculus, although it might not seem like it at first. A gem hidden in the rough, all in all, although I'd say that the latter two weeks were far from being as easy as the first two. I'd recommend...
  • Anonymous
    I am a materials engineering undergrad from IIT Bhubaneswar, India. I am in the 5th semester. I took this course to revise the vector calculus I learnt in my 1st semester of engineering. This course was very well balanced and conceptual. I got to know...
  • Anonymous
    As to a review any such assessment of a course turns upon what one had in terms of objectives for selecting the course. In my case the objectives were attained although I did not complete all of the exercises. I was able to do so but it didn't seem worth...
  • Anonymous

    Anonymous completed this course.

    The content is fairly comprehensive and along the way you get to revise things you may have learnt elsewhere and you will also be challenged. The content is for the most part very practical and straightforward but some more optional practice problems...
  • Anonymous

    Anonymous completed this course.

    Professor Jeff explained well the topics covered. I learned a lot from him especially on the application of the Kronecker delta and levi-civita identities. All of the topics discussed on this course are very important when handling subjects like engineering...
  • Anonymous

    Anonymous completed this course.

    This course is really helpful for someone who wants an easy approach for vector calculus. The included exercises have few questions each, but really test your concept application. Thus, they are not stressful, while remaining worthy of solving. As an...
  • Profile image for Jim Campbell
    Jim Campbell
    This is a great refresher and introduction to calculus. Professor Jeff Chasnov speaks clear English and formatted the course well. The notes accompany the lecture so multiple reviews are available. I had been away from introductory Calculus 35-40 years...
  • Anonymous
    Many years ago I had a course in Electromagnetism as part of my physics degree. The course was saturated with vector calculus, but I hadn't any training in that. Accordingly, I had to pick up tricks/procedures to get through the course. I passed the...
  • Anonymous

    Anonymous completed this course.

    I took the course as complementary material to my vector calculus class, since I felt many things were skipped and not talked about in my class. I enjoyed that the course encourages you to do derivations and practice by yourself, however, I felt certain...
  • Anonymous
    This is one of the best courses I have taken so far. Not only the subject is essential in many scientific areas, but the course is extremely organized, facilitating follow-up. In my opinion, a few details that make this course one of the best are: (1) the introductory videos, giving a broader view of the following content; (2) the use of examples to explain the contents; (3) the review at the end of each video; (4) the PDF file available, presenting the content of the videos, exercises and solutions; and (5) the lenght of the videos, which are short enough not to make them exausting. In addition, this course is useful beyond engineering. I am actually a PhD student in chemistry and I have benefited a lot from this course!
  • Anonymous

    Anonymous completed this course.

    My experience during the progress of this course has been enlightening and refreshing. The words used were very relatable, and the mode of teaching was also commendable. It has been a highly exciting journey, and I am proud of what I have learnt.
    Vector Calculus for Engineers on Coursera superseded my expectations because I have had some lessons on vectors and I have not been so enlightened as I am. My work here has been a very fantastic journey and I sincerely just want to say that Coursera should keep up the excellent work. It has been a tremendous journey with you, and I hope to experience more like this as I progress.
  • Profile image for Eftekhar Sadik
    Eftekhar Sadik
    Well, before joining this course , my knowledge in vector calculus was not mentionable. But I have learnt pretty much good stuffs in this course and obviously it is going to help me a lot , in my engineering career. Specifically, to say I have enjoyed the use of divergence theorem and the stokes theorem in Maxwell's equation . I kind of knew about the Maxwell's equation but I was totally unware of the fact that it can be converted to the differential form from the integral form . And , above all , truly to speak ,all the learnings boosted up my interest in vector calculus .Thank you , Sir , for your valuable knowledge.

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