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Vector Calculus for Engineers
  • Level Foundation
  • المدة 30 ساعات hours
  • الطبع بواسطة The Hong Kong University of Science and Technology
  • Offered by

عن

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 https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf Watch the promotional video from the link https://youtu.be/qUseabHb6Vk

الوحدات

Welcome

    • 1 Videos
    • 3 Readings
    • 1 Assignment
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1 Assignment

  • Diagnostic Quiz

1 Videos

  • Course Overview

3 Readings

  • Welcome and Course Information
  • Certificate or Audit?
  • How to Write Math in the Discussions using MathJax

Introduction to Week One

    • 1 Videos
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1 Videos

  • Week One Introduction

Vectors

    • 4 Videos
    • 10 Readings
    • 1 Plugin
    • 1 Assignment
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1 Assignment

  • Vectors

4 Videos

  • Vectors | Lecture 1
  • Cartesian Coordinates | Lecture 2
  • Dot Product | Lecture 3
  • Cross Product | Lecture 4

10 Readings

  • Associative Law
  • Triangle Midpoint Theorem
  • Newton's equation for the force between two masses
  • Commutative and Distributive Properties
  • Dot Product between Standard Unit Vectors
  • Law of Cosines
  • Do you know matrices?
  • Commutative and Distributive Properties
  • Cross Product Between Standard Unit Vectors
  • Associative Property

Analytic Geometry

    • 2 Videos
    • 2 Readings
    • 1 Assignment
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1 Assignment

  • Analytic Geometry

2 Videos

  • Analytic Geometry of Lines | Lecture 5
  • Analytic Geometry of Planes | Lecture 6

2 Readings

  • Parametric Equation for a Line
  • Equation for a Plane

Vector Algebra

    • 4 Videos
    • 12 Readings
    • 1 Plugin
    • 1 Assignment
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1 Assignment

  • Vector Algebra

4 Videos

  • Kronecker Delta and Levi-Civita Symbol | Lecture 7
  • Vector Identities | Lecture 8
  • Scalar Triple Product | Lecture 9
  • Vector Triple Product | Lecture 10

12 Readings

  • Levi-Civita Identities
  • The Levi-Civita Symbol and the Cross Product
  • Kronecker-Delta Identities
  • Levi-Civita and Kronecker-Delta Identities
  • Optional Parentheses
  • Scalar Triple Product with any Two Vectors Equal
  • Swapping the Position of the Operators
  • Scalar Triple Product of the Unit Vectors
  • Jacobi Identity
  • Scalar Quadruple Product
  • Lagrange's Identity in Three Dimensions
  • Vector Quadruple Product

Fields

    • 1 Videos
    • 1 Readings
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1 Videos

  • Scalar and Vector Fields | Lecture 11

1 Readings

  • Examples of Scalar and Vector Fields

Supplemental Videos

    • 2 Videos
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2 Videos

  • Matrix Addition and Multiplication
  • Matrix Determinants and Inverses

Quiz

    • 2 Assignment
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2 Assignment

  • Week One Assessment (audit)
  • Week One Assessment

Introduction to Week Two

    • 1 Videos
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1 Videos

  • Week Two Introduction

Partial Derivatives

    • 5 Videos
    • 6 Readings
    • 1 Assignment
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1 Assignment

  • Partial Derivatives

5 Videos

  • Partial Derivatives | Lecture 12
  • The Method of Least Squares | Lecture 13
  • Chain Rule | Lecture 14
  • Triple Product Rule | Lecture 15
  • Triple Product Rule: Example | Lecture 16

6 Readings

  • Computing Partial Derivatives
  • Taylor Series Expansions
  • Least-squares Method
  • Chain Rule
  • Triple Product Rule for a Linear Function
  • Quadruple Product Rule

The Del Operator

    • 4 Videos
    • 6 Readings
    • 1 Assignment
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1 Assignment

  • The Del Operator

4 Videos

  • Gradient | Lecture 17
  • Divergence | Lecture 18
  • Curl | Lecture 19
  • Laplacian | Lecture 20

6 Readings

  • Computing the Gradient
  • The Gradient of the Position Vector
  • Computing the Divergence
  • Computing the Curl
  • The Vorticity in Two Dimensions
  • Computing the Laplacian

Vector Calculus Algebra

    • 3 Videos
    • 3 Readings
    • 1 Assignment
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1 Assignment

  • Vector Calculus Algebra

3 Videos

  • Vector Derivative Identities | Lecture 21
  • Vector Derivative Identities (Proof) | Lecture 22
  • Electromagnetic Waves | Lecture 23

3 Readings

  • Vector Derivative Identities
  • The Material Acceleration
  • Wave Equation for the Magnetic Field

Quiz

    • 2 Assignment
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2 Assignment

  • Week Two Assessment (audit)
  • Week Two Assessment

Introduction to Week Three

    • 1 Videos
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1 Videos

  • Week Three Introduction

Multidimensional Integration

    • 2 Videos
    • 2 Readings
    • 1 Assignment
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1 Assignment

  • Multidimensional Integration

2 Videos

  • Double and Triple Integrals | Lecture 24
  • Example: Double Integral with Triangle Base | Lecture 25

2 Readings

  • Computing the Mass of a Cube
  • Volume of a surface above a parallelogram

Polar Coordinates

    • 6 Videos
    • 9 Readings
    • 1 Assignment
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1 Assignment

  • Polar Coordinates

6 Videos

  • Polar Coordinates (Gradient) | Lecture 26
  • Polar Coordinates (Divergence and Curl) Lecture 27
  • Polar Coordinates (Laplacian) |Lecture 28
  • Central Force | Lecture 29
  • Change of Variables (Single Integral) | Lecture 30
  • Change of Variables (Double Integral) | Lecture 31

9 Readings

  • Cartesian Unit Vectors
  • Cartesian Partial Derivatives
  • Some Common Two-Dimensional Vectors
  • Computing the Divergence and Curl in Polar Coordinates
  • Pipe Flow
  • Angular Momentum
  • Velocity Dot Acceleration
  • Mass of a Disk
  • Gaussian Integral

Cylindrical and Spherical Coordinates

    • 3 Videos
    • 13 Readings
    • 1 Assignment
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1 Assignment

  • Cylindrical and Spherical Coordinates

3 Videos

  • Cylindrical Coordinates | Lecture 32
  • Spherical Coordinates (Part A) | Lecture 33
  • Spherical Coordinates (Part B) | Lecture 34

13 Readings

  • Del in Cylindrical Coordinates
  • Divergence of a Unit Vector
  • Divergence and Curl of the Unit Vectors
  • Center-of-Mass of a Uniform Solid Cone
  • Spherical and Cartesian Unit Vectors
  • Change-of-variables Formula for Spherical Coordinates
  • Integrating a Function that only Depends on Distance from the Origin
  • Mass of a Sphere when the Density is a Linear Function
  • Derivatives of the Unit Vectors
  • Divergence and Curl of the Unit Vectors
  • Laplacian of a Vector Field in Spherical Coordinates
  • Laplacian of 1/r
  • Laplacian of a Vector Field with Inverse Square Law

Quiz

    • 2 Assignment
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2 Assignment

  • Week Three Assessment (audit)
  • Week Three Assessment

Introduction to Week Four

    • 1 Videos
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1 Videos

  • Week Four Introduction

Line Integrals of Scalar Fields

    • 2 Videos
    • 3 Readings
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2 Videos

  • Line Integral of a Scalar Field | Lecture 35
  • Arc Length | Lecture 36

3 Readings

  • Circumference of a Circle
  • Computing the Mass of a Wire
  • Approximating the Perimeter of an Ellipse

Line Integrals of Vector Fields

    • 2 Videos
    • 3 Readings
    • 1 Assignment
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1 Assignment

  • Line Integrals

2 Videos

  • Line Integral of a Vector Field | Lecture 37
  • Work-Energy Theorem | Lecture 38

3 Readings

  • Line Integral around a Square
  • Line Integral around a Circle
  • Mass Falling Under Gravity

Surface Integrals of Scalar Fields

    • 2 Videos
    • 3 Readings
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2 Videos

  • Surface Integral of a Scalar Field | Lecture 39
  • Surface Area of a Sphere | Lecture 40

3 Readings

  • Surface Area of a Cylinder
  • Surface Area of a Cone
  • Surface Area of a Paraboloid

Surface Integrals of Vector Fields

    • 2 Videos
    • 2 Readings
    • 1 Assignment
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1 Assignment

  • Surface Integrals

2 Videos

  • Surface Integral of a Vector Field | Lecture 41
  • Flux Integrals | Lecture 42

2 Readings

  • Surface Integral over a Cylinder
  • Mass Flux Through a Pipe

Quiz

    • 2 Assignment
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2 Assignment

  • Week Four Assessment (audit)
  • Week Four Assessment

Introduction to Week Five

    • 1 Videos
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1 Videos

  • Week Five Introduction

Gradient

    • 3 Videos
    • 3 Readings
    • 1 Assignment
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1 Assignment

  • Gradient Theorem

3 Videos

  • Gradient Theorem | Lecture 43
  • Conservative Vector Fields | Lecture 44
  • Conservation of Energy | Lecture 45

3 Readings

  • Gradient Theorem
  • Conservative Vector Fields
  • Escape Velocity

Divergence

    • 4 Videos
    • 8 Readings
    • 1 Assignment
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1 Assignment

  • Divergence Theorem

4 Videos

  • Divergence Theorem | Lecture 46
  • Divergence Theorem: Example I | Lecture 47
  • Divergence Theorem: Example II | Lecture 48
  • Continuity Equation | Lecture 49

8 Readings

  • Divergence Theorem for a Sphere
  • Test the Divergence Theorem for a Cube
  • Divergence Theorem for a Cube
  • Test the Divergence Theorem for a Sphere
  • Flux Integral of the Position Vector
  • Source Flow
  • Continuity Equation
  • Electrodynamics Continuity Equation

Curl

    • 2 Videos
    • 5 Readings
    • 1 Assignment
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1 Assignment

  • Stokes' Theorem

2 Videos

  • Green's Theorem | Lecture 50
  • Stokes' Theorem | Lecture 51

5 Readings

  • Test Green's Theorem for a Square
  • Test Green's Theorem for a Circle
  • Stokes' Theorem in Two Dimensions
  • Test Stokes' Theorem
  • Point Vortex

Applications

    • 2 Videos
    • 3 Readings
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2 Videos

  • Meaning of the Divergence and the Curl | Lecture 52
  • Maxwell's Equations | Lecture 53

3 Readings

  • The Navier-Stokes Equation
  • Electric Field of a Point Charge
  • Magnetic Field of a Wire

Quiz

    • 2 Assignment
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2 Assignment

  • Week Five Assessment (audit)
  • Week Five Assessment

Farewell

    • 1 Videos
    • 2 Readings
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1 Videos

  • Concluding Remarks

2 Readings

  • Please Rate this Course
  • Acknowledgements
Jeffrey R. Chasnov
Instructor

Jeffrey R. Chasnov