Vector Calculus for Engineers
        Lecture Notes for
         Jeffrey R. Chasnov
               The Hong Kong University of Science and Technology
                     Department of Mathematics
                     Clear Water Bay, Kowloon
                        HongKong
                Copyright ©2019-2022 by Jeffrey Robert Chasnov
     This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view
     a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to
     Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
    Preface
    View the promotional video on YouTube
    These are the lecture notes for my online Coursera course, Vector Calculus for Engineers.
    Students who take this course are expected to already know single-variable differential
    and integral calculus to the level of an introductory college calculus course. Students
    should also be familiar with matrices, and be able to compute a three-by-three determi-
    nant.
      I have divided these notes into chapters called Lectures, with each Lecture correspond-
    ing to a video on Coursera. I have also uploaded all my Coursera videos to YouTube, and
    links are placed at the top of each Lecture.
      There are some problems at the end of each lecture chapter. These problems are
    designed to exemplify the main ideas of the lecture. Students taking a formal university
    course in multivariable calculus will usually be assigned many more problems, some of
    them quite difficult, but here I follow the philosophy that less is more. I give enough
    problems for students to solidify their understanding of the material, but not so many
    that students feel overwhelmed. I do encourage students to attempt the given problems,
    but, if they get stuck, full solutions can be found in the Appendix. I have also included
    practice quizzes as an additional source of problems, with solutions also given.
                                   Jeffrey R. Chasnov
                                       HongKong
                                      October 2019
        Contents
        I   Vectors                                                                   1
        1  Vectors                                                                    2
        2  Cartesian coordinates                                                      4
        3  Dot product                                                                6
        4  Cross product                                                              8
           Practice quiz: Vectors                                                    10
        5  Analytic geometry of lines                                                11
        6  Analytic geometry of planes                                               13
           Practice quiz: Analytic geometry                                          15
        7  Kronecker delta and Levi-Civita symbol                                    16
        8  Vector identities                                                         18
        9  Scalar triple product                                                     20
        10 Vector triple product                                                     22
           Practice quiz: Vector algebra                                             24
        11 Scalar and vector fields                                                   25
        II   Differentiation                                                         27
        12 Partial derivatives                                                       28
        13 The method of least squares                                               30
        14 Chain rule                                                                32
        15 Triple product rule                                                       34
        16 Triple product rule (example)                                             35
           Practice quiz: Partial derivatives                                        37
        17 Gradient                                                                  38
                                               iv