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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


This course covers both the basic theory and applications of Vector Calculus. 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 contains 53 short lecture videos, with a few problems to solve following each lecture. 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


4.8 rating, based on 228 Class Central reviews

4.8 rating at Coursera based on 1251 ratings

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
    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
    The biggest challenge for me is that I spent about half a year to finish this course, and I often have to go back to what was taught a few weeks or months ago. One thing that could help is a condensed version of the lecture notes with important formulas/notations, sort of a cheat sheet (I was trying to create one myself but lost most progress due to a software glitch, not trivial to write all the latex code). That said, the course itself is very well taught and moderately challenging. Thanks for providing such a great course online!
  • Anonymous
    Perfect course for engineering students! This course mainly teaches you how to apply vector, differentiation, and integration in engineering stuff such as 2D and 3D substances, moving motions, and some proofs of physics formulae by using vector calculus. Although you may find it difficult at the beginning or throughout the course, you will understand easier if you pay more effort on it. I strongly believe that the knowledge I learnt in this course definitely benefit my way in studying engineering in the future!
  • Anonymous
    F​or Senios´s Engineers like me, who have made their careers in other fields and want to update maths, it is an excellent course. To me, the most difficult part is the algebra of vectors since it´s not as intuitive as arithmetic, hence it needs a lot of training. I understand the course is directed to Engineers, for those like me that would like to deep their Math knowledge on the theory of Matrces and Vectors, it would be very helpful if you can advise on further online courses or texts to visit
  • 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

    Your review helps other learners like you discover great courses. Only review the course if you have taken or started taking this course.
    Your review helps other learners like you discover great courses. Only review the course if you have taken or started taking this course.
    Your review helps other learners like you discover great courses. Only review the course if you have taken or started taking this course.
  • Adán Eumir Torres Moreno
    Aprendí las caracteristicas generales incluyendo el tema cientifico de "Cálculo Vectorial". Y los resolvi intentando lo más que pude.
  • 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.
  • Anonymous was great , thank you
  • Anonymous
    This course is incredibly well done. It filled in all sorts of gaps from my MIT/Harvard education. The structure and pace of the course is done is such a way each topic can be learned and mastered before moving on to the next topic.
  • 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
    It made a great contribution to me. I think that it has increased my knowledge of calculus at an advanced level, even though it is complicated in some topics which helped me to understand the information better.
  • Anonymous
    All the math is well-explained, and the physical applications (particularly to fluid mechanics and electricity/magnetism) made the math less abstract and easier to understand.
  • 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
    Another great course from the Math for Eng specialization at coursera. The concepts are well explained, and the exercises allow to complement and assimilate the theory.
  • Anonymous
    This is an excellent course! I highly recommend his to anyone who wants to learn vector calculus for the first time, or to anyone who just wants to review the material.
  • Anonymous
    Excellent explanation,new learning materials,nice examples .If you are keen its easy to understand and achieve your goal
  • Anonymous
    I cover most of the information you need to understand and apply multivariate calculus to engineering problems.

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