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 Stokesâ€™ theorem. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics.

This course is part of the Mathematics for Engineers Specialization

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## About this Course

A course in single variable calculus

## What you will learn

Vectors, the dot product and cross product

The gradient, divergence, curl, and Laplacian

Multivariable integration, polar, cylindrical and spherical coordinates

Line integrals, surface integrals, the gradient theorem, the divergence theorem and Stokes' theorem

A course in single variable calculus

## Offered by

### The Hong Kong University of Science and Technology

HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world.

## Syllabus - What you will learn from this course

**7 hours to complete**

## 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.

**7 hours to complete**

**5 hours to complete**

## 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.

**5 hours to complete**

**6 hours to complete**

## 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.

**6 hours to complete**

**4 hours to complete**

## 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.

**4 hours to complete**

## Reviews

- 5 stars83.41%
- 4 stars14.40%
- 3 stars1.13%
- 2 stars0.64%
- 1 star0.40%

### TOP REVIEWS FROM VECTOR CALCULUS FOR ENGINEERS

Professor Chasnov is a great instructor. I strongly recommend this course (and others from his). Thank you so much for making such great quality content available for everyone no matter where.

Great course all around. Just one suggestion would be to solve at least 1 example of each type of theorem if possible as it makes problem solving much easier.

Well taught, high-paced. It needs effort to understand the material but there is nothing wrong with that. Much clearer than in my first degree, which was many years ago!

This course is very well organized and well explained. I am very much thankful to Prof Jeffrey R. Chasnov for his fruitful videos which help us to update our knowledge in this area.

## About the Mathematics for Engineers Specialization

This specialization was developed for engineering students to self-study engineering mathematics. We expect students are already familiar with single variable calculus and computer programming. Students will learn matrix algebra, differential equations, vector calculus and numerical methods. MATLAB programming will be taught. Watch the promotional video!

## Frequently Asked Questions

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