AIMS: Problem Solving | 2024-Fall

Course Name
AIMS: Problem Solving
Course Number
AIMS 23
Year/Term
2024
Display other available years here.
Course Credits
0
Class meeting times
30 hours total
Prerequisites
TBA
Course description
Physicists and other physical scientists and engineers routinely use algebra and calculus, including vector calculus, in problem-solving. But the hard part of physics problem solving is often at the beginning and end of the problem, namely getting to an algebraic expression from a description of a physical situation in words or other representations, and interpreting an algebraic expression as a statement about what will happen in the real world. This course will use examples from electromagnetism involving scalar and vector fields in the three-dimensional world to explore a variety of problem-solving methods, including using information from experimental data, approximations, idealizations, and visualizations. The emphasis will be on how geometry can help.

Topic/Day

Activities

Resources

Homework Due

W1 D1
Introduction to the Teaching Team
Corinne Manogue
Tevian Dray
Yannick Awountsa
Marry Thekhwe
Mina Monadjem
W1 D1 Practice w/Solution
Introduction to the Course
Introduction to Problem Solving
Using Gradescope
On your own:
Review of basic calculus
Basic Calculus Review
Read BEFORE class:
GSF: Review of Single Variable Differentiation
GSF: Review of Single Variable Integration
Densities
Acting Out Charge Densities
Read AFTER class:
GSF: Densities
Student Introductions
Representations of Scalar Fields
Surfaces Intro
The Park
Topographic Maps
Read AFTER class:
GSF: Scalar Fields
W1 D2
Introduction to the Assignments
Course Homepage at AIMS
Description of the Course Assignments
HW 0 (Required)
W1 D2 Practice w/Solution
Student Introductions
Contour Maps
Visualizing Contour Maps
What is Charge?
More about Density
The Functions \(1/r\) and \(1/r^2\)
The Functions $1/r$ and $1/r^2$
Electrostatic & Gravitational Potential and Potential Energy
Gravitational and Electrostatic Potential
Read AFTER class:
GSF: Electrostatic and Gravitational Potentials and Potential Energies
The Superposition Principle
Visualization of Potentials
Drawing Equipotential Surfaces
Using Technology to Visualize Potentials
Read AFTER class:
GSF: Visualization of Potentials
GSF: Using Technology to Visualize Potentials
W1 D3
Student Introductions
W1 D3 Practice w/Solution
Power Series Introduction
Power Series Notation
Read BEFORE class:
GMM: Definition of Power Series
GMM: Dimensions in Power Series
Derivation of Power Series Coefficients
Read BEFORE class:
GMM: Calculating Power Series Coefficients
Calculating Coefficients of Power Series
Calculating Coefficients for a Power Series
Read AFTER class:
GMM: Calculating Power Series Coefficients
Using Technology to Explore Power Series Approximations
Using Technology to Explore Power Series Approximations
Read AFTER class:
GMM: Using Technology to Explore Power Series Approximations
GMM: Convergence of Power Series
Visualizing the Relationship between Power Series and their Graphs
Visualization of Power Series Approximations
Read AFTER class:
GMM: Visualization of Power Series
Guessing Power Series
Guess the Power Series
Read AFTER class:
GMM: Guessing Power Series from Graphs
Power Series Approximations
Sensemaking from Graphs of Power Series II
More Power Series Information
Read BEFORE class:
GMM: Common Power Series
GMM: Theorems about Power Series
W1 D4
Student Introductions
Tutorial 1 (Required) w/Solution
Definition of the Position Vector
Read BEFORE class:
GSF: Definition of the Position Vector
Optional reading:
GSF: The Position Vector as a Vector Field
Dot Product
Dot Product Review
Read AFTER class:
GSF: The Dot Product
GEM 1.1.1
Tetrahedron
The Tetrahedron
The Distance Formula
The Distance Formula (AIMS version)
Read AFTER class:
GSF: The Distance Formula
GEM 1.1.2
Law of Cosines
Superposition for Scalar Fields
Electrostatic Potential Due to a Pair of Charges (without Series)
Read AFTER class:
GSF: Superpositions from Discrete Sources
GSF: Two Point Charges
Limiting Cases for a Pair of Charges
Electrostatic Potential Due to a Pair of Charges (with Series)
Read AFTER class:
GSF: Power Series for Two Point Charges
Position Vector in Multiple Coordinate Systems
Which Way is North? [AIMS]
W1 D4 Tutorial (4 pm)
Making Clear Justifications
Tutorial 1 Group
W1 D5
Student Introductions
Compare Series and Visualization
Read AFTER class:
GSF: Using Technology to Visualize Series Expansions
4ChargeExpansion
Vectors
Representations of Vectors
Read BEFORE Class:
GSF: Vectors
GSF: Bases
GSF: Unit Vectors
Addition Formulas
Read AFTER Class:
GMM: Addition Formulas
Curvilinear Coordinates
Curvilinear Coordinates Introduction
Read BEFORE Class:
GSF: Curvilinear Coordinates
GSF: Change of Coordinates
GEM 1.4
W2 D1
Curvilinear Basis Vectors
Curvilinear Basis Vectors
Read AFTER Class:
GSF: Orthonormal Basis Vectors
GSF: The Position Vector in Curvilinear Coordinates
GEM 1.1.4
Vector Fields
Draw Vector Fields
Read AFTER Class:
GVC: Vector Fields for Mathematicians
GSF: Vector Fields for Physicists
Vector Fields in Curvilinear Coordinates
Draw Vector Fields (Curvilinear)
Introduction to the Lorentz Force Law
Read AFTER Class:
GSF: The Lorentz Force Law
The Magnetic Field Due to a Current Carrying Wire
W2 D2
Geometric Introduction to Differentials
Tutorial 2 (Required) w/Solution
Vector Differential
Vector Differential--Rectangular
Vector Differential--Polar
Vector Differential--Curvilinear
Pineapples and Pumpkins
GSF: The Vector Differential
GSF: Other Coordinate Systems
GSF: Calculating Line Elements in Cylindrical and Spherical Coordinates
Circulation
The Wire
Electrostatic Potential from Continuous Charge Distributions
Electrostatic Potential Due to a Ring of Charge
GSF: Potentials from Continuous Charge Distributions
GSF: Potential Due to a Uniformly Charged Ring
GEM 2.3.4
W2 D2 Tutorial (2 pm)
More Power Series
W2 D3
Limiting Cases for Potential Due to a Ring
Definitions of Multipoles
Linear Quadrupole Follow-up
W2 D4
Charge on a Cube
Scalar Line, Surface, Volume Elements
Scalar Surface and Volume Elements
GEM 1.3.1
Total Charge
Total Charge: Spheres \& Cylinders
GSF: Total Charge
W2 D4 (4-6 pm)
Individual/Group Test
W2 D5
Proportional Reasoning
Modeling Nonuniform Density
Review of Derivatives
Zapping with d
Practice Zapping with d
Watch some short video:
Rules for Differentials
Product Rule
Chain Rule
and/or Read:
GMM: Differentials
GMM: Rules for Differentials
GMM: Properties of Differentials
Introduction to Use What You Know
Introduction to Use What You Know
GSF: Using \(d\vec{r}\) in Non-Rectangular Coordinate Systems
Use What You Know
Charge on a Parabola
The Helix
The Helix
W3 D1
Vector Surface Elements
Vector Surface and Volume Elements
GSF: Vector Surface Elements
Flux Definition
Acting Out Flux
GSF: Flux
Current Density
Acting Out Current Density
Total Current
GSF: Current
Flux Calculation
Flux through a Cylinder
W3 D2
Cross Product
Cross Product Review
GMM: Cross Product
GEM 1.1.1-1.1.3
Tutorial 3 (Required) w/Solution
Flux Calculation
Flux through a Paraboloid
GSF: Highly Symmetric Surfaces
GSF: Less Symmetric Surfaces
Visualization of Divergence
Visualization of Divergence
GSF: Exploring the Divergence
GSF: Exploring the Divergence in Polar Coordinates
GSF: Visualizing Divergence
Using Divergence and Curl
(Make bathtub an exam problem?)
W3 D2 (Tutorial) 4 pm
More Surface Integrals
W3 D3
Partial Derivatives
Generalized Leibniz Notation
Partial Derivatives from a Contour Map
GEM 1.2.1
Derivatives of Vector Fields
Visualizing the Gradient 1
The Hill
The Hillside (Updated)
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
Visualizing Gradient 2
Acting Out the Gradient
Visualising the Gradient
GSF: Visualizing the Geometry of the Gradient
GSF: Using Technology to Visualize the Gradient
Directional Derivatives
GSF: Directional Derivatives
Gradient in Curvilinear Coordinates
GSF: The Gradient in Curvilinear Coordinates
GSF: Formulas for Div, Grad, Curl
The Multivariable Differential
GMM: The Multivariable Differential
Electric Field Due to a Point Charge
W3 D4
Using Technology to Visualize the Gradient
Visualizing Flux
Visualizing Flux through a Cube
GSF: Flux through a Cube
Divergence
GSF: The Definition of Divergence
GSF: The Divergence in Two Dimensions
GSF: The Divergence in Curvilinear Coordinates
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
W3 D4 (4-6 pm)
In-class Individual/Group Test
W3 D5
Gauss's Law in Integral Form
Gauss's Law in Symmetric Situations
GSF: Gauss's Law
GSF: Gauss's Law for High Symmetry
GSF: Gauss's Law on Cylinders and Spheres
Ampère's Law in Integral Form
Amp\`ere's Law in Symmetric Situations
GSF: Ampère's Law
GSF: Ampère's Law and Symmetry
GSF: Ampère's Law on Cylinders
Curl
Visualization of 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
Review Test Questions
Geometric Reasoning
Geometric Problem Solving
Temp2
Step Functions
GMM: Step Functions
Delta Functions
GMM: Delta Functions
GEM 1.5
Conductors
Conductors
GSF: Conductors
OPTIONAL
Maxima/Minima
Constraints
The Roller Coaster
Optimization using gradient
Optimization using Differentials
The material from this point on will be covered if time allows.
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
Electrostatic Energy Due to Discrete Charges
Electrostatic Energy of Discrete Charges
GEM 2.4.1-2.4.2
Lorentz Force Law
Lorentz Force and Work Done on a Rectangular Loop
Magnetic Vector Potential
Magnetic Vector Potential Due to a Spinning Charged Ring
Electric Field of a Point Charge
Electric Field of a Point Charge
GSF: Electric Field of a Point Charge
Superposition for Electric Fields
Drawing Electric Field Vectors for Discrete Charges
GSF: Superposition for the Electric Field
GSF: The Geometry of Electric Fields
Electric Field for Two Point Charges
Electric Field Due to a Pair of Charges (without Series)
Power Series for the Electric Field Electric Field for Two Point Charges
Electric Field Due to a Pair of Charges (with Series)
Electric Fields from Continuous Charge Distributions
Electric Field Due to a Ring of Charge
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
Potential Due to a Charged Line Segment
GSF: Potential Due to a Finite Line
GSF: Potential Due to an Infinite Line
Magnetic Field \(\vec{B}\) from Magnetic Vector Potential \(\vec{A}\)
Biot Savart Law
Magnetic Field Due to a Spinning Ring of Charge
Curl-Free Vector Fields
GSF: Curl-Free Vector Fields
Relationship of Fields
V, $\vec{E}$, U, $\vec{F}$
GEM 2.3.1-2.3.2
Work
GEM 1.3.2-1.3.3
Presentations
Course Summary
Electrostatic Potential Due to a Ring of Charge
Electric Field Due to a Ring of Charge
Magnetic Field Due to a Spinning Ring of Charge
W1 D4.5,5.0
New Row
F 17 Nov
Vector Line and Surface Elements