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Vector Calculus for Engineers

Approx. 28 hours to complete

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Course Summary

This course teaches Vector Calculus, an essential mathematical tool for engineers, scientists, and mathematicians. Students will learn how to apply vector calculus to real-world problems.

Key Learning Points

Vector calculus is a powerful tool for solving complex engineering problems
This course covers topics such as vector fields, line integrals, and surface integrals
Students will learn how to apply vector calculus in a variety of real-world scenarios

Learning Outcomes

Understand the principles and applications of vector calculus
Be able to apply vector calculus to solve real-world engineering problems
Develop critical thinking and problem-solving skills

Prerequisites or good to have knowledge before taking this course

Calculus I and II
Familiarity with linear algebra

Course Difficulty Level

Intermediate

Course Format

Online
Self-paced
Video Lectures

Similar Courses

Differential Equations for Engineers
Numerical Methods for Engineers

Notable People in This Field

Grant Sanderson
Salman Khan

Related Books

Description

Vector Calculus for Engineers covers both basic theory and applications. 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 Stokes’ theorem. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics.

Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. Two semesters of single variable calculus (differentiation and integration) are a prerequisite. The course is organized into 53 short lecture videos, with a few problems to solve following each video. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of five weeks to the course, and at the end of each week there is an assessed quiz. Download the lecture notes: http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Watch the promotional video: https://youtu.be/qUseabHb6Vk

Knowledge

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

Outline

Vectors
Promotional Video
Course Overview
Week One Introduction
Vectors | Lecture 1
Cartesian Coordinates | Lecture 2
Dot Product | Lecture 3
Cross Product | Lecture 4
Analytic Geometry of Lines | Lecture 5
Analytic Geometry of Planes | Lecture 6
Kronecker Delta and Levi-Civita Symbol | Lecture 7
Vector Identities | Lecture 8
Scalar Triple Product | Lecture 9
Vector Triple Product | Lecture 10
Scalar and Vector Fields | Lecture 11
Matrix Addition and Multiplication
Matrix Determinants and Inverses
Welcome and Course Information
How to Write Math in the Discussions using MathJax
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
Parametric Equation for a Line
Equation for a Plane
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
Jacobi Identity
Scalar Quadruple Product
Lagrange's Identity in Three Dimensions
Vector Quadruple Product
Examples of Scalar and Vector Fields
Diagnostic Quiz
Vectors
Analytic Geometry
Vector Algebra
Week One Assessment

Differentiation
Week Two Introduction
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
Gradient | Lecture 17
Divergence | Lecture 18
Curl | Lecture 19
Laplacian | Lecture 20
Vector Derivative Identities | Lecture 21
Vector Derivative Identities (Proof) | Lecture 22
Electromagnetic Waves | Lecture 23
Computing Partial Derivatives
Taylor Series Expansions
Least-squares Method
Chain Rule
Triple Product Rule for a Linear Function
Quadruple Product Rule
Computing the Gradient
Computing the Divergence
Computing the Curl
The Vorticity in Two Dimensions
Computing the Laplacian
Vector Derivative Identities
The Material Acceleration
Wave Equation for the Magnetic Field
Partial Derivatives
The Del Operator
Vector Calculus Algebra
Week Two Assessment

Integration and Curvilinear Coordinates
Week Three Introduction
Double and Triple Integrals | Lecture 24
Example: Double Integral with Triangle Base | Lecture 25
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
Cylindrical Coordinates | Lecture 32
Spherical Coordinates (Part A) | Lecture 33
Spherical Coordinates (Part B) | Lecture 34
Computing the Mass of a Cube
Volume of a surface above a parallelogram
Cartesian Unit Vectors
Cartesian Partial Derivatives
Some Common Two-Dimensional Vectors
Computing the Divergence and Curl in Polar Coordinates
Pipe Flow
Angular Momentum
Mass of a Disk
Gaussian Integral
Del in Cylindrical Coordinates
Divergence of a Unit Vector
Divergence and Curl of the Unit Vectors
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 1/r
Multidimensional Integration
Polar Coordinates
Cylindrical and Spherical Coordinates
Week Three Assessment

Line and Surface Integrals
Week Four Introduction
Line Integral of a Scalar Field | Lecture 35
Arc Length | Lecture 36
Line Integral of a Vector Field | Lecture 37
Work-Energy Theorem | Lecture 38
Surface Integral of a Scalar Field | Lecture 39
Surface Area of a Sphere | Lecture 40
Surface Integral of a Vector Field | Lecture 41
Flux Integrals | Lecture 42
Parametrization of the Curve y=y(x)
Circumference of a Circle
Line Integral around a Square
Line Integral around a Circle
Mass Falling Under Gravity
Surface Area of a Cylinder
Surface Area of a Cone
Surface Area of a Paraboloid
Surface Integral over a Cylinder
Mass Flux Through a Pipe
Line Integrals
Surface Integrals
Week Four Assessment

Fundamental Theorems
Week Five Introduction
Gradient Theorem | Lecture 43
Conservative Vector Fields | Lecture 44
Conservation of Energy | Lecture 45
Divergence Theorem | Lecture 46
Divergence Theorem: Example I | Lecture 47
Divergence Theorem: Example II | Lecture 48
Continuity Equation | Lecture 49
Green's Theorem | Lecture 50
Stokes' Theorem | Lecture 51
Meaning of the Divergence and the Curl | Lecture 52
Maxwell's Equations | Lecture 53
Concluding Remarks
Gradient Theorem
Conservative Vector Fields
Escape Velocity
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
Volume Integral of the Laplacian of 1/r
Continuity Equation
Electrodynamics Continuity Equation
Test Green's Theorem for a Square
Test Green's Theorem for a Circle
Stokes' Theorem in Two Dimensions
Test Stokes' Theorem
The Navier-Stokes Equation
Electric Field of a Point Charge
Magnetic Field of a Wire
Please Rate this Course
Acknowledgements
Gradient Theorem
Divergence Theorem
Stokes' Theorem
Week Five Assessment

Summary of User Reviews

Discover the world of vector calculus with Coursera's course for engineers. Users have praised the course for its comprehensive approach and clear explanations. Overall, the course is highly recommended for anyone interested in mastering vector calculus.

Key Aspect Users Liked About This Course

Comprehensive approach and clear explanations

Pros from User Reviews

Thorough coverage of vector calculus concepts
Engaging lectures and interactive quizzes
Easy to follow along with
Great for beginners and advanced learners alike

Cons from User Reviews

Some users found the pace too slow
Not enough practice problems for some users
Some users found the quizzes too easy
Limited interaction with the instructor
Some users found the course too theoretical

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