[Note: For the MAT 5530 syllabus, please go here.]
Prerequisites: Calculus 3 (MAT 2130), Linear Algebra (MAT 2240), and some proof writing background (MAT 2110).
Texts: We will begin by "reviewing" Calculus 3, then we will look at the final chapter of Susan Colley's Vector Calculus, and finally we will move on to Renteln's Manifolds, Tensors, and Froms.
Web Page: My webpage is located here:
https://BillCookMath.com
and
our course webpage is located here: https://BillCookMath.com/index.html?page=./courses/math4010-fall2023.
Meeting times: We meet Mondays, Wednesdays, & Fridays from 11:00am until 11:50am in Walker 310.
Final Exam: We will have final project presentations (tentatively in our usual classroom) during our final exam period. Our final exam period is Friday, December 8th from 11:00am until 1:30pm.
Lecturer:
| Name: | Dr. William (Bill) Cook | |
| Office: | Walker Hall 345 | |
| Office Hours: |
Monday, Wednesday, & Friday 8-9am & 10am-11am (Other times by appointment) |
|
| Phone: | (828) 262-2367 | |
| Email: | cookwj@appstate.edu | |
| Webpage: | https://BillCookMath.com |
Technology: You are welcome to use any technology at your disposal to complete out of class assignments. We will periodically use Maple in class (Maple is a “computer algebra system”) and possibly have some Maple based homework sets. Maple should be available on all campus lab computers. If you would like a personal copy, download and activation information are available on our ASULearn page. Calculators and other computer technology will not be allowed on exams or quizzes. This includes cell phones. Your cell phone should never be out during an exam or quiz.
Course topics:
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.
In more detail, I plan to start by reconsidering parts of multivariable calculus now that we can assume some
basic linear algebra background. We will be taking a more sophisticated view of several topics
touched on in Calculus 3. This will include a more careful discussion of open/closed sets as
well as connected and compact sets (i.e., a bit of topology).
Next, we will take a new look at vector calculus using tools such as differential forms and
exterior derivatives. In our quest to better understand these vector calculus tools, we will
need to look at a few topics just beyond the end of introductory linear algebra. I plan to
explore: quotient spaces, dual spaces, tensor products, and exterior algebras.
Finally, we turn our attention to the general notion of a manifold. After exploring the
definition and some examples, we will look at smooth maps, tangent spaces, vector fields,
differential forms, and exterior derivatives, and the generalized Stokes' theorem in this
more general context.
If there is time, we can turn our attention to topics that interest the class. These
could include bits of differential geometry (like curvature tensors, Riemannian geometry,
and the math of general relativity). Or we could explore deRham cohomology and get a taste
of algebraic topology. Or look at fiber bundles or...the possibilities are endless.
A (mostly undetermined) very tentative course schedule can be found at:
https://BillCookMath.com/courses/math4010-fall2023/schedule.html.
Grades: Your term grade will be based on the results of your tests and final presentation as well as your scores on quizzes and homework and class participation. Here is more information about the individual components of your grade:
Tests: There will be two tests. Each test will make up 25% of your term grade. I have not attempted to fix dates for these tests presently. The actual dates will be negotiated/announced in class. It is likely that both tests will be take home exams or at least involve a significant take home portion.
Homework & Quizzes: I will regularly assign sets of homework problems to be turned in for a grade. We may have a few quizzes (if needed). I encourage you to work on your homework with your classmates. However, you must write up your solutions yourself. Do NOT merely copy your collaborators work and turn it in as your own. The homeworks (and quizzes) will make up 30% of your term grade.
Participation & Final Presentations:
Instead of having a final exam we will have final presentations. Everyone will pick a topic to present (this topic
will have to be approved and should be something related to what we covered in class). I will require that you create
a nice handout to go with your presentation. We will have presentations during the final exam period.
Also, I except you to come to class and participate in discussions. I may periodically ask (especially grad students) to present a homework problem. Participation as well as your final presentation and handout will make up 20% of your term grade.
Here are the components of the term grade with their weights:
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Differences between 4010 and 5530: This course is dual listed with Mat 5530. We have the same meeting times. However, graduate exams will include extra problems (most likely given as additional take-home portions of the exam) and most homework assignments will include additional problems just for 5530 students. In addition, their presentations and presentation handouts will be held to a higher standard.
Attendance: Don't miss class. If you miss class, you are responsible for the material covered during your absence. If you miss a quiz, test/exam, or workshop, you must bring in documentation proving that your absence is excusable or otherwise receive a zero. If a make-up quiz/test/exam is granted, it must be made up before the next quiz/test/exam.
Help! If you need help, please come to my office hours. If you are in Walker Hall and my office door is open, please feel free to stop by and ask questions – even if it's not during my posted office hours.
Fine Print: Copies of the academic integrity code, disability services information, religious observance policies can be found at https://academicaffairs.appstate.edu/resources/syllabi-policy-and-statement-information.