Course Name
The Geometry of Maxwell's Equations
Course Number
aims21
Year/Term
Fall-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 Mon 29 Nov
Introduction to the Course
Using Gradescope
On your own:
Review of basic calculus
Introduction to the Lorentz Force Law
Vectors
Electrostatic & Gravitational Potential and Potential Energy
2 Tues 30 Nov
Curvilinear Basis Vectors
Electric Field of a Point Charge
On Your Own (Optional):
Limiting Cases for a Pair of Charges
3 Wed 1 Dec
Dot Product
Position Vector
The Distance Formula
Electric Field for Two Point Charges
4 Thurs 2 Dec
Power Series Introduction
Power Series Notation
Guessing Power Series
Power Series Approximation Demo
Derivation of Power Series Coefficients
Calculating Coefficients of Power Series
Visualizing the Relationship between Power Series and their Graphs
Power Series Approximations
5 Fri 3 Dec
Power Series for the Electric Field Electric Field for Two Point Charges
6 Mon 6 Dec
7 Tues 7 Dec
Scalar Line, Surface, Volume Elements
GEM 1.3.1
Densities
Total Charge
Proportional Reasoning
Assignment I
8 Wed 8 Dec
Electrostatic Potential from Continuous Charge Distributions
Use What You Know
9 Thurs 9 Dec
Cross Product
GMM: Cross Product
GEM 1.1.1-1.1.3
Vector Surface Elements
Flux Definition
10 Fri 10 Dec
Review of Derivatives
Partial Derivatives
GEM 1.2.1
The Multivariable Differential
Properties of Gradient
Directional Derivatives
11 Mon 13 Dec
Visualization of Divergence
12 Tues 14 Dec
Divergence Theorem
Visualizing Flux
Magnetic Field \(\vec{B}\) from Magnetic Vector Potential \(\vec{A}\)
Current Density
Total Current
Biot Savart Law
Assignment II
13 Wed 15 Dec
Circulation
Stokes' Theorem
Differential Form of Ampère's Law
14 Thurs 16 Dec
Electric Field Due to a Point Charge
Curl-Free Vector Fields
Relationship of Fields
GEM 2.3.1-2.3.2
Work
GEM 1.3.2-1.3.3
15 Fri 17 Dec
Presentations
Step Functions
Delta Functions
Course Summary
16 Tues 21 Dec
Assignment III