1. Clicking ▼&► to (un)fold the tree menu may facilitate locating what you want to find. 2. Videos embedded here do not necessarily represent my viewpoints or preferences. 3. This is just one of my several websites. Please click the category-tags below these two lines to go to each independent website.
Due to the new embedding mechanism implemented by YouTube, it becomes too difficult for me to carry on the web project on this site all by myself. I decided to stop all the scheduled editing works here for the time being. The project will resume only when a substantial help appears. This web project does need help. Please help spread the word if you can. Bye for now. Take care!
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source: James Cook 2016年1月19日
Matrix Lie Groups
You might also call this a course in Naive Lie Groups. The focus is on matrix group examples and a minimum of background in topology and manifold theory is needed. In particular, we follow Stillwell's text "Naïve Lie Groups"
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source: James Cook 2015年8月11日
Matrix Groups: crash course 2015
Here I give a series of little talks that walk you through my notes from reading Tapp's "Matrix Groups for Undergraduates". We see a good amount of detail about the complex linear matrices or quaternionic linear matrices which are special matrices of real numbers which encode the multiplication of complex or quaternionic matrices. Then, the later part of the talks outlines the rudiments of Lie Theory of Groups and Algebras in the context of matrices. Here probably too much detail is missing, so don't be discouraged if it's hard to follow. Finally we do some sight-seeing about the classification theory of compact connected Lie groups and the theory of maximal tori.
I assume the audience has some experience with manifold theory or differential geometry and a willingness to learn some group theory on the fly. Almost everything I say in these talks is also found directly in Tapp where you will also find further discussion and much more depth. Ideally, these videos might convince you to study further.
Matrix Groups: Part 1 13:42 In this talk we cover pages 1 to 4 of my notes where the K-notation for K=R,C,H is explained and the general linear group of nxn matrices over K is defined. We also begin to explain mapping which embeds nxn complex matrices in 2nx2n real matrices. Part 2 20:16 Part 3 15:26 Part 4 13:47 Part 5 16:52 Part 6 13:34 Part 7 13:47 Part 8 5:18 Part 9 21:28 Part 10 7:09 Part 11 9:12 Part 12 7:54 Part 13 25:32 Part 14 11:01
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source: James Cook 2015年3月24日
Lie Algebra: talks paired with Erdmann and Wildon's text
not much to see here. At some point I hope to make a proper series of videos. Basic Lie Algebras is one of the most interesting and pure applications of linear algebra. It's a great playground to hone your skills in theoretical linear algebra. Erdmann and Wildon's text is a good place to start.
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source: James Cook 2015年8月24日
Complex Analysis of 2015
This is the playlist for lectures captured from Math 331 of Liberty University of Lynchburg VA during the Fall 2015 semester. This course follows part of Gamelin's Complex Analysis. This introduction to complex analysis is intended for someone who has completed the calculus sequence and has some experience with proof. However, there is no real analysis prerequisite so we do not chase all the rabbits... just some. Anyway, see: http://www.supermath.info/Complex.html.
In a nutshell, we study complex number models, basic topology and limit theory in the complex plane, nth roots of unity, the complex exponential, log, sine, cosine, cosh, sinh, the CR equations and the relation of complex-linearity of the differential, holomorphic functions, conformal functions, basic idea of analytic continuation, Riemann Sphere, harmonic functions and conjugates, Green's Theorem complexified, Cauchy integral theorem and formulas, Goursat Theorem, Lioville Theorem, basic theory of complex power series, Laurent series, residue theory, techniques of complex integration, conformal mapping and fluid flow (perhaps from Fisher's text which is available as a Dover edition) , Mandelbrot and Julia set, and select topics TBD in part by interest of students.
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source: James Cook 2017年2月2日
Math 495: PDEs and Geometry
These are videos gathered from a Math 495 course I'm running in the Spring 2017 Semester. We're looking at some assorted topics which we don't usually cover in the required DEqns course then we'll turn to differential geometry in a few weeks. Many meetings I don't have a video here since we're working on a paper in this meeting time. The PDE part is largely based on Haberman's 3rd edition PDE book and the 4th edition of Nagle Saff and Snider's introductory Differential Equations text.
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source: James Cook 2015年2月24日
Number Theory of 2015
These lectures are from my first attempt at teaching number theory in the Spring 2015 semester at Liberty University. We used Stillwell's Elements of Number Theory text. We covered most of the book, with the exception of a couple technical results.
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source: James Cook 2016年1月21日
Undergraduate Topology: These are from a short course based on Manetti's text. We hope to cover the essential topics in point set topology.
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source: James Cook 2016年8月24日
Elementary PDEs with applications to Physics
These are from Fall 2016 semester. This course will examine the standard introductory PDEs; the heat, wave and Laplace equations. We'll see how Fourier decompositions are used to match initial conditions. We also consider characteristics, Green's functions, Laplace and Fourier transforms. I plan to work from Logan's Third Edition of Applied Partial Differential Equations. Also, the texts of Haberman, Weinberg and Gustafson. I also used Nagle Saff and Snider to prepare the notes I sometimes reference. See http://www.supermath.info/DifferentialEqns.pdf for my elementary differential equations notes. The last chapter is on PDEs, I'll cover it in these videos. Basically, we'll be bouncing from book to book as the semester unfolds.
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source: James Cook 2015年1月15日
These are my Lectures from the Spring 2015 offering of Physics 231 at Liberty University. This is a course in University Physics which covers the physics of motion from a calculus-based perspective. Moreover, we use vectors to communicate physical law. This course covers the topics of Kinematics, linear and rotational dynamics, linear and angular momentum, Gravity and whatever else time allows. My hand-written Lectures (not in-sync with the numbering below) are found at: http://www.supermath.info/PhysicsI.html
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source: James Cook 2015年8月25日
Advanced Calculus of 2015
This is the playlist for Math 332 as captured at Liberty University in Lynchburg VA during the Fall 2015 semester. This is not a real analysis course even though it is very much about analysis of real-valued functions of a real-variable. In particular, this course focuses first on the theory of differentiation for mappings on a finite-dimensional (mostly) normed linear space. We include sketches of the implicit and inverse function theorems as well as a presentation of the multivariate Talyor series and theory of optimization. Also, a brief introduction to classical variational calculus is given. Differentiating under the integral is also studied. Essentially, I try to make good on the title "advanced" calculus, I seek to present all manner of computational techniques which you probably did not see in your earlier coursework.
However, I also intend to build a bridge to higher mathematics here. I do intend to present the concept of a manifold together with some of the basic calculus of curved spaces. So, the second half of the course centers around differential forms which provide the natural calculus in higher-dimensional space. We study multilinear maps built from the tensor product, the wedge product, the exterior derivative, push-forward and pull-back and a few deeper things like the Generalize Stokes' Theorem (no proof offered here!) and the Poincare Lemma (proof given based closely on argument found in Flander's text). Also, I present a 5-dimensional electrostatics where we see how the number of ambient spatial dimensions determines the field-strength.
All of the things we study in the second half of the course have abstractions and formalizations which are deep, abstract, and beyond this course. I merely hope to get your attention and for the interested student take some of the sting out of more advanced treatments.
Finally, I have a few lectures left at the end to do something really novel. TBD by my semester and of course the interest of the students. The official text for the course is Edward's Advanced Calculus text available as a Dover. Almost everything I gloss over is treated carefully in the body of Edwards. In there you'll find a technically precise proof of the implicit and inverse mapping theorems replete with the contraction mapping technique. Also, Generalized Stokes' Theorem is established with some care.
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source: James Cook 2015年8月24日
Multivariate Calculus
These are lectures captured from Math 231 at Liberty University in Lynchburg VA. Much more for this course can be found at the course website: http://www.supermath.info/MultivariateCalculus.html
This is the third semester in the usual calculus sequence. In a nutshell, we study of vector algebra, curves in 3D including Frenet-Serret equations, limits of several variables, partial differentiation, local extrema, Lagrange multipliers, closed set test, double and triple integrals, Jacobians, curvelinear coordinates, gradients, level curves, line and surface integrals, Green's Theorem, Stokes' Theorem and Gauss' Theorem. Some attention is paid to a few nontraditional topics: we do discuss the general derivative, cylindrical and spherical coordinate frame fields and the formulae for calculation of grad, div, curl and Laplacian in noncartesian setting. Finally, I'll probably include some discussion of differential forms if time permits. Some proofs are given in this course, but, deeper theorems are just sketched. Past this course, for the student who also has some experience with linear algebra, I recommend study of my advanced calculus course if you like my take on calculus III.
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source: James Cook 2017年1月16日
These are the Lectures from my section of Math 231 from the Spring 2017 Semester. While my notes are based on several calculus texts (Stewart, Thomas, Salas and Hille and Etgen etc.) these Lectures follow my notes primarily. I am doing my best to make the notes self-contained so there is no need to purchase additional materials. That said, it's probably wise to get an old edition of Salas and Hille or something for the sake of having many additional homework examples to ponder.
My understanding of multivariate calculus stems in part from my background in physics. I have much higher expectations for what multivariate calculus should encompass. On the other hand, I may spend a bit less time on certain issues of analysis than other math professors. I take up those issues in greater generality in the advanced calculus class (audience permitting).
Topics: vectors, calculus and geometry of paths, Frenet frames, limits of functions of several variables, partial derivatives, chain rules, integration over areas or volumes, optimization, Lagrange multipliers, line integrals, surface integrals, theorems of vector calculus including Green, Stokes and Divergence. Time permitting, the theory of scalar and vector potentials. We also hope to see how to present various constructions in noncartesian coordinate frames.
