Date Chapter/Topics or Page/Problems
W Jan 20 Lecture 1: Introduction: Ordinary Differential Equations
and the Calculus of Variations
M 25 Lecture 2: Length and Curvature (one-dimensional variational problems)
W 27 Lecture 3: Calculus of Variations
Assignment 1 Due Monday February 8, 2021
M Feb 1 Lecture 4: The fundamental lemma
(open sets and compactly supported functions)
Assignment 2 = Exam 1 Due Monday February 22, 2021
W 3 Lecture 5: The fundamental lemma and the first variation
M 8 Lecture 6: General framework and examples of variational problems
W 10 Lecture 7: Vector valued functions
M 15 Lecture 8: Constraints and Lagrange multipliers
Assignment 3 Due Monday March 8, 2021
W 17 Lecture 9: Calculus of variations and Partial Differential Equations
M 22 Lecture 10: Partial Derivatives
W 24 Lecture 11: Introduction to Functions of Several Variables
M Mar 1 Lecture 12: The Divergence
Assignment 4 = Exam 2 Due Monday March 22, 2021
W 3 Lecture 13: More Topics in Integration and PDE
M 8 Lecture 14: The Mean Value Property for Harmonic Functions
W 10 Lecture 15: Other Properties of Harmonic Functions (smoothing/mollification)
M 15 Lecture 16: Weak Solutions
Assignment 5 Due Monday April 5, 2021
W 17 Lecture 17: The Maximum Principle; Other PDEs
M 22 Lecture 18: First Order Linear PDE
W 24 No Class/Lecture
M 29 Lecture 19: Calculus Review/Overview
Assignment 6 Due Monday April 19, 2021
W 31 Lecture 20: Multivariable Calculus
M April 5 Lecture 21: Gradient Flow and the Heat Equation
W 7 Lecture 22: Heat Equation
M 12 Lecture 23: Introduction to the wave equation
W 14 Lecture 24: Postponed due to technical difficulties
Final Assignment Due Wednesday April 30, 2021
M 19 Lecture 24: Regularity and Propogation Speed (heat versus waves)
W 21 Lecture 25: Review/Questions
26-28 worry free days
T 27 Post Final Assignment/Challenge Due Friday May 7 (optional)
Mathematica Notebook (Challenge Problem 2(a)(iv))
Solution for Challenge Problems
W Apr 30 Final Exam 11:20-2:10
===================Old Schedule from Spring 2020===================
The assignments below are from Spring 2020. They are nice assignments,
and you can probably learn a lot from them, but ror Spring 2021, please
work on the assignments above rather than these.
Date Chapter/Topics or Page/Problems
M Jan 6 Introduction
Assignment 1 Due Monday January 13
Lecture 1:
Introduction and Overview
Differentiation and preliminaries concerning regularity
Preliminary consideration of the Cauchy-Riemann equations
W 8 Differentiation and Integration (Function Spaces)
Lecture 2:
Existence and uniqueness for ODEs
Something about existence and uniqueness for PDEs (sketchy)
Preliminaries concerning sets: open, closed, boundary
M 13 Power series (not covered this day---covered in lecture 4 under Hints for Assignment 2)
Lecture 3:
Preliminaries concerning sets---the domains of functions
connected, closed, paths
sup norm and continuity
normed vector spaces
metric spaces (without vector space structure)
Assignment 2 Due Wednesday January 22
W 15 Lecture 4:
Spaces of functions
compact and connected sets
simply connected sets
Assignment 1 (review)
Hints for Assignment 2
M 20 holiday
Assignment 3 Due Wednesday January 29
W 22 Lecture 5:
Examples of functions (continued)
Lipschitz and H"older conditions
Why PDEs are different from ODEs (sketchy)
M 27 Lecture 6:
Hints for Assignment 3
Boas Problem 4.6.10-11 (from Assignment 5)
1-D heat equation with homogeneous boundary conditions (semi-intro)
Assignment 4 Due Monday February 3
W 29 Lecture 7:
Fourier series solution of 1-D heat equation with homogeneous boundary conditions
Fourier series and L^2
Integrability (+ an aside on measurability)
Green's function for the heat operator (with homogeneous boundary conditions) on [0,L]
M Feb 3 Lecture 8:
Overview/Summary/Organization (the heat equation and other PDEs)
Exam 1 Due Wednesday February 12
W 5 Lecture 9:
Weak derivative of the Green's function
Assignment 5 Due Monday February 17
M 10 Lecture 10:
Weak derivatives and weak solutions of PDE/ODE
W 12 Lecture 11:
Exam Review: Uniqueness of solutions for Laplace's/Poisson's Equation
Identification of the Green's function for Poisson's ODE
Exam 1 due
Exam 1 Solution
Reading Assignment
Assignment 6 Due Monday February 24
M 17 Lecture 12:
Weak solutions of Poisson's equation
W 19 Lecture 13:
Distributional derivatives and distributional solutions
Assignment 7 Due Monday March 2
Assignment 7 selected solutions
Assignment 7 Problem 4 solution
M 24 Lecture 14:
Homework Hints and Exam Review: ODEs and first order PDE
Scaling and the Chain Rule
W 26 Lecture 15:
Approximation of a distributional solution by weak solutions
Assignment 8 Due Monday March 9
M Mar 2 Lecture 16:
Physical quantities and units
Exam 2 Due Wednesday April 1 (Spring Spring break break)
W 4 Lecture 17:
Physical interpretation/derivation of the heat equation (cont.)
The divergence theorem
M 9 Lecture 18:
Physical interpretation/derivation of the heat equation (cont.)
Assignment 9 Due Monday March 30 (Spring Spring break break)
W 11 Lecture 19:
Green's Function for the Heat Equation
M 16 Spring Break
W 18 Spring Break
M 23 Spring Spring break break
Assignment 10 Due Monday April 6 (Spring Spring break break)
Unit Balls in R^3
W 25 Spring Spring break break (formerly Exam 2 due)
M 30 Lecture 20 Existence and Uniqueness of Weak Solutions
Assignment 11 Due Monday April 13
W Apr 1 Lecture 21 Riesz Representation on Euclidean Space
Exam 2 due
Exam 2 (solution)
Assignment 12 Due Monday April 20
M 6 Lecture 22 Riesz Representation on Hilbert Space
Final Assignment Due Wednesday April 29
Problem 2 (comments/draft)
reference
Problem 3 (comments/draft)
reference Note the awning in violation of this policy on the right in Figure 9 (Yeah for non-compliance)
Problem 4 (comments/draft)
Problem 4 (Mathematica Notebook)
Problem 4 (Second Mathematica Notebook (error))
Problem 4 (Jinbang Notebook)
W 8 Lecture 23 Poincare' inequalities and
Overview/review of existence/uniqueness for weak solutions
M 13 Lecture 24 Assignment 11 Problem 2 (mollification)
W 15 Lecture 25 Mollification (general discussion)
M 20 Last Monday of classes
Lecture 26 Green's function relation
W 22 Take a ride on the Reading
Office Hours---mostly about Problem 2 of Assignment 12
W 29 Final Exam 2:40-5:30