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The Hong Kong University of Science and Technology

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.

Download the lecture notes:

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  • 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. 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 204 reviews

Start your review of Vector Calculus for Engineers

  • Profile image for Chris Harding
    Chris Harding
    Although I earned a BS degree in chemical engineering in 1999 and have taken multivariable calculus, Professor Jeffrey Chasnov’s Vector Calculus for Engineers was a great challenging learning process. I found the time needed to complete the course could...
  • Anonymous
    This course covers all essential concept of partial, line and surface integral, gradient, divergence, curl, laplacian which are the useful mathematical tools for convert the abstraction of physics theory to nice, able to evaluated equations.

    Overall, the content of this course is more difficult than the general conception of matrix algebra and differential equation course , the formulas is complicated and its application is abstract and theoretical. It takes more time to digest these new knowledge! More challenging more attractive of the world of mathematics. Worth your time to enroll this course!
  • Anonymous
    Starting from the very basic, the course takes to the advanced concepts on Vector calculus. I took this course as a refresher and found it very helpful. The large number of reading problems helped strengthen the understanding. For some topics, when the professor mentions something but doesn't go at length to explain, some secondary complementary resources could be useful. I used khan academy videos to fill in the gaps.
  • Profile image for Jorge Luis Dominguez Martinez
    Jorge Luis Dominguez Martinez
    It is an outstanding course based on five weeks. It includes all basics you need to know to be involved in Vector Calculus. From Vectors, Operators, Differentiation, Integration (Line and Surface integrals), Curvilinear Coordinates, and Fundamental Theorems. Also, it provides a book with detailed information on each topic. So then, Jeffrey Chasnov thanks for this amazing journey.
  • 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...
  • 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...
  • Anonymous
    Professor Chasnov, Thank you very much for such awesome courses. You helped my career very much! I changed the direction of my career a little bit (from construction to the engineering seismology), but new area is related to vectors, matrices and differential...
  • Anonymous
    This is an axcellent course with illustrative examples and relevant calculation algorythms, it is a complex introduction to vector calculus, however, it is not for the faint of heart. Unless you have a very solid foundation in mathematics (unlike me),...
  • 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...
  • Lisa Frey
    I think this class was extremely well taught, but recommend more specific examples that tie in with the theory. Admittedly, I have been out of school for some time now and for this reason, this class was challenging to me. A very good cover of topics, however, and I found it to prove me with the opportunity to refresh my knowledge. Thank you for the opportunity to take this class with such a professional and knowledgeable professor. I hope to take another class with him sometime in the near future.
  • 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
    One of the best courses with the best teaching method from Professor Chasnov. Thank so much Professor Chasnov and Coursera for giving me the opportunity to learn such difficult concepts of vector calculus. I am pretty sure that these concepts would be really handy in learning new concepts such as the displacement field of laminated composites with defects, potential fields for smart materials, magnetic fields for energy harvesting.
  • 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
    Professor Chasnov is excellent at explaining and very organized. His great enthusiasm is noticeable when he is explaining each topic. The course presents all the basic topics. In the last few weeks I felt that the contents are very superficial, so to solve the assigned problems it is necessary to practice a little more. Thank you for all professor Jeffrey Chasnov.
  • Anonymous
    THIS Is a great fundamental course, before taking this course, I always struggled to read the equations in fluid mechanics, and this course helps me to review some of the basic vector calculus, after finishing all the course materials and all the practice problems, I've gained more confidence to read and conduct some fluid mechanics equations.
  • Prashant Kanuru
    Gets in Deep into abstractions. The 3rd and 4th week are deep but great. I would be happy if some more depth in terms of optimization is added up through Lagrange Multipliers and if some more depth by bringing in Vector Spaces is done, would be great.

    All The Best!!!
  • 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
    The course is really good. Yet, there aren't enough practical problems from different engineering disciplines. I expected the course to be tackling more applications to consolidate the material taught. but in general it is a really good course
  • Anonymous
    The course is designed good .It helps the engineering students .It is very helpful to me.I am retired Maths teacher.I know from course a new idia.

    If you give more examples that's better for students.

    I really enjoy.

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