The Geometry of Maxwell's Equations | 2021-Spring

Course Name
The Geometry of Maxwell's Equations
Course Number
aims21
Year/Term
Spring-2021
Course Credits
0
Class meeting times
3 hours of lecture per week
Prerequisites
Listed below is the background knowledge you should have for this course. Physics: We do not expect you to know any physics. We will motivate the mathematical techniques using physics examples and terminology, we will teach the necessary physics content as part of the course. For those of you who may have seen some of the physics topics before, there should still be plenty of new material. Mathematics: We expect that you have taken a first course in calculus, covering the basics of single-variable differentiation and integration. (It's OK if you need to refresh your memory.) Previous exposure to multivariable calculus (partial derivative and multiple integrals) would be an advantage, but is not necessary.
Course description
Electromagnetism is beautifully described using vector calculus, yet most treatments of vector calculus emphasize algebraic manipulation, rather than the geometric reasoning that underpins Maxwell's equations. This course attempts to bridge that gap, providing a unified view of both electro- and magneto-statics and the underlying vector calculus.
Topic/Day
Activities
Resources
Homework Due
1 3/1 Mon
Introduction to the Course
Electrostatic & Gravitational Potential and Potential Energy
GSF: Electrostatic and Gravitational Potentials and Potential Energies
Visualization of Potentials
GSF: Visualization of Potentials
GSF: Using Technology to Visualize Potentials
On your own:
Review of basic calculus
On Your Own (Optional):
Series Notation
Power Series
GMM: Power Series
2 3/2 Tues
Vectors
Position Vector
GMM: The Position Vector
GEM 1.1.4
Dot Product
GMM: The Dot Product
GEM 1.1.1
The Distance Formula
GMM: The Distance Formula
GEM 1.1.2
Superposition
GSF: Superpositions from Discrete Sources
GSF: Two Point Charges
On Your Own (Optional):
Power Series Approximations
GSF: Power Series for Two Point Charges
On Your Own (Optional):
Limiting Cases for a Pair of Charges
GSF: Power Series for Two Point Charges
3 3/3 Wed
Zapping with d
Prep Review how to find total differentials
Watch some short video:
Rules for Differentials
Product Rule
Chain Rule
and/or Read:
GMM: Differentials
GMM: Rules for Differentials
GMM: Properties of Differentials
GMM: The Multivariable Differential
Curvilinear Coordinates
GSF: Curvilinear Coordinates
GSF: Change of Coordinates
GEM 1.4
Curvilinear Basis Vectors
GSF: Orthonormal Basis Vectors
Using Gradescope
4 3/4 Thurs
Vector Fields
GVC: Vector Fields for Mathematicians
GSF: Vector Fields for Physicists
Vector Differential
GSF: The Vector Differential
GSF: Other Coordinate Systems
GSF: Calculating Line Elements in Cylindrical and Spherical Coordinates
5 3/5 Fri
Scalar Line, Surface, Volume Elements
GEM 1.3.1
Densities
GSF: Densities
Total Charge
GSF: Total Charge
Use What You Know
GSF: Using \(d\vec{r}\) in Non-Rectangular Coordinate Systems
6 3/8 Mon
More Use What You Know
Partial Derivatives
GEM 1.2.1
Definition of Gradient
GSF: The Geometry of the Gradient
GSF: The Gradient in Rectangular Coordinates
GEM 1.2.2-1.2.3
Properties of Gradient
GSF: Properties of the Gradient
Directional Derivatives
GSF: Directional Derivatives
7 3/9 Tues
Practice with Gradient
Visualizing Gradient
GSF: Visualizing the Geometry of the Gradient
GSF: Using Technology to Visualize the Gradient
Gradient in Curvilinear Coordinates
GSF: The Gradient in Curvilinear Coordinates
GSF: Formulas for Div, Grad, Curl
Assignment I
8 3/10 Wed
Potential Due to a Charged Line Segment
GSF: Potential Due to a Finite Line
GSF: Potential Due to an Infinite Line
9 3/11 Thurs Part 1
Superposition for Electric Fields
GSF: Superposition for the Electric Field
GSF: The Geometry of Electric Fields
Electric Field Due to a Point Charge
9 3/11 Thurs Part II
Cross Product
GMM: Cross Product
GEM 1.1.1-1.1.3
Flux Definition
GSF: Flux
Vector Surface Elements
GSF: Vector Surface Elements
Flux Calculation
GSF: Highly Symmetric Surfaces
GSF: Less Symmetric Surfaces
10 3/12 Fri
Visualizing Flux
GSF: Flux through a Cube
Divergence
GSF: Divergence
GSF: The Divergence in Curvilinear Coordinates
GSF: Visualizing Divergence
GEM 1.2.4
Divergence Theorem
GSF: The Divergence Theorem
GEM 1.3.4
Differential Form of Gauss's Law
GSF: Differential Form of Gauss's Law
GSF: The Divergence of a Coulomb Field
11 3/15 Mon
Current Density
Total Current
GSF: Current
Step Functions
GMM: Step Functions
Delta Functions
GMM: Delta Functions
GEM 1.5
12 3/16 Tues
Circulation
Curl
GEM 1.2.5
GSF: Curl
GSF: Visualizing the Curl II
Stokes' Theorem
GEM 1.3.5
GSF: Stokes' Theorem
Differential Form of Ampère's Law
GSF: Ampère's Law
GSF: Differential Form of Ampère's Law
Assignment II
13 3/17 Wed
Gauss's Law in Integral Form
GSF: Gauss's Law
GSF: Gauss's Law for High Symmetry
GSF: Gauss's Law on Cylinders and Spheres
14 3/18 Thurs
Ampère's Law in Integral Form
GSF: Ampère's Law
GSF: Ampère's Law and Symmetry
GSF: Ampère's Law on Cylinders
Biot Savart Law
15 3/19 Fri
Presentations
Course Summary
16 3/20 Saturday
Assignment III
OPTIONAL
The material from this point on will be covered if time allows.
Magnetic Field \(\vec{B}\) from Magnetic Vector Potential \(\vec{A}\)
Product Rule
Integration by Parts
Second Derivatives
Laplace's Equation
GSF: Product Rules
GSF: Integration by Parts
GSF: Second Derivatives
GSF: Laplace's Equation
GEM 1.2.6-1.2.7, 1.3.6
GEM 2.3.3
Compare Series and Visualization
Limiting Cases
Work
GEM 1.3.2-1.3.3
Relationship of Fields
GEM 2.3.1-2.3.2
Electrostatic Energy Due to Discrete Charges
GEM 2.4.1-2.4.2
Curl-Free Vector Fields
GSF: Curl-Free Vector Fields
Lorentz Force Law
Magnetic Vector Potential
Conductors
GSF: Conductors
Introduction to the Lorentz Force Law
Electrostatic Potential in Curvilinear Coords
GSF: Potentials from Continuous Charge Distributions
GSF: Potential Due to a Uniformly Charged Ring
GEM 2.3.4
Electric Fields from Continuous Charge Distributions
GSF: Electric Field from Continuous Charge Distributions
GSF: Electric Field Due to a Uniformly Charged Ring
Other Continuous Sources
GSF: Potential Due to a Finite Line of Charge
GSF: Potential Due to an Infinite Line of Charge
GSF: The Electric Field of a Uniform Disk
Boundary Conditions