Vector Calculus for Engineers 

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Course Overview

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

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

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

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

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

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

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Acknowledgments

Gradient Theorem

Divergence Theorem

Stokes' Theorem

Week Five Assessment