James Cook's Manifold Theory Homepage
Welcome, I intend to post some new things here once I create them.

• Manifold Theory (Math 497, special topics course) of Spring 2024 Playlist on You Tube:


• Course Planner (updated 3-16-24)
• Course Notes for last story arc of course (relevant to Mission 9 and 10)
• Elementary Differential Geometry ala Frames and Forms much more detail than time will allow for us this term, if you want more examples etc. It's all here.

Missions, Calculations and Supplements from 2024 Manifolds:

• Mission 1: and Solution: problems from Jeffrey Lee's text on coordinate charts.
• Mission 2: and Solution: problems from John Lee's text on smooth maps and tangent spaces
• Equivalence of tangent space viewpoints also has pushforward of coordinate derivation to partial derivative on Euclidean space
• Chain Rule calculations, a conversation with Ernesto ultimately this is easier with velocity vector viewpoint, but I try to show how to lift chain rule on Euclidean space up to the manifold chain rule by an explicit coordinate-based calculation here.
• Mission 3: problems and Solution: and video overview of solution from John Lee's text on submanifolds and local diffeomorphisms.
• Mission 4: problems and Solution: and video overview of solution from John Lee's text on Lie Groups.
• Mission 5: problems and Solution: and video overview of solution from John Lee's text on vector fields, flows and Lie Algebras.


• Mission 6: problems and Solution: from John Lee's text on tensor fields and metrics.
• Mission 7: problems and Solution: from John Lee's text on differential forms.
• Mission 8: problems and Solution: from John Lee's text on distributions, foliations and a hint of homogenous spaces.
• Mission 9: problems and Solution: from on Frenet Frame, curvature, torsion, isometry of Euclidean Space, congruence, Shape Operator, Gaussian curvature and Mean Curvature.
• Mission 10: problems and Solution: on the intrinsic calculus of surfaces, Gauss' Theorem of Awesomeness, Gauss Bonnet Theorem, local isometry, global isometry, non-Euclidean geometries.


• Take Home Part of Final Exam: method of Cartan continued.

Select Conversations:
Here I collect a few conversations I've had with students on the topic of advanced calculus and associated real analysis. I sometimes foolishly try to teach students outside the framework of formal courses.

• a picture gallery of an intensive 2hr lecture I once gave to a certain student who never followed up on the work...
  • best hits of advanced calculus; a condensed remix.
  
  
• a series of discussions with several students over the summer "break"(these conversations partly inspired the 2013 revision of the Advanced Calculus course. )
1. (2013)advanced calculus, DEqns, manifolds
2. DEqns and limits
3. Multivariate MVT
4. Contraction Mapping, Newton's Method
5. tensors, wedges, multilinear algebra
6. tensors remix, differential forms
7. manifolds and tangent spaces


• push-forward example on matrix manifold (from Fall 2011)
• Based on Rosenlicht's Introduction to Analysis text. From a focused hybrid 332/422 course with a student who went on to graduate school:
  • (2010)advanced calculus with some analysis

Additional Examples:
A few calculations which are advanced calculus we did not precisely cover:

• My course notes from Math 430 at NCSU. This course introduces Maxwell's Equations in terms of differential forms on Minkowski space.
• My notes from a course on manifold theory:
1. part 1 outline of course, inverse function theorem,
2. part 2 examples of manifolds, atlas, topology,
3. part 3 stereographic projection, smooth maps between manifolds,
4. part 4 connected implies path connected on manifold, theorems on curves in a manifold,
5. part 5 a bit more on bumps, derivations and differentiation on manifold,
6. part 6 contrasting views of tangent vectors, the differential,
7. part 7 tangent bundle,
8. manifolds my study guide from the notes above
• Notes from a summer I read about Quantum Mechanics and Symmetries from Greiner's text:
1. Chapter 3 and 4 on exponentiation of Lie algebra, Lie groups, generation of symmetries in quantum mechanics
2. Chapter 5 on isospin and representations, the adjoint representation.
3. Chapter 6 and 7 on hypercharge and SU(3) representations as derived from SU(2) subalgebras.
4. Chapter 8 on quarks built from SU(3) representations
5. my talk from graduate school on Quarks built from SU(3)
• When I find an interesting question at the Math Stack Exchange then I sometimes put a little effort into producing an answer:
1. (form vs. area integrals)
2. (spherical frame exhibition)
3. (linearization and differential contrasted)
4. (chain-rule and method of differentials)
5. (a differential form calculation)
6. (inuition on push-forward and pull-back)
7. (concerning what a tangent vector is)
8. (more fun with pull-backs)
9. (heurstics on calculus on a manifold)
10. (on the adapted coordinate chart of tangent bundle)
11. (transformation law for n-form and determinant)
12. (on pull-back and how to differentiate on manifold)
13. (concerning orientations, a question from Morita)
14. (product rule for vector-valued forms)
15. (orientiation on manifold)
16. (on induced orientations)
17. (Lie bracket calculation)
18. (bilinearity of two-form)
19. (wedge and pull-back commute)
20. (pull-back of one-form)
21. (distributivity of tensors)
22. (exterior product of matrix)
23. (how to generate translation)
24. (real Jordan form explained)
25. (explicit calculation of exp(M) for matrix M)
26. (change of basis for simple linear transformation)
27. (rank-nullity proof)
28. (isometries on Minkowski)
29. (Weingarten map vs. the metric for the geometry of surfaces in three dimensions)
30. (concerning analytic continuation and annuli, a hard problem)
31. (fun from Complex Analysis 2014)
32. (a word on Wirtinger derivatives)
33. (covariance and contravariance)
34. (good old-fashioned tensor calculus)
35. (fun with tensor calculus contractions)
36. (index shifting)

Last modified 8-10-2024

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