MATH 332: Advanced Calculus
James Cook's Advanced Calculus Homepage
Welcome, I include here the most recent and the most ancient materials on this course. Enjoy.
Useful Materials and Links:
Advanced Calculus of Fall 2015:
We are mostly following my Lecture Notes (2015) which are based on "Advanced Calculus of Several Variables," by C.H.Edwards, Jr. which is
available from Dover. However, I am influenced by many other sources. In particular, I'm looking at Renteln's "Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists" for some interesting new homework ideas. I'm also hopeful to implement the musical morphisms and coordinate change partly from a few sections in Jeffrey M. Lee's "Manifolds and Differential Geometry" (Graduate Studies in Mathematics). In any event, you can follow along at Advanced Calculus You Tube Playlist (Lectures from Fall 2015).
There were two Tests,
Problems and Exams with some Solutions from past years:
- homework, tests from Fall 2013:
- homework, tests from Fall 2011:
- homework, exams and some solutions from Spring 2010:
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.
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:
Course Notes from Fall 2013:
- (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
I post below my third version of Math 332 notes. This updates the 2011 version to include more on normed linear spaces. Also, I make
an effort to be more inuitive and less formal in the introduction of manifolds.
the third version of my Math 332 course notes
Course Notes from Fall 2011:
I post below my second version of Math 332 notes. This was a fairly major revision of the first version. In particular, I no longer
delay proof of the implicit and inverse function theorem. Instead, I sketch the proofs and refer the reader to Edwards.
the second version of my Math 332 course notes
Course Notes from Spring 2010:
In my first run through this course we used C.H. Edward's Advanced Calculus text. I partly used Hildebrand for the variational calculus portion of the course as well. What is posted below are the notes I provided during that semester. They are partly based on my Math 430 notes and in Chapter 14 my notes from a Junior level classical mechanics course I took at NCSU as an undergraduate.
the first version of my Math 332 course notes
Notice the blank spots are filled in in the pdfs below.
- Chapter 2 (version 2) with what was said in lecture more or less.
- Chapter 3 with what was said in lecture more or less.
- Chapter 4 with what was said in lecture more or less.
- Coriolis effect adjoined to end of chapter.
- Chapter 6 with what was said in lecture more or less.
- Chapter 6 with what was said in lecture more or less.
- Chapter 7 on local extrema from lense of multivariate Taylor and the joy of quadratic forms.
- Chapter 8 on manifolds as level sets and Lagrange multipliers
- Chapter 9 on generalized Newton's method and how it justifies the implicit and inverse function and mapping theorems (wow).
- Chapter 10 on basics of abstract manifold theory. Just the basics.
- Chapter 11 is on exterior algebra of forms on Rn ( part of this )
- Chapter 12 is on the Generalized Stokes Theorem ( part of this )
- Chapter 13 is on Hodge duality ( part of this )
- Chapter 14 is on Variational Calculus: the main chapter and some extra and the central force problem.
A few calculations which are advanced calculus we did not precisely cover:
Notes from a summer I read about Quantum Mechanics and Symmetries from Greiner's text:
- 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
When I find an interesting question at the Math Stack Exchange then I sometimes put a little effort into producing an answer:
- 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)
Last modified 5-23-2016
- (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)
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