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.

- 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.

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...
- a series of discussions with several students over the summer "break"(these conversations partly inspired the 2013 revision of the Advanced Calculus course. )
- (2013)advanced calculus, DEqns, manifolds
- DEqns and limits
- Multivariate MVT
- Contraction Mapping, Newton's Method
- tensors, wedges, multilinear algebra
- tensors remix, differential forms
- 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:

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:
- part 1 outline of course, inverse function theorem,
- part 2 examples of manifolds, atlas, topology,
- part 3 stereographic projection, smooth maps between manifolds,
- part 4 connected implies path connected on manifold, theorems on curves in a manifold,
- part 5 a bit more on bumps, derivations and differentiation on manifold,
- part 6 contrasting views of tangent vectors, the differential,
- part 7 tangent bundle,
- manifolds my study guide from the notes above
- Notes from a summer I read about Quantum Mechanics and Symmetries from Greiner's text:
- Chapter 3 and 4 on exponentiation of Lie algebra, Lie groups, generation of symmetries in quantum mechanics
- Chapter 5 on isospin and representations, the adjoint representation.
- Chapter 6 and 7 on hypercharge and SU(3) representations as derived from SU(2) subalgebras.
- Chapter 8 on quarks built from SU(3) representations
- 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:
- (form vs. area integrals)
- (spherical frame exhibition)
- (linearization and differential contrasted)
- (chain-rule and method of differentials)
- (a differential form calculation)
- (inuition on push-forward and pull-back)
- (concerning what a tangent vector is)
- (more fun with pull-backs)
- (heurstics on calculus on a manifold)
- (on the adapted coordinate chart of tangent bundle)
- (transformation law for n-form and determinant)
- (on pull-back and how to differentiate on manifold)
- (concerning orientations, a question from Morita)
- (product rule for vector-valued forms)
- (orientiation on manifold)
- (on induced orientations)
- (Lie bracket calculation)
- (bilinearity of two-form)
- (wedge and pull-back commute)
- (pull-back of one-form)
- (distributivity of tensors)
- (exterior product of matrix)
- (how to generate translation)
- (real Jordan form explained)
- (explicit calculation of exp(M) for matrix M)
- (change of basis for simple linear transformation)
- (rank-nullity proof)
- (isometries on Minkowski)
- (Weingarten map vs. the metric for the geometry of surfaces in three dimensions)
- (concerning analytic continuation and annuli, a hard problem)
- (fun from Complex Analysis 2014)
- (a word on Wirtinger derivatives)
- (covariance and contravariance)
- (good old-fashioned tensor calculus)
- (fun with tensor calculus contractions)
- (index shifting)

Last modified 8-10-2024

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