This course is part of the Mathematics for Engineers Specialization

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Jeffrey R. Chasnov

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Mathematics for Engineers SpecializationThe Hong Kong University of Science and Technology

About this Course

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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. Solutions to the problems and practice quizzes can be found in the instructor-provided lecture notes. The course spans five weeks and at the end of each week, there is an assessed quiz.

Download the lecture notes from the link
https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf

Watch the promotional video from the link
https://youtu.be/qUseabHb6Vk

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Course 3 of 5 in the

Mathematics for Engineers Specialization

Beginner Level

A course in single variable calculus

Approx. 29 hours to complete

English

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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

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Course 3 of 5 in the

Mathematics for Engineers Specialization

Beginner Level

A course in single variable calculus

Approx. 29 hours to complete

English

Could your company benefit from training employees on in-demand skills?

Try Coursera for Business

Instructor

Instructor rating

4.83/5 (388 Ratings)

Jeffrey R. Chasnov
Top Instructor

Professor

Department of Mathematics

171,706 Learners

16 Courses

Offered by

The Hong Kong University of Science and Technology

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Syllabus - What you will learn from this course

Content Rating97%(7,116 ratings)

Week1

Week 1

7 hours to complete

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.

7 hours to complete

15 videos (Total 139 min), 28 readings, 6 quizzes

15 videos

28 readings

Welcome and Course Information1mCertificate or Audit?1mHow to Write Math in the Discussions using MathJax1mAssociative Law5mTriangle Midpoint Theorem10mNewton's equation for the force between two masses10mCommutative and Distributive Properties10mDot Product between Standard Unit Vectors5mLaw of Cosines10mDo you know matrices?1mCommutative and Distributive Properties10mCross Product Between Standard Unit Vectors5mAssociative Property10mParametric Equation for a Line5mEquation for a Plane10mLevi-Civita Identities10mThe Levi-Civita Symbol and the Cross Product5mKronecker-Delta Identities5mLevi-Civita and Kronecker-Delta Identities10mOptional Parentheses5mScalar Triple Product with any Two Vectors Equal5mSwapping the Position of the Operators5mScalar Triple Product of the Unit Vectors10mJacobi Identity5mScalar Quadruple Product10mLagrange's Identity in Three Dimensions5mVector Quadruple Product5mExamples of Scalar and Vector Fields5m

6 practice exercises

Diagnostic Quiz5mVectors15mAnalytic Geometry15mVector Algebra10mWeek One Assessment (audit)Week One Assessment30m

Week2

Week 2

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 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.

5 hours to complete

13 videos (Total 122 min), 14 readings, 5 quizzes

13 videos

14 readings

Computing Partial Derivatives10mTaylor Series Expansions10mLeast-squares Method10mChain Rule10mTriple Product Rule for a Linear Function10mQuadruple Product Rule10mComputing the Gradient10mComputing the Divergence5mComputing the Curl10mThe Vorticity in Two Dimensions5mComputing the Laplacian10mVector Derivative Identities10mThe Material Acceleration10mWave Equation for the Magnetic Field10m

5 practice exercises

Partial Derivatives15mThe Del Operator15mVector Calculus Algebra15mWeek Two Assessment (audit)Week Two Assessment30m

Week3

Week 3

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. 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.

6 hours to complete

12 videos (Total 112 min), 22 readings, 5 quizzes

12 videos

Week Three Introduction1mDouble and Triple Integrals | Lecture 249mExample: Double Integral with Triangle Base | Lecture 259mPolar Coordinates (Gradient) | Lecture 2612mPolar Coordinates (Divergence and Curl) Lecture 2716mPolar Coordinates (Laplacian) |Lecture 286mCentral Force | Lecture 2914mChange of Variables (Single Integral) | Lecture 309mChange of Variables (Double Integral) | Lecture 3110mCylindrical Coordinates | Lecture 328mSpherical Coordinates (Part A) | Lecture 336mSpherical Coordinates (Part B) | Lecture 346m

22 readings

Computing the Mass of a Cube10mVolume of a surface above a parallelogram10mCartesian Unit Vectors5mCartesian Partial Derivatives10mSome Common Two-Dimensional Vectors5mComputing the Divergence and Curl in Polar Coordinates10mPipe Flow10mAngular Momentum5mVelocity Dot Acceleration10mMass of a Disk10mGaussian Integral10mDel in Cylindrical Coordinates5mDivergence of a Unit Vector5mDivergence and Curl of the Unit Vectors5mCenter-of-Mass of a Uniform Solid Cone10mSpherical and Cartesian Unit Vectors5mChange-of-variables Formula for Spherical Coordinates10mIntegrating a Function that only Depends on Distance from the Origin5mMass of a Sphere when the Density is a Linear Function10mDerivatives of the Unit Vectors5mDivergence and Curl of the Unit Vectors5mLaplacian of 1/r5m

5 practice exercises

Multidimensional Integration15mPolar Coordinates15mCylindrical and Spherical Coordinates15mWeek Three Assessment (audit)Week Three Assessment45m

Week4

Week 4

4 hours to complete

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.

4 hours to complete

9 videos (Total 75 min), 10 readings, 4 quizzes

9 videos

10 readings

Parametrization of the Curve y=y(x)5mCircumference of a Circle5mLine Integral around a Square5mLine Integral around a Circle5mMass Falling Under Gravity5mSurface Area of a Cylinder10mSurface Area of a Cone10mSurface Area of a Paraboloid10mSurface Integral over a Cylinder10mMass Flux Through a Pipe10m

4 practice exercises

Line Integrals15mSurface Integrals20mWeek Four Assessment (audit)Week Four Assessment45m

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TOP REVIEWS FROM VECTOR CALCULUS FOR ENGINEERS

by SLMay 26, 2021

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!

by GMJan 25, 2021

Excellent course for reviewing undergraduate vector calculus, including gradient, divergence, curl and Laplacian, all the way to Stokes' Theorem and Maxwell's Equations.

by NLDec 5, 2021

Great overview of Vector Calculus, I have confidence to tutor my son on this subject now. It's been many decades since I first learn the subject. Prof Chasnov made the class very clear.

by AOAug 1, 2020

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.

View all reviews

About the Mathematics for Engineers Specialization

This specialization was developed for engineering students to self-study engineering mathematics. We expect students to already be familiar with single variable calculus and computer programming. Through this specialization, students will learn matrix algebra, differential equations, vector calculus, numerical methods, and MATLAB programming. This will provide them with the tools to effectively apply mathematics to engineering problems and be well-equipped to pursue a degree in engineering. To get a better understanding of what this specialization has to offer, be sure to watch the Promotional Video!

Frequently Asked Questions

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Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I subscribe to this Specialization?
When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
Is financial aid available?
Yes. In select learning programs, you can apply for financial aid or a scholarship if you can’t afford the enrollment fee. If fin aid or scholarship is available for your learning program selection, you’ll find a link to apply on the description page.

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Vector Calculus for Engineers

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