Math 4010 & 5530 (Fall 2023) Homepage

News & Announcements

12/08
Final Presentations are done!
[Note to future self: We did 16 presentations in 2 hours - but felt a little rushed at times.]

A final grade report with feedback is forthcoming.

Also, just because this class is over, don't feel like you can no longer ask me questions. If you ever have a question mathematical or otherwise, please stop by my office or send me an email. I'm happy to help if I can! I hope you all have a fantastic Christmas break!

11/10
Final Project & Presentation Guidelines and Suggested Topics:

Instead of a Final Exam, we will have a final "project" of sorts. I would like everyone to select a topic somehow related to this class (I have suggestions listed below). This needs to be something that we didn't cover or at least didn't cover in detail.

  • Pick and topic and clear it with me (in person or via email).
  • Study your topic.
  • Create a handout. This should be at least one page front and back.
    • Your classmates are your audience. Build off of background from class.
    • Let us know where to go for further reading somewhere in your handout.
      [Cite any valuable resources - like websites and textbooks - that you found.]
    • You might want to give a brief historical blurb.
    • Do worked out examples help us understand your topic? If so, include a few.
    • Is your topic about a "big theorem"? Include its proof or a sketch of its proof.
  • During the final exam period, everyone will give a brief presentation of their topic. 16 people x 10 minutes = about 2 hrs. and 40 mins. So too long! Plan 5-10 minutes while trying to keep it closer to 5.
  • The presentation could use slides (like tex or powerpoint) or just use the board and your handout.
  • Several people can work on related topics. If so, you'll need to coordinate presentations and handouts. However, everyone should be creating their own unique handout (that work should be individual not group work).
  • I can print off copies of your handout for you [if you send your handout to me in time].

My Fall 2020 MAT 2110 page has some sample slides.
Or Slides (.pdf) [Source: (.tex)] from a recent talk I gave.

Handout? Why not model after one of my monstrosities. For example:
Differential Algebra and Liouville's Theorem (.pdf) [Source: (.tex)]
Some differential Galois Theory (.pdf) [Source: (.tex)]

Suggested topics/ideas:
  • Infinite Dimensional Stuff - Vector Spaces? Manifolds?
  • Hodge Dual
  • Other Cross Products
  • Quaternions/Octonions/Hurwitz Theorem
  • Complex Manifolds
  • Proofs of Multivariate Analysis Theorems
  • Applications of Stokes etc.
  • Other coordinate systems - Example: Hyperbolic
  • Volume of n-ball / Gamma function
  • Poincare's Lemma / Constructing Vector Potentials
  • DeRham Cohomology
  • Physics connections - Relativity?
  • Matrix Groups / Lie Groups.
  • How do Manifolds show up in Differential Equations?
  • Jet Spaces
  • Lie Derivative
  • Geometric Manifolds - Curvative etc.
  • Vector Bundles
  • Jacob A. - Cross Product(s)
  • Jake A. - Complex Manifolds
  • Sam A. - ODEs, Invariant/Stable Submanifolds
  • Josie B. - ODEs, Invariant/Stable Submanifolds
  • Kyle C. - Hodge Dual
  • Gavin C. - Quaternions/Octonions/Hurwitz stuff
  • Carolina D. - Big Theorems from Analysis
  • Hank E. [i.e. Fake Blake] - Quaternions/Octonions/Hurwitz stuff
  • Blake H. [i.e. Fake Hank] - Inertia Tensor
  • Collin H. [i.e. Imposter Wade] - Laplace's Equation in Other Coordinates
  • Emily L. - Poincare's Lemma
  • Jacob L. - Hyberbolic Geometry & Manifolds
  • Michael S. [not Bolton] - Gamma function / volume of n-ball
  • Jack H. - Jet Spaces
  • Brody M. - deRham Cohomology
  • Deborah N. - Applications of Stokes' Theorem
11/03
Homework #8 [Source: (.tex)] is due Friday, November 10th Wednesday, November 15th.

Semi-relevant topology papers:
• Introduction, Connectedness, & the Intermediate Value Theorem:
   "The Simplicity and Beauty of Topology:
         Connecting with the Intermediate Value Theorem"
• Compactness & the Extreme Value Theorem:
   "Fitting the Extreme Value Theorem into a Very Compact Paper"

Note: These are preliminary versions of papers that were published in
   MathAMATYC Educator in Sept. 2016 & Feb. 2018 respectively.

10/27
Homework #7 [Source: (.tex)] is due Friday, November 3rd.

10/25
Half-baked Handout:
• Upstairs vs. Downstairs [Source: (.tex)]

10/18
Homework #6 [Source: (.tex)] is due Wednesday, October 25th.

Relevant or semi-relevant linear algebra handouts:
• RREF & Linear Correspondence [Source: (.tex)]
• Spanning, Indepence, Dimension & Direct Sum Notes [Source: (.tex)]
• Coordinate Matrix Example [Source: (.tex)]
• Kernel, Range, & Composition Example [Source: (.tex)]
• Coordinates vs. Coordinate Matrices [Source: (.tex)]
• Dual Spaces [Source: (.tex)]

10/04
Homework #5 [Source: (.tex)] is due Wednesday, October 11th.

09/22
The in class portion of Test #1 is Friday, September 29th. I will then distribute a take home protion that will be due Wednesday, October 4th.

09/15
Homework #4 [Source: (.tex)] is due Friday, September 22nd.

09/12
As we "review" integration and vector calculus material, some calculus 3 materials are worth pointing out. Examples from old MAT 2130 exam answer keys might be helpful. In particular, Test #3s have integration problems and Final Exams have vector calculus problems.

In addition, here are various relevant or semi-relevant Calculus 3 handouts:
• Divergence, Curl, and Forms [Source: (.tex)]
• Supplemental Vector Calculus Problems [Source: (.tex)]
• The Big Theorems [Source: (.tex)]
• Verify Stokes' Example [Source: (.tex)]
• Spherical Cap Example (.pdf) [Maple: (.mw)]
• Examples of centroids handout [Source: (.tex)]
• Triple Integral Example (.pdf) [Maple: (.mw)]
• Triple Integral Example (.pdf) [Maple: (.mw)]

09/08
Homework #3 [Source: (.tex)] is due Friday, September 15th.

09/01
Homework #2 [Source: (.tex)] is due Friday, September 8th.

Painful Polar Partials: Derivatives in Polar [Source: (.tex)]

08/28
Handouts:
• Differentiability [Source: (.tex)]
• Some Multivariate Analysis Theorems [Source: (.tex)]
• Taylor and the Derivative Test(s) [Source: (.tex)]

Semi-relevant References:
• Taylor Polynomials [Source: (.tex)]
• Quadratic Approximations [Source: (.tex)]
        and Maple worksheet with Problem Answers (.mw) [Export: (html)]
• Maple Examples: Quad. Approx. and Critical Pts. (.mw) [Export: (html)]

08/23
Homework #1 [Source: (.tex)] is due Wednesday, August 30th.

08/21
Handouts:
• $\mathbb{R}^n$'s Structures [Source: (.tex)]
• Volumes of Parallelotopes [Source: (.tex)]

Semi-relevant/Future Refences:
• Determinants (.pdf) [Source: (.tex)]
• Cross Product Paper: Generalized Pythagorean Theorem [Related Slides]
• Topology Papers: Connected and Compact
• Characterizing the Reals [Source: (.tex)]
• Axioms for the Reals [Source: (.tex)]
• Inner Product Space Defs [Source: (.tex)]

08/11
The 4010 syllabus, 5530 syllabus, tentative schedule, & ASULearn have been updated.

MAT 4010-101 & MAT 5530-101:
"Calculus 4: An Introduction to Manifold Theory"
We will extend topics introduced in Calculus 3 (MAT 2130). In particular, we will study derivatives (i.e., Jacobian matrices) of functions of several variables and generalize from parameterized curves and surfaces to manifolds covered by a single coordinate chart. Here we also generalize vector fields to differential forms, curl and divergence to the exterior derivative, and our big theorems to the generalized Stokes' theorem. From there we will begin developing the theory of manifolds including concepts such as the tangent bundle, tensor fields, exterior algebras, orientations, and integration on manifolds.

Meeting Times: Mondays, Wednesdays, & Fridays at 11-11:50am in Walker Hall room 302 Prerequisites: Calculus 3 (MAT 2130), Linear Algebra (MAT 2240), and some proof writing background (MAT 2110)

If you have any questions specifically about this course, please feel free to contact Dr. Bill Cook at: cookwj@appstate.edu