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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 theoretical foundations and practical applications of Vector Calculus. During the first week, students will learn about scalar and vector fields. In the second week, they will differentiate fields. The third week focuses on multidimensional integration and curvilinear coordinate systems. Line and surface integrals are covered in the fourth week, while the fifth week explores the fundamental theorems of vector calculus, including the gradient theorem, the divergence theorem, and Stokes' theorem. These theorems are essential for subjects in engineering such as Electromagnetism and Fluid Mechanics.

Note that this course may also be referred to as Multivariable or Multivariate Calculus or Calculus 3 at some universities. A prerequisite for this course is two semesters of single variable calculus (differentiation and integration).

The course includes 53 concise lecture videos, each followed by a few problems to solve. After each major topic, there is a short practice quiz. At the end of each week, there is an assessed quiz. Solutions to the problems and practice quizzes can be found in the instructor-provided lecture notes.

Download the lecture notes from the link

Watch the promotional video from the link


  • Vectors
    • 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.
  • 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 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.
  • Integration and Curvilinear Coordinates
    • 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.
  • Line and Surface Integrals
    • 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.
  • 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 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.

Taught by

Jeffrey R. Chasnov


4.8 rating, based on 243 Class Central reviews

4.8 rating at Coursera based on 1312 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…
  • Anonymous
    The course on Vector Calculus was indeed a great challange to me but at the end of the day i have acquired a new and profound knowladge on the course as far as engineering is concerned. He is an outstanding Prof. I thought online learning might be difficult base on the fact that the lecturer might not break down the explanation to everyone. But the Prof explanation was so good to me and i think for many others. I have really improve in my mathematic knowladge of on differential equations mostly on the second derivative which i usually have issues.
  • Anonymous
    Este curso me ha ayudado a reforzar mis conocimientos de cálculo vectorial, ya que soy académica de universidad y doy cursos de física y matemáticas. También me ayudo a ver otras formas de enseñar.
  • Anonymous
    Excellent course. The material is presented in an accessible, logical way. Interesting tasks and tests. You can study at your own pace. Super course.
  • Anonymous
    This course, as well as the others in this series, was great. Prof. Chasnov is a fantastic teacher, and explained everything very well.
  • 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 some…
  • 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
    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…
  • 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.
  • 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 was a good course and it surely helped clear my Fluid Mechanics and Multi-variable Calculus. As a mechanical engineering student, I was pleasantly surprised by the depth and practicality of the course content. This course, without a doubt, deserves a five-star rating for its outstanding quality. The video lectures are engaging, and the instructors' passion for the subject matter shines through their teaching. Additionally, the course materials provided, including lecture notes, supplementary readings, and problem sets, are well-organized and greatly contribute to the learning experience.
  • Anonymous
    It is challenging, but it is great helpful. there are many lectures I have watched several times.
    From the lecture to the practice problem design, and the solution provided, very well organised.
    If I find something I can not understand, I usually check carefully to find the solution.
    Highly recommend Jeff's other courses for engineers. Actually, I start with the numerical method for engineers and find this one helps a lot in understanding the concepts in numerical simulation.
    This series course helps me strengthen the foundation of mathematics. I do appreciate it.
  • USG
    It covers the fundamentals of vector calculus and provide very useful lecture notes containing the topics explained in the on line lessons. Generally very clear explanations of the arguments, to have at least a glimpse of the matter. There is a very very good "companion" course - "Matrix Algebra for Engineers" - from the same teacher. Thanks for your free courses.
  • Anonymous
    El curso Vector Calculus for Engineers dictado por el profesor Jeffrey R. Chasnov , para mi criterio fue excelente, en el cual la teoria y las aplicaciones de la diferenciación e integración desde el punto de vista vectorial, la transformación de coodenadas rectangulares a cilindricas o esféricas es imprescindible para analizar las integrales de línea y de superficie, los teoremas de Stokes - Gradiente con las respectivas ecuaciones de Navier-Stokes, Maxwell, etc, que nos detallan la importancia del Cálculo Vectorial en las diversas áreas de las Ciencias Físicas. Pero se requiere de conceptos claros de la derivada e integración con la terminología matemática respectiva.
  • Anonymous
    Өте керемет курс. Менің өзіме қатты ұнады. Алдағы уақытта басқа да студенттерге осы курсқа қатысу керек екеніне кеңес беріп жүретін боламын.
  • Anonymous
    Барлығы өте керемет сабақтар дақсы түсіндірілген. Өзіме қатты ұнады. Джеффрей профессор өте жақсы кісі екен. Керемет түсіндіреді
  • Anonymous
    its pretty awesome to see the applications tha it goes, the teacher explains very well and i all of the terms its a good course, i personally don't like the demostrations that much but its very satisfying when u complete a proof.
  • Anonymous
    The material covers a large part of vector calculus. Very useful is the use of the Kronecker and Levi-Civita symbols for vector identities. Highly recommended.
  • Anonymous
    Very good class. It help me refresh some old calculus 2 & 3 material. It also help me with polar and spherical coordinate systems

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