Department of Mathematics and Statistics, Binghamton University people:grads:eppolito

Discrete Maths Section 1 (Spring 2020)

2022-08-21T14:30:35-04:00

Discrete Maths Section 1 (Spring 2020)

This is the official website of math314-01-s20.¹⁾ Please read our syllabus…

Due to the COVID-19 pandemic, our course has gone entirely online. Please carefully read the updated syllabus and the site below.

THIS PAGE NO LONGER RECEIVES UPDATES

General Information

Meetings: Meetings take place during our old class time (08:00 - 09:30 on MWF).

Office Hours: By appointment on Zoom. Appointments should be made for reasonable times of the day (“reasonable” as defined by me).

Textbook: Discrete Mathematics and its Applications (8e) by Kenneth Rosen.

Grading: See the course syllabus for a grade distribution.

Content: Propositional logic, methods of proof, naive set theory, functions and relations, induction and recursion, counting, and basic graph theory.

Content Delivery

Videos: I will upload videos to replace some lectures. You are to watch these videos, take notes, and answer any questions posed in them.²⁾

Notes: I will sometimes post PDFs for you to read. These are assigned reading; take notes on these PDFs, as they are fair game for homework and the final exam.

Readings: I will still assign reading from the textbook.

Problem Sessions: Roughly twice a week, I will host a problem session on Zoom during our old class meeting time. You are expected to attend these sessions.

Homework

Here is a quick look at what you will need to submit homework for the remainder of the course.

Submission

Homework is to be submitted as a PDF through Gradescope; the company has made their service free-to-use for this semester. See the Gradescope guide for students and watch the Gradescope video on submitting homework.

I will not accept homework in any form other than PDF.

I will not accept homework via email. Use Gradescope.

Miscellany

I offer you 1 homework bonus point per homework turned in using the LaTeX typesetting language; you must typeset exercises legibly. I wrote a short PDF on how to use LaTeX; you can use its source code as a startup template. The easiest way to get started is to make an account on Overleaf so you can view and compile .tex files; if you end up using LaTeX more frequently, you will probably want a different setup.

If you choose not to learn LaTeX, you can use a PDF scanning application to submit your homework. I think “Tiny Scanner” is nice; you can download it on your phone with either Android or iOS.

Here are documents on basic proof techniques and proof-writing style for your reference.

Participation

As a portion of your grade, you must contribute to our class notes. This can be done in one of the following two ways.

Notes: Typing a portion of our notes from lectures (either one or two lectures).
Project: Typing a short report on some topic related to our course.

All contributions must be approved by me in advance. The typing can be done either as plain-text or in $\LaTeX{}$; in either case students must discuss formatting with me before beginning their work.

Schedule

22 January 2020 W

Review Syllabus
Basic Logic and Sets
- Propositions
- Logical Connectives
Homework:
- Read textbook sections 1.1, 1.3.1 - 1.3.5

24 January 2020 F

Basic Logic and Sets
- Translation of Sentences
- Logical Equivalence
Homework:
- Read textbook sections 1.4 - 1.5

27 January 2020 M

Basic Logic and Sets
- Predicates and Quantifiers
Homework:
- Read textbook sections 2.1 - 2.2 and 1.6

29 January 2020 W

Basic Logic and Sets
- Sets and Set Operations
- Rules of Inference
Homework:
- Read textbook section 1.6
- Think about the relationship between the logic we know and the basic operations of set theory.

31 January 2020 F

Basic Logic and Sets
- Natural Deduction
Homework:
- Complete these exercises (Due: 10 February 2020)
- Enjoy the Superb Owl! (more information here if you're unfamiliar)

3 February 2020 M

Elementary Properties of Divisibility
- Some careful mathematical proofs
Homework:
- Read textbook section 4.3; pay special attention to sections 4.3.2, 4.3.4, and 4.3.7-4.3.8.
- Attempt problems from the textbook.
Add\Drop Deadline is TODAY!

5 February 2020 W

Modular Arithmetic
- Constructed $\mathbb{Z}/n\mathbb{Z}$
- Studied more basic properties of divisibility
Homework:
- Read textbook section 4.4; pay special attention to sections 4.4.2 and 4.4.5.
- Attempt problems from the textbook.

7 February 2020 F

sNOw Classes
- Homework is still due on Monday.
- Also still do the reading for Monday.

10 February 2020 M

Greatest Common Divisor
- Euclid's Algorithm
Modular Arithmetic
- Units in $\mathbb{Z}/n\mathbb{Z}$
- Solving Equations in $\mathbb{Z}/n\mathbb{Z}$
Homework:
- Collected Homework 1
- Due Wednesday: Prove the following proposition.
  - Let $a \in \mathbb{Z}$ and $m \in \mathbb{Z}^+$. Integer $a$ is a unit modulo $m$ if and only if $\gcd(a, m) = 1$.
- Attempt problems from the textbook.

12 February 2020 W

Primes and Factorization
- Prime numbers and prime sieves
- Fundamental Theorem of Arithmetic
Homework:
- Read textbook section 4.6
- Watch this video from KhanAcademy.org.
- Attempt problems from the textbook.

14 February 2020 F

Application:
- RSA Cryptography
- RSA Problems
Homework:
- Read textbook section 5.1.
- Attempt problems from the textbook.

17 February 2020 M

Mathematical Induction
- Introduction (with dominoes!)
- Many simple examples
Homework:
- Attempt problems from the textbook.

19 February 2020 W

Mathematical Induction
- Fibonacci Numbers
- More proofs by induction
Homework:
- Attempt problems from the textbook.

21 February 2020 F

Review Session for Exam 1
- Student questions only
Homework: Study!!!

24 February 2020 M

Midterm Exam 1 (In Class)

26 February 2020 W

Mathematical Induction
- Review of Weak Induction
- Strong Induction
Homework:
- Read textbook section 5.2
- Watch this video on strong induction examples.
- Attempt problems from the textbook.

28 February 2020 F

Mathematical Induction
- More Strong Induction
- The Well Ordering Principle
Homework:
- Due Monday: Find (with proof!) all integers of the form $n = 5s + 9t$ for some $s, t \in \mathbb{N}$.
- Read textbook section 5.3
- Watch this video on comparing weak induction, strong induction, and well ordering (you can stop watching around the 5:45 marker).
- Attempt problems from the textbook.

2 March 2020 M

Recursion
- Some interesting relationships among on Fibonacci numbers
- NB: Recursion is an extremely important concept; much of what we do in the rest of the course requires understanding this fundamental topic…
Homework:
- Due Wednesday: Prove that every Fibonacci number with odd index (i.e. $f_{2k+1}$ for $k \in \mathbb{N}$) can be expressed as a sum of squares of Fibonacci numbers.
- Read textbook section 6.1
- Attempt problems from the textbook.

4 March 2020 W

Enumeration
- Cardinality of Sets
- Basic Counting Techniques
  - Product Principle: If $A$ and $B$ are finite sets, then $\#(A \times B) = \#A \cdot \#B$.
  - Sum Principle: If $A$ and $B$ are disjoint finite sets, then $\#(A \cup B) = \#A + \#B$.
    - NB: Take note that $A$ and $B$ must be disjoint!
  - Correspondence Principle: Let $A$ and $B$ be finite sets. If there is a rule of assignment $R \subseteq A \times B$ such that both $A = \{a : (a, b) \in R \text{ for some } b \in B\}$ and $\#\{b : (a, b) \in R\} = 1$ for all $a \in A$, and $B = \{b : (a, b) \in R \text{ for some } a \in A\}$ and $\#\{a : (a, b) \in R\} = 1$ for all $b \in B$, then $\#A = \#B$.
- Many Examples
Homework:
- Read textbook sections 6.3 and 6.4
- Attempt problems from the textbook.

6 March 2020 F

No Class (Winter Break)

9 March 2020 M

Enumeration
- Binomial Coefficients
  - Proved several identities via counting
- Binomial Theorem: Let $n \in \mathbb{N}$. For all $x, y \in \mathbb{R}$ we have $(x + y)^n = \sum_{k = 0}^n \binom{n}{k} x^k y^{n-k}$.
Homework:
- Read section 8.5 (NB: Yes, I mean “8.5”; it is not a typo)
- Due Wednesday Give two proofs that for all $n \in \mathbb{Z}^+$ we have $\sum_{k = 0}^n (-1)^k \binom{n}{k} = 0$.
  - Use the Binomial Theorem.
  - Use a direct counting argument (Hint: move the negative terms to the other side before you start counting).
- Attempt problems from the textbook.

11 March 2020 W

Enumeration
- Inclusion-Exclusion Principle: For all finite sets $A$ and $B$ we have $\#(A \cup B) = \#A + \#B - \#(A \cap B)$.
  - NB: Compare this with the Sum Principle…
Homework:
- Attempt problems from the textbook.

13 March 2020 F

Enumeration
- NB: This lecture taught over Zoom.
- Problem session on enumeration.
Homework:
- Attempt problems from the textbook.

Note on Coronavirus Changes

In response to the COVID-19 pandemic, the remainder of the course went online. I have grouped the remainder of the semester by week (because this is more natural for an online course).

Old Lecture Notes

Here are lecture notes taken and plain-text typed by students–with editing and further LaTeX formatting done by me–for lectures before the class went online.

Week 9 (15 March 2020 -- 22 March 2020)

Topics: Enumeration
- Pigeonhole Principle and Generalized Pigeonhole Principle
  - Textbook section 6.2.
  - Lecture notes on the Pigeonhole Principle.
  - Video on the Generalized Pigeonhole Principle.
  - Lecture notes on the Generalized Pigeonhole Principle.
- Anagrams
  - Textbook section 6.5.
  - Lecture notes.
  - Video on counting anagrams.
- Counting Poker Hands (for practice; this is a different kind of counting cards…).
Homework:
- Submit a PDF to Gradescope with the text “This is a test homework. I took this opportunity to familiarize myself with Gradescope and making PDF submissions.” (Due 23 March 2020)
  - See the homework section for more information.
- Complete these counting problems (Due 27 March 2020).

Week 10 (22 March 2020 -- 28 March 2020)

Topics: Functions and Relations.
- Introduction to Relations
  - Textbook sections 9.1 and 9.3.
  - Lecture notes.
  - Video on relations: reflexive, symmetric, and transitive (PS. this guy's videos are pretty great!).
- Equivalence Relations and Closures
  - Textbook sections 9.5 and 9.4.
  - Lecture notes.
  - Video on equivalence relations (this YouTube channel is also pretty nice!).
- Functions
  - Textbook section 2.3.
  - Notes on functions and relations.
  - Video on injective functions.
  - Video on surjective functions.
Homework:
- Complete these homework problems (Due 13 April 2020).

Week 11 (29 March 2020 -- 4 April 2020)

Topics: Graph Theory.
- Introduction to Graphs
  - Textbook sections 10.1 – 10.3 (just for reference, see the lecture first).
  - Lecture Notes: Introduction to Graphs.
  - Lecture Notes: Proof of the Handshake Lemma.
  - Lecture Notes: Subgraphs and Counting Subgraphs of $K_n$.
- Connection in Graphs
  - Textbook section 10.4.
  - Lecture Notes: Graph Connection.

Spring Break (5 April 2020 -- 11 April 2020)

No Scheduled Meetings.

Week 12 (12 April 2020 -- 18 April 2020)

Topics: Graph Theory
- Eulerian and Hamiltonian Graphs
  - Textbook section 10.5.
  - Lecture Notes on Eulerian and Hamiltonian Graphs.
- Graph Coloring
  - Textbook section 10.8.
  - Lecture Notes on Graph Coloring.

Week 13 (19 April 2020 -- 25 April 2020)

Topics: Trees
- Characterizing Trees
  - Textbook section 11.1.
  - Lecture Notes: Characterizing Trees.
- Spanning Trees
  - Textbook sections 11.4 and 11.5.
  - Lecture Notes: Spanning Trees.
- Storing Labeled Trees
  - Lecture Notes: Pruefer Code.
Homework:
- Complete these graph theory problems (Due 8 May 2020).
- Optional: Implement the Pruefer code algorithm and its inverse in your favorite programming language.

Week 14 (26 April 2020 -- 2 May 2020)

Topics: Models of Computation
- Finite State Machines
  - Textbook section 13.2.
  - Lecture Notes.
- Deterministic Finite State Automata
  - Textbook section 13.3.
  - Lecture Notes.
- Noneterministic Finite State Automata
  - Textbook section 13.4.
  - Lecture Notes.

Week 15 (3 May 2020 -- 9 May 2020)

Review for Final.

FINAL EXAM

Due to the COVID-19 pandemic, final exams will be administered orally over Zoom.

Here a list of topics to guide your studying for your oral final.

¹⁾ If you have an idea to improve this space, please email eppolito-at-math-dot-binghamton-dot-edu with your suggestion; I would like this space to be as useful to students as possible…

²⁾ If you have trouble viewing the videos through Safari, you can either use another browser or try this solution from reddit.

Discrete Math (Spring 2019)

2022-08-21T14:28:33-04:00

Discrete Math (Spring 2019)

This is the official website of math314-02-s19.¹⁾ Please read our syllabus…

Our Final Exam is scheduled for 13 May 2019.

Here are documents on basic proof techniques and proof-writing style for your own reference.

THIS PAGE NO LONGER RECEIVES UPDATES

General Information

Meetings: Monday, Wednesday, Friday 8am - 9:30am in WH G02

Office Hours: Tuesday 10am - 1pm in WH236 (or by appointment in WH310)

Textbook: Mathematics for Computer Science (Lehman, Leighton, Meyer)

Grading: See the course syllabus for a grade distribution.

Content: Propositional logic, methods of proof, naive set theory, functions and relations, induction and recursion, counting, and basic graph theory.

Schedule

W 23 Jan (C1)

Discussed Syllabus
Brief introduction to propositional logic (what is meant by the terms statement, connective, etc.)
HW: Read textbook section 3.1 (4 pages)

F 25 Jan (C2)

Translation of English statements into the formal language
Truth tables (optional: read textbook section 3.2)
Practice: problem set 1
HW: Read textbook sections 3.3 and 3.4 (2 pages and 5 pages)

M 28 Jan (C3)

Quiz: Truth table construction
Validity, satisfiability, and logical equivalence
Developed several basic equivalences of propositional statements.

W 30 Jan (C4)

Disjunctive Normal Form and Conjunctive Normal Form
Algebra of propositions (textbook section 3.4)
Basic rules of equivalence
Practice: problem set 2
HW: Read textbook section 3.6 (5 pages)

F 1 Feb (C5)

Quantified predicate logic (textbook section 3.6)

M 4 Feb (C6)

Basics of sets (textbook section 4.1)
Example proofs involving sets (direct proof)

W 6 Feb (C7)

More proofs of set theoretic identities (proof by cases, proof via a string of equivalent statements).
Introduced general relations (lots of examples)
Defined equivalence relations
HW: Read about functions and relations for Friday's class (caution: I'm NOT following the textbook here!)

F 8 Feb (C8)

Equivalence relations (many examples)
Introduction to functions (defined injective, surjective, and bijective)
Written Homework 1: Complete this list of problems (Due 15 Feb 2019)

M 11 Feb (C9)

Quiz: Equivalence relations
Images and preimages, right and left inverses (many examples)
Various propositions, examples, and practice problems concerning functions
HW: Read textbook section 5.1 through section 5.1.4 (5 pages)

W 13 Feb (C10)

Quiz: Functions and equivalence relations
Relationship between functions inverses and injectivity, surjectivity, and bijectivity
Introduction to the Principle of Mathematical Induction (textbook section 5.1.1-5.1.4)

F 15 Feb (C11)

Collected Written Homework 1
Quiz: Induction and Well Ordering
More proofs by induction
A false proof by induction (to illustrate the importance of the base case)!
Number Theory: Properties of Divisibility (textbook section 9.1.1)

M 18 Feb (C12)

Recursion and the relationship to induction
Fibonacci numbers and relations therebetween (more induction proofs!)
Here is a list of cool problems involving Fibonacci numbers!

W 20 Feb (C13)

The Quotient-Remainder Theorem (i.e. the Division Algorithm, i.e. textbook Theorem 9.1.4 “The Division Theorem”)
Proof of the Quotient-Remainder Theorem (for the case n and d are natural numbers)
HW: Finish the proof of the Quotient-Remainder Theorem for n and d integers.

F 22 Feb (C14)

Quiz: Quotient-Remainder Theorem
Using the Quotient-Remainder Theorem
Modular Arithmetic
HW: Read my notes on basic number theory
Written Homework 2: Complete this list of problems (Due 4 Mar 2019)

M 25 Feb (C15)

Greatest common divisor
Bezout's Identity (one proof given in this pdf)
Euclid's (Extended) Algorithm
Examples involving Euclid's Algorithm

W 27 Feb (C16)

Quiz: Modular Arithmetic and Euclid's Algorithm
Using Euclid's Algorithm to solve modular equations
Prime numbers
- Definition
- Prop: Every natural number n > 1 is divisible by some prime number.
- Prop: There are infinitely many prime numbers.

F 1 Mar (C17)

Proved Euclid's Lemma
Proved the Fundamental Theorem of Arithmetic via Strong Induction
- Exercise: Translate the proof into a Well Ordering Principle type of proof
Introduced the Sieve of Eratosthenes
NB: I updated the number theory notes.

M 4 Mar (C18)

Collected Written Homework 2.
Computed the list of primes not exceeding 50 via the Seive of Eratosthenes.
Briefly discussed some ways of improving the Seive of Eratosthenes.
Discussed the RSA Cryptosystem (including a small example).
HW: Read textbook section 9.11 (on RSA Cryptography)

W 6 Mar (C19)

More work on RSA (try these problems).
HW (assigned in class, sent also by email): Let p = 11, q = 19, and e = 13.
- What is the corresponding public key?
- Encrypt m = 19 using RSA encryption.
- Compute the private key d in this encryption scheme.
- Decrypt an encoded message m' = 47 using RSA.
Question: Why does RSA work? (Answer: Euler's Totient Theorem!)
Set the stage to study Euler's Totient Function

F 8 Mar (C20)

Studied Euler's Totient Function, laying groundwork for the proof
Proof of Correctness of RSA (assuming Euler's Totient Theorem)
HW: Study for our Midterm Exam
- Material for the Midterm stops at RSA
- Know definitions and major theorems and be ready to use them
- Be sure to look over your notes
- By popular request, here is a practice midterm.
  - Disclaimer: The practice midterm DOES NOT pretend to be comprehensive, and I make no claims about its fitness as a study guide.

M 11 Mar (C21)

Exam Review (questions from students)
Proof of Euler's Totient Theorem
- Remember that this implies correctness of the RSA Cryptosystem

NB: I leave for a visit to the IAS early Tuesday morning–I return after the midterm.

W 13 Mar (C22)

Exam Review (led by guest lecturer David Cervantes Nava)
Student questions on the Practice Exam
Here are solutions to the practice midterm

F 15 Mar--Midterm Exam (in class)

Happy Spring Break!
- HW: Read textbook sections 15.1 - 15.3 on basic counting techniques

M 25 Mar (C23)

Returned Midterm Exams
Basics of counting
- Addition Principle
- Product Principle
- Bijective Counting

W 27 Mar (C24)

More counting (with many examples and some recursive counting)

F 29 Mar (C25)

Problem session on basic counting techniques
- Counting with binomial coefficients
HW: Finish the in-class problems for Monday
HW: Complete these problems for 5 Apr.

M 1 Apr (C26)

More counting with binomial coefficients
- Algebraic Formula for the binomial coefficients
- Counting anagrams of a word
Learned the methods “Count Dracula” and the method of Monte Cristo²⁾

T 2 Apr--Withdrawal Deadline

W 3 Apr (C27)

Quiz: Binomial coefficients and counting
More advanced counting techniques
- Inclusion-Exclusion Principle
- Pigeonhole Principle
More counting!

F 5 Apr (C28)

Quiz: Pigeonhole principle
Collected Homework on counting poker hands
Simple graphs (textbook chapter 12)
- Basic definitions
- Many examples (the Petersen graph is my favorite)
- Handshake Lemma

M 8 Apr (C29)

Quiz: examples of simple graphs
Matchings in graphs
- Examples
- Basic results
- Hall's Marriage Theorem for bipartite graphs
  - Statement
  - Proof of sufficiency

W 10 Apr (C30)

Quiz: perfect matchings
Hall's Marriage Theorem
- Proof of necessity
Written Homework 3: Complete this list of problems (due 26 Apr 2019)

F 12 Apr (C31)

Quiz: Hall's Marriage Theorem
Algorithms to compute matchings in bipartite graphs
- Algorithm suggested by Hall's Marriage Theorem
- Augmenting Paths Algorithm
Connection in graphs

M 15 Apr (C32)

Quiz: Augmenting Paths Algorithm
Graph coloring
- Many examples
- Basic properties

W 17 Apr (C33)

Quiz
More graph coloring
- Brooks's Theorem
  - Every finite simple graph has chromatic number bounded above by one more than its maximum degree.

F 19 Apr No Classes :/

M 22 Apr (C34)

Group Quiz (discussion led by guest lecturer Kunle Abawonse)
- Solutions to these problems

W 24 Apr (C35)

Quiz: Graph coloring
More on connection in graphs
- Reducing problems on graphs to problems on their connected components
- Exercise: Prove that the chromatic number of a graph is the maximum of the chromatic numbers of its components.
Eulerian graphs (short introduction)

F 26 Apr (C36)

Quiz: Euler trails
Collected Homework 3
Eulerian graphs
- Proved Euler's characterization of Eulerian graphs
  - A graph has an Euler trail if and only if it has at most one component with edges and at most two vertices of odd degree.
Hamiltonian graphs
- No nice equivalent conditions are known
- Homework: Does the Petersen graph have a Hamilton cycle?
  - Solution is in this pdf
You should read about Leonhard Euler...

M 29 Apr (C37)

Quiz
Trees (Textbook section 12.11)
- Equivalent descriptions of trees
- Spanning trees
- Minimum weight spanning trees
  - Kruskal's Algorithm

W 1 May (C38)

Quiz: Kruskal's Algorithm
Algorithms and state machines (Textbook chapter 6)
- State machine definitions and examples
- Evaluations
- Preserved Invariant Lemma (Textbook “Preserved Invariant Principle”)
Here are my notes on state machines.

F 3 May (C39)

Quiz
More algorithms and state machines
- Examples encoding algorithms into state machines
- Examples interpreting state machines as algorithms

M 6 May (C40)

The fast exponentiation algorithm
- State machine model
- Proof of correctness

W 8 May (C41)

Review for Final
- Student questions
Gave comment forms

F 10 May (C42)

Review for Final
- Student questions
I will hold extra office hours today!
- Room: WH 236
- Time: 11am - 4pm

FINAL EXAM

Date: Monday 13 May 2019
Time: 8am - 10am
Room: S1 149

¹⁾ If you have an idea to improve this space, please email eppolito-at-math-dot-binghamton-dot-edu with your suggestion; I would like this space to be as useful to students as possible…

²⁾ APRIL FOOLS!

Calculus III (Fall 2019, Sections 05 and 06)

2022-08-21T14:47:25-04:00

Calculus III (Fall 2019, Sections 05 and 06)

This is a website built for math323-05-f19 and math323-06-f19. If you have an idea to improve this space, please email eppolito-at-math-dot-binghamton-dot-edu with your suggestion; I would like this space to be as useful to students as possible…

THIS PAGE IS NO LONGER UPDATED

General Information

The official course page is at the following link:

https://www2.math.binghamton.edu/p/calculus/math_323/start.

Its contents are our syllabus.

Contact Information

Your instructor is Chris Eppolito (hey, that's me!). Email me at eppolito-at-math-dot-binghamton-dot-edu for anything regarding this class. Be sure to include your name and section number in the email.

Meetings

Section 5: MWF 14:50 - 16:20 in LH 3

Section 6: MWF 16:40 - 18:10 in CW 323

Office Hours

Our help room for this class is WH 233. Please see the schedule for my hours. I can also meet by appointment (email me to set up an appointment).

You can visit any Calculus III instructor in the help room, and I encourage you to do so! Different folks think differently; maybe someone else thinks more like you, and you can get help from them too.

Textbook

Multivariable Calculus (8e) by Stewart

The textbook (as an e-book) comes with a subscription to WebAssign, so just get that…

Grading

See the official grading distribution.

Content

Reasoning in three dimensions. Calculus of functions in several variables. We will cover most of chapters 12 - 16 in the textbook, together with some small excursions as I see fit.

Homework

This class uses WebAssign for online homework assignments. You are expected to check WebAssign daily, and to know when assignments are due. I will NOT notify you of assignment due dates (all due dates are already posted).

Email me if you need the key to our class in WebAssign.

Written Homework

I sometimes assign written homework to collect in class.

Practice Problems

Here are some practice problems (updated 16 November 2019) I like. Check back for updates…

Quizzes

Quizzes will be given frequently throughout the course. Quizzes are intended to check that you understand basic concepts from previous classes and from assigned reading.

I typically grade quiz questions on a 5-point scale. The following is a ROUGH guide to interpreting each score.

5: Understands concept, with minor errors
4: Mostly understands concept, with errors in execution or minor misunderstanding
3: Conceptual misunderstanding present, but not yet serious
2: Major conceptual misunderstandings, with shallow understanding (if any)
1: Mostly nonsense, with serious errors
0: Demonstrates no understanding of the topic

Useful Software

Here are some useful technological tools for this class, together with some descriptions.

Mathpix is a browser 3D graphing calculator. The interface is self explanatory, produces very nice pictures, and doesn't use a lot of computing power.
GeoGebra is a math learning software (that link is to the browser 3D graphing calculator). You can download a copy (for Linux, Mac, or Windows) and write your own sheets, too! There is a learning curve…
Academo is a browser 3D graphing calculator. It provides nice shading, but has a limited rotation.

Schedule

This section is a record of what I have covered and plan to cover in my sections. See also the official schedule.

21 August 2019 W

First-day stuff
- Discussed the syllabus
- Introductions from students
Geometry in three dimensions (textbook section 12.1)
- Proved the Distance Formula.
Homework
- Read: Textbook sections 12.1 and 12.2
- Do: Attempt practice problems

23 August 2019 F

Vectors and Vector Operations (textbook section 12.2)
- Algebraic and geometric descriptions of vectors
- Addition and scalar multiplication operations on vectors
- Algebraic properties of vector addition and scalar multiplication
Homework
- Read: Textbook section 12.3
- Do: Attempt practice problems

26 August 2019 M

The Dot Product (textbook section 12.3)
- Algebraic definition of the dot product
- Algebraic properties of the dot product
- Geometric interpretation of the dot product and consequences
The Cross Product (textbook section 12.4)
- Briefly constructed the cross product as a particular vector orthogonal to two given vectors
Homework
- Read: Textbook sections 12.4 and 12.5
- Do: Attempt practice problems

28 August 2019 W

The Cross Product (textbook section 12.4)
- Construction as a “determinant”
- Algebraic Properties
- Geometric Properties
Lines and Planes in 3-Space (textbook section 12.5)
- Lines
  - Vector parametrizations
  - Coordinate functions
  - Symmetric equations
- Planes
  - Vector sums of non-parallel vectors
  - Normal vector to a plane
Homework
- Do: Attempt practice problems

30 August 2019 F

Brief review of lines and planes in 3-space
Problem Session on Chapter 12
Homework
- Read: Textbook sections 12.6 and 13.1
- Do: Play with quadric surfaces and curves in 3-space–use a 3D graphing calculator as you read!
Written Homework: Complete these exercises--Due 6 September 2019

2 September 2019 M

No Classes (Labor Day: Don't wear white after today?)

4 September 2019 W

Quadratic Surfaces (textbook section 12.6)–The links below are to GeoGebra sheets I made for class; play with the parameters to get a feel for quadratic surfaces.
Vector Functions and Curves in 3-Space (textbook section 13.1)
- Vector function definition
- Limits of vector functions
- Examples
  - Helix
  - Trefoil Knot
Homework:
- Read: Textbook sections 13.1 and 13.2
- Do: Attempt practice problems

6 September 2019 F

Collected Written Homework 1
Recap of Vector Functions (textbook section 13.1)
Calculus of Vector Functions (textbook section 13.2)
- Running Example: Moment Curve
- Properties of the Limit
- Definition and Properties of the Derivative
- Definition and Properties of the Integral
Homework
- Read: Textbook section 13.3 (with special attention to arc length)
- Do: Attempt practice problems
- Study: Start studying for Midterm 1

9 September 2019 M

Arc Length of Space Curves (textbook section 13.3)
- Derivation of Formula
- Examples to Computing Arc Length
- Reparametrizing a Curve by Arc Length
Homework
- Do: Attempt practice problems
- Study: Study (harder?) for Midterm 1

11 September 2019 W

Motion in Space (textbook section 13.4)
- Problem Session on Vector Functions and Space Curves
NB: End material for Midterm 1.
Homework
- Do: Attempt practice problems
- Study: Study for Midterm 1.

13 September 2019 F

Multivariable Functions (textbook section 14.1)
Limits and Continuity of Multivaraible Functions(textbook section 14.2)
- Definition of the Limit (with much motivation!)
Homework
- Study: Study for Midterm 1.

16 September 2019 M

Review Session for Midterm 1
- Student Questions Only
- Study: Study for Midterm 1

18 September 2019 W

Midterm 1 in Class
- Content: Chapters 12 and 13 (omitting curvature and normal/binormal vectors)

20 September 2019 F

Limits and Continuity of Multivaraible Functions(textbook section 14.2)
- Review the definition of the limit
- The Curves Criterion for Limits
- Lots of examples
- Here are two more examples...
Homework
- Read: Textbook sections 14.1 and 14.2; get started on 14.3

23 September 2019 M

Derivatives of Multivariate Functions
- Partial Derivatives (textbook section 14.3)
  - Definition and intuition
  - Computational aspects
  - Examples
- Directional Derivatives (textbook section 14.6)
  - Definition and intuition
  - Examples
- Differentiability of Multivariate Functions
  - Definition and intuition
Homework
- Do: Attempt practice problems
- Read: Textbook sections 14.3 and 14.6

25 September 2019 W

More on Derivatives of Multivariate Functions
- Directional Derivatives in terms of Partial Derivatives
- Clairaut's Theorem on Mixed Partial Derivatives
Homework
- Do: Attempt practice problems
- Read: Textbook section 14.4

27 September 2019 F

Tangent Planes and Linear Approximation (textbook section 14.4)
- Problem session
Homework
- Do: Attempt practice problems
- Read: Textbook section 14.5

30 September 2019 M

No class (Rosh Hashanah: Happy New Year!)

2 October 2019 W

Multivariable Chain Rule (textbook section 14.5)
- Derivation for the 2-variable case
- Statement of the chain rule
- Many examples
Homework
- Do: Attempt practice problems
- Read: Textbook sections 14.6 (on gradient) and 14.7

4 October 2019 F

Gradient (textbook section 14.6)
- The gradient maximizes the directional derivative!
Maxima and Minima (textbook section 14.7)
- Finding critical points
- Classifying critical points with the Second Derivative Test
- Many examples
Homework
- Do: Attempt practice problems
- Read: Textbook section 14.8

7 October 2019 M

Lagrange Multipliers (textbook section 14.8)
- Statement of the method
- Examples applying the method
- Here is another example.
Homework
- Do: Attempt practice problems
- Read: Textbook sections 15.1, 15.2, and 15.3
- Study: Start studying (harder?) for Midterm 2

9 October 2019 W

No Class (Yom Kippur: Sorry, no class today…)

11 October 2019 F

Another Lagrange Multipliers example
Double Integrals
- Rectangular Domain (textbook section 15.1)
Homework
- Do: Attempt practice problems
- Read: Textbook sections 15.1, 15.2, and 15.3
- Study: Study for Midterm 2

14 October 2019 M

Double Integrals
- General Domain (textbook section 15.2)
- Polar Coordinates (textbook section 15.3)
Homework
- Do: Attempt practice problems
- Read: Textbook sections 15.1, 15.2, and 15.3
- Study: Study for Midterm 2
NB: End content for Midterm 2

16 October 2019 W

Double Integrals
- Problem Session
Homework
- Study: Study for Midterm 2

18 October 2019 F

Triple Integrals (textbook section 15.6)
- Too many examples
- Application to Physics: Center of Mass
Homework
- Study: Study for Midterm 2

21 October 2019 M

Review Session for Midterm 2
- Student Questions Only
Homework
- Study: Study for Midterm 2

23 October 2019 W

Midterm 2 in Class
- Content: Chapters 14 and 15.1–15.3

25 October 2019 F

Cylindrical Coordinates (textbook section 15.7)
Spherical Coordinates (textbook section 15.8)
Homework
- Do: Attempt practice problems
- Read: Textbook sections 15.6, 15.7, and 15.8; get started reading textbook section 15.9

28 October 2019 M

Change of Variables (textbook section 15.9)
- Unifying the differentials with Jacobians!
Homework
- Do: Attempt practice problems
- Read: Textbook section 15.9

30 October 2019 W

Change of Variables (textbook section 15.9)
- Lots of examples
- Here are solutions...
Homework
- Do: Attempt practice problems
- Read: Textbook sections 16.1

1 November 2019 F

Vector Fields (textbook section 16.1)
Line Integrals (textbook section 16.2)
Homework
- Do: Attempt practice problems
- Read: Textbook sections 16.2 and 16.3

4 November 2019 M

More on line integrals
- Too many examples
Fundamental Theorem of Line Integrals (textbook section 16.3)
- Do: Problem assigned in class
- Read: Textbook section 16.4

6 November 2019 W

Green's Theorem (textbook section 16.4)
- Statement and Explanation (proof omitted)
- Many Examples
Homework
- Do: Attempt practice problems
- Read: Textbook section 16.5

8 November 2019 F

Curl and Divergence (textbook section 16.5)
- Definitions
- Many examples
- Fundamental properties
NB: End content for Midterm 3
Homework
- Do: Attempt practice problems
- Read: Textbook sections 16.6 and 16.7
- Study: Start studying (harder?) for Midterm 3

11 November 2019 M

Parametric Surfaces (textbook section 16.6)
- Examples
- Tangent planes from parametrizations
- Surface areas from parametrizations
Homework
- Do: Attempt practice problems
- Study: Study for Midterm 3

13 November 2019 W

Surface Integrals (textbook section 16.7)
- Examples
Homework
- Study: Study for Midterm 3

15 November 2019 F

Review Session for Midterm 3
- Student Questions Only
Homework
- Study: Study for Midterm 3

18 November 2019 M

Midterm 3 in Class
- Content: Chapters 15.6–15.9 and 16.1–16.5

20 November 2019 W

Stokes's Theorem (textbook section 16.8)
Homework
- Do: Attempt practice problems from the textbook
- Read: Textbook section 16.8
- Study: Study for the CUMULATIVE Final

22 November 2019 F

More on Stokes's Theorem
- Problem session
Homework
- Do: Attempt practice problems from the textbook
- Read: Textbook sections 16.9
- Study: Study for the CUMULATIVE Final

25 November 2019 M

Divergence Theorem (textbook section 16.9)
Homework
- Do: Attempt practice problems from the textbook
- Study: Study for the CUMULATIVE Final

27 November 2019 W

No Class (Find-a-T(of)urkey Day: Safe travels!)

29 November 2019 F

No Class (Recover-from-Turkey Day: Heat up those leftovers…)

2 December 2019 M

Classes Cancelled by the University for SNOW reason

4 December 2019 W

Review Session for Final
- Student Questions Only
- Study: Study for CUMULATIVE Final

6 December 2019 F

No Class (Reading Day: Read your favorite calculus textbook/notebook!)

12 December 2019 R

FINAL EXAM!
- Exam Room: LH 001
- Exam Time: 15:15 - 17:15

Differential Equations (Summer 2019)

2022-08-21T14:50:24-04:00

Differential Equations (Summer 2019)

This is the official website of math324-01-su19.¹⁾

THIS PAGE IS NO LONGER UPDATED

General Information

The syllabus is posted, and you should definitely read it; here is some of the content for quick reference.

Meetings

Every Weekday 9:50am - 11:50am in SL 210

Office Hours

By Appointment in WH310

Textbook

A First Course in Differential Equations with Modeling Applications (11e) by Zill

Grading

See the course syllabus for a grade distribution.

Content

Basic techniques and theory in solving simple ordinary differential equations, initial value problems, and boundary value problems. Time permitting we will also cover numerical methods for approximating solutions and some real-world applications of our methods in various disciplines.

Learning Tools

This course uses the following free learning tools for students.

Khan Academy is useful for videos and practice problems. I will assign videos from time to time; join our class by following the link I sent to our email listserv.

Paul's Online Notes are fairly good; be warned that they don't match our class perfectly.

GeoGebra is a freely available computer algebra system; it will open all files below with the .ggb extension. I write GeoGebra sheets using GeoGebra Classic 6.

GNU Octave is an open source version of MatLab and is mostly compatible with the MatLab commands. I use Octave for numerical methods.

Schedule

28 May 2019 T

Discussed the syllabus
Introduction to ODEs
- Terminology
- Verifying solutions
Homework 1 (Mostly Calculus 1 review in our new language…)

29 May 2019 W

Went over two homework problems (with brief Calculus 1 review)
Brief review of partial derivatives (in an)
Initial Value Problems
- A theorem on existence and uniqueness of solutions to first-order IVPs
- An example from Calculus 1
Visualizing First-Order Equations
- Direction fields (very brief discussion)
Homework 2

30 May 2019 R

Direction fields (for first-order ODEs)
- An example on the board
- Several examples (including approximated solutions) in this GeoGebra sheet
  - NB: Mine is a slightly modified version of this GeoGebra sheet
Separable First-Order Equations
- Solved several separable ODEs
- Developed the Separation of Variables method
Homework 3 (use graph paper for the first problem)

31 May 2019 F

Exact First-Order Equations (assigned Khan Academy videos)
- Exactness Criterion
- Solving an exact equation
- Using integrating factors to transform nonexact ODEs into exact ODEs
Linear First-Order Equations
- Case 1: Separable
- Case 2: Simple integrating factor
Solutions by Substitutions
- Bernoulli's Equation
NB: I've assigned two Khan Academy videos on homogeneous equations (relevant for problem 7 below).
Homework 4 (long, but you have the weekend…)

3 June 2019 M

Solutions by Substitutions
- Homogeneous First-Order Equations (assigned Khan Academy videos)
Review for Midterm 1 (student questions)
Written Assignment 1 (Due at Midterm 1)

4 June 2019 T (MIDTERM 1)

Midterm 1 Administered
Collected Written Assignment 1
Applications
- Linear Models
  - Bacterial Growth
  - Radioactive Decay
  - Newton's Law of Cooling
  - Kirchoff's Current Law

5 June 2019 W

Returned Midterm 1
Higher-Order Linear Equations
- Homogeneous Linear Equations
  - Superposition Principle (i.e. linear combinations of solutions are solutions)
  - Linear Independence and the Wronskian
  - Fundamental Sets of Solutions
  - The General Solutions Theorem
- Nonhomogeneous Equations
  - The General Solutions Theorem
NB: Gave lots of motivation from linear algebra to make the theorems plausible; linear algebra will not be tested.
Homework 5

6 June 2019 R

Derived Euler's Formula via Maclaurin Series
Brief Review of General Solutions to Linear Equations
Solving Homogeneous Linear Equations with Constant Coefficients
- Using Auxiliary Equation
  - Case 1: Distinct real roots
  - Case 2: Complex roots
  - Case 3: Repeated roots
Reduction of Order
- Brief Introduction
Homework 6

7 June 2019 F

Reduction of Order
- Second-order examples
Method of Undetermined Coefficients (via Differential Operators)
- Factoring differential operators
- Constructing operators for given functions
- Using differential operators to find particular solutions
NB: See Paul's Online Notes for an alternative “guess-and-check”-type approach–that method can fail spectacularly (e.g. when the guess is a solution of the associated homogeneous ODE)…
Homework 7 (long, but you have the weekend…)

10 June 2019 M

More on the Method of Undetermined Coefficients
- Another example
Variation of Parameters
- Idea of method (partial derivation)
- Several second-order examples
- Third-order example
Cauchy-Euler Equations
- Brief introduction to auxiliary equation
Homework 8

11 June 2019 T

Cauchy-Euler Equations
- Reduction to constant coefficients case
Review for Midterm 2 (student questions)
Written Assignment 2 (Due at Midterm 2)

12 June 2019 W (MIDTERM 2)

Midterm 2 Administered
Collected Written Assignment 2
Review of Power Series
- Definitions of Interval and Radius of Convergence
- Basic Properties of Power Series
- Table of Useful Maclaurin Series

13 June 2019 R

Power Series Operations
Power Series Methods in ODEs
- Solved Three ODEs via Power Series
Homework 9

14 June 2019 F

Quiz
- Went over quiz problem together
Ordinary Points and Singular Points
- Definitions and brief discussion of differences
Homework 10

17 June 2019 M

Series Solutions near Singular Points
- Frobenius Method
- Several Examples
Homework 11

18 June 2019 T

Numerical Methods
- Power Series Approximations
- Euler's Method
- Improved Euler's Method
- Runge-Kutta Methods
Homework 12

19 June 2019 W

Review for Midterm 3 (student questions)

20 June 2019 R (MIDTERM 3)

Midterm 3 Administered
Laplace Transforms
- Definition
- Three Examples

21 June 2019 F

Laplace Transforms
- Basic Properties of Laplace Transforms
- Many Examples
- The Inverse Laplace Transform
Homework 13

24 June 2019 M

Solving ODEs with Laplace Transforms
- Two IVPs
- Group Quiz
Homework 14

25 June 2019 T

Linear Systems of ODEs
- Laplace Transforms to Solve Linear ODE Systems
- Annihilator Technique to Solve Linear ODE Systems
Homework 15

26 June 2019 W

Review for Final (student questions)

27 June 2019 R

Review for Final (student questions)

28 June 2019 F (FINAL)

Final Administered

¹⁾ If you have an idea to improve this space, please email eppolito-at-math-dot-binghamton-dot-edu with your suggestion; I would like this space to be as useful to students as possible…

Chris Eppolito

2026-04-14T05:10:34-04:00

Chris Eppolito

I am no longer at Binghamton University, as of Summer 2022.

My homepage has moved to https://eppo-math.codeberg.page/. I plan to keep that page updated into the foreseeable future.

THIS PAGE IS NO LONGER UPDATED