Vector Calculus for Engineers - Hong Kong University of ...

1 Vector Calculus for EngineersLecture Notes forJeffrey R. ChasnovThe Hong Kong University of Science and TechnologyDepartment of MathematicsClear Water Bay, KowloonHong KongCopyright 2019-2022 by Jeffrey Robert ChasnovThis work is licensed under the Creative Commons Attribution Hong Kong License. To view a copy of thislicense, visit or send a letter to Creative Commons, 171 SecondStreet, Suite 300, San Francisco, California, 94105, the promotional video on YouTubeThese are the lecture notes for my online Coursera course, Vector Calculus for Engineers . Studentswho take this course are expected to already know single-variable differential and integral Calculus tothe level of an introductory college Calculus course.

2 Students should also be familiar with matrices,and be able to compute a three-by-three have divided these notes into chapters called Lectures, with each Lecture corresponding to avideo on Coursera. I have also uploaded all my Coursera videos to YouTube, and links are placed atthe top of each are some problems at the end of each lecture chapter. These problems are designed toexemplify the main ideas of the lecture. Students taking a formal University course in multivariablecalculus will usually be assigned many more problems, some of them quite difficult, but here I followthe philosophy that less is more.

3 I give enough problems for students to solidify their understandingof the material, but not so many that students feel overwhelmed. I do encourage students to attemptthe given problems, but, if they get stuck, full solutions can be found in the Appendix. I have alsoincluded practice quizzes as an additional source of problems, with solutions also ChasnovHong KongOctober 2019iiiContentsI Vectors11 Vectors32 Cartesian coordinates53 Dot product74 Cross product9 Practice quiz: Vectors115 Analytic geometry of lines136 Analytic geometry of planes15 Practice quiz: Analytic geometry177 Kronecker delta and Levi-Civita symbol198 Vector identities219 Scalar triple product2310 Vector triple product25 Practice quiz.

4 Vector algebra2711 Scalar and Vector fields29II Differentiation3112 Partial derivatives3313 The method of least squares3514 Chain rule3715 Triple product rule3916 Triple product rule (example)41 Practice quiz: Partial derivatives4317 Gradient45ivCONTENTSv18 Divergence4719 Curl4920 Laplacian51 Practice quiz: The del operator5321 Vector derivative identities5522 Vector derivative identities (proof)5723 Electromagnetic waves59 Practice quiz: Vector Calculus algebra61 III Integration and Curvilinear Coordinates6324 Double and triple integrals6525 Example: Double integral with triangle base67 Practice quiz: Multidimensional integration6926 Polar coordinates (gradient)7127 Polar coordinates (divergence & curl)7328 Polar coordinates (Laplacian)7529 Central force7730 Change of variables (single integral)7931 Change of variables (double integral)81 Practice quiz: Polar coordinates8332 Cylindrical coordinates8533 Spherical coordinates (Part A)8734 Spherical coordinates (Part B)89 Practice quiz.

5 Cylindrical and spherical coordinates91IV Line and surface integrals9335 Line integral of a scalar field9536 Arc length9737 Line integral of a Vector field99viCONTENTS38 Work-energy theorem101 Practice quiz: Line integrals10339 Surface integral of a scalar field10540 Surface area of a sphere10741 Surface integral of a Vector field10942 Flux integrals111 Practice quiz: Surface integrals113V Fundamental Theorems11543 Gradient theorem11744 Conservative Vector fields11945 Conservation of energy121 Practice quiz: Gradient theorem12346 Divergence theorem12547 Divergence theorem (example 1)12748 Divergence theorem (example 2)12949 Continuity equation131 Practice quiz: Divergence theorem13350 Green s theorem13551 Stokes theorem137 Practice quiz: Stokes theorem13952 Meaning of the divergence and the curl14153 Maxwell s equations143 Appendix145A Matrix addition and multiplication147B Matrix determinants and inverses149C Problem and practice quiz solutions151 Week IVectorsIn this week s lectures, we learn about vectors.

6 Vectors are line segments with both length and direction, andare fundamental to engineering mathematics. We will define vectors, how to add and subtract them, and howto multiply them using the scalar and Vector products (dot and cross products). We use vectors to learn someanalytical geometry of lines and planes, and introduce the Kronecker delta and the Levi-Civita symbol to provevector identities. The important concepts of scalar and Vector fields are 1 | VectorsView this lecture on YouTubeWe define a Vector in three-dimensional Euclidean space as having a length (or magnitude) and adirection.

7 A Vector is depicted as an arrow starting at one point in space and ending at another vectors that have the same length and point in the same direction are considered equal, no matterwhere they are located in space. (Variables that are vectors will be denoted in print by boldface, andin hand by an arrow drawn over the symbol.) In contrast, scalars have magnitude but no can either be a scalar or a Vector and has zero magnitude. The negative of a Vector reverses itsdirection. Examples of vectors are velocity and acceleration; examples of scalars are mass and can be added to each other and multiplied by scalars.

8 A simple example is a massmactedon by two forces 1and 2. Newton s equation then takes the formm = 1+ 2, where is theacceleration Vector of the mass. Vector addition is commutative and associative: + = + ,( + ) + = + ( + );and scalar multiplication is distributive:k( + ) =k +k .Multiplication of a Vector by a positive scalar changes the length of the Vector but not its addition can be represented graphically by placing the tail of one of the vectors on the head ofthe other. Vector subtraction adds the first Vector to the negative of the second. Notice that when thetail of and are placed at the same point, the Vector points from the head of to the headof , or equivalently, the tail of to the head of.

9 Vector additionVector subtraction34 LECTURE 1. VECTORSP roblems for Lecture graphically that Vector addition is associative, that is,( + ) + = + ( + ). vectors, prove that the line segment joining the midpoints of two sides of a triangle is parallelto the third side and half its to the ProblemsLecture 2 | Cartesian coordinatesView this lecture on YouTubeTo solve a physical problem, we usually impose a coordinate system. The familiar three-dimensionalx-y-zcoordinate system is called the Cartesian coordinate system. Three mutually perpendicularlines called axes intersect at a point called the origin, denoted as(0, 0, 0).

10 All other points in three-dimensional space are identified by their coordinates as(x,y,z)in the standard way. The positivedirections of the axes are usually chosen to form a right-handed coordinate system. When one pointsthe right hand in the direction of the positivex-axis, and curls the fingers in the direction of thepositivey-axis, the thumb should point in the direction of the Vector has a length and a direction. If we impose a Cartesian coordinate system and place thetail of a Vector at the origin, then the head points to a specific point. For example, if the Vector hashead pointing to(A1,A2,A3), we say that thex-component of isA1, they-component isA2, andthez-component isA3.