from the editor

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!

Matrix Lie Groups by James Cook at Liberty University

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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"

Matrix Lie Groups: Lecture 1 part 1: complex and quaternions 59:51 We are working through Stillwell's Naive Lie Groups. There is a second part to this.
Lecture 1 part 2: complex and quaternion 25:38
Lecture 2 part 1: more quarternions, group theory 59:51
Lecture 2 part 2: more quarternions, group theory 28:25
Feb 1, chapter 2 (part 1) 59:51
Feb 1, chapter 2 and isometries over R,C and H (part 2) 31:21
Feb 8, chapter 3 path connectedness for SO(n) and SU(2) (part 1) 59:51
Feb 8, chapter 3 path connectedness for SO(n) and SU(2) (part 2) 27:36
Feb 22, Lie Groups and tangent space at I (part 1) 59:51
Feb 22, Lie Groups and tangent space at I (part 2) 34:54
March 8, what is the logarithm (part 1) 59:51
March 8, what is the logarithm (part 2) 29:27
March 21, on the derivative of a homomorphism 30:27
March 30, adjoint and Lie Algebra (part 1) 59:51
March 30, adjoint and Lie Algebra (part 2) 32:01
April 4, Lie algebra example, 2nd order BCH (part 1) 59:51
April 4, Lie algebra example, 2nd order BCH (part 2) 25:49
April 11, BCH identity (part 1) 59:51
April 11, BCH identity (part 2) 18:56
April 18, calculation of BCH (part 1) 59:51
April 18, BCH calculation (part 2) 30:21
universal covers and Lie's Theorem, April 25 (part 1) 59:51
covering groups and Lie's Theorem, April 25 (part 2) 12:57
May 2, group orbits and homogeneous space (part 1) 59:51
May 2, group orbits and homogeneous space (part 2) 36:08

Matrix Groups: crash course 2015 by James Cook

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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

Lie Algebra: talks paired with Erdmann and Wildon's text by James Cook

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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.

Solvable and Semisimple (part 1) 59:51 here is a painfully slow presentation of the first part of Chapter 4 from Erdmann and Wildon. Here we study the derived series of a Lie Algebra which leads us to define the radical as the largest solvable ideal. We conclude by defining semisimple. This is part 1 of 3 (there is about 2 hours of this)
Solvable and Semisimple (part 2) 6:48
Solvable and Semisimple (part 3) 37:17
nilpotent and big picture ahead part 1 59:51
nilpotent and big picture ahead part 2 5:16

Differential Geometry (2015) by James Cook

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source: James Cook      2015年6月2日

Differential Geometry: Lecture 1: overview 5:23
Lecture 2 part 1: points, vectors, directional derivative 23:30
Lecture 2 part 2: points, vectors, directional derivative 39:18
Lecture 3 Part 1: differential forms 39:42
Lecture 3 Part 2: differential forms 12:21
Lecture 3 Part 3: differential forms 2:56
Lecture 4 part 1: curves and velocity 32:59
Lecture 4 part 2: Jacobian and push-forward 36:29
Lecture 5 part 1: frames and components in R3 53:32
Lecture 5 part 2: attitude matrix and calculus along curve 46:28
Lecture 6 part 1: Frenet Serret Equations 57:01
Lecture 6 part 2: Frenet Serret for nonconstant speed 20:09
Lecture 7 part 1: covariant derivative in R3 31:08
Lecture 7 part 2: connection form in R3 17:50
Lecture 7 part 3: matrices of forms 32:26
Lecture 8: coframes and structure equations for R3 59:51
Lecture 9 part 1: Euclidean geometry of Rn 25:00
Lecture 9 part 2: push forward of isometries 28:30
Lecture 10: Frenet Curves in Rn 34:46
Lecture 11 Part 1: pushing around vectors in R3 27:28
Lecture 11 Part 2: congruence of curves in R3 51:13
Lecture 12 part 1: surfaces in R3 11:00
Lecture 12 part 2: 19:03
Lecture 12 part 3: surfaces 8:38
Lecture 12 part 4: calculus on surface 16:17
Lecture 12 part 5: TpM the tangent space 49:12
Lecture 13 part 1: differential forms on surface in R3 40:23
Lecture 13 part 2: push-forward of cartesian frame 5:19
Lecture 13 part 3: 41:33
Lecture 13 part 4: diffeomorphism of surfaces 10:49
Lecture 13 part 5 19:08
Lecture 13 part 6 11:16
Lecture 14 part 1: topological trivia for surfaces 43:03
Lecture 14 part 2: proof and manifolds 15:25
Lecture 15 part 1: Shape Operator Defined 41:05
Lecture 15 part 2: normal curvature 35:05
Lecture 15 part 3: Gaussian and Mean curvature 38:50
Lecture 16: calculation of K and H 35:02
Lecture 17: on principal,aymptotic and geodesic curves 56:22
Lecture 18: adapted frame fields of surfaces in R3 41:47
Lecture 19: theorems for surfaces 13:32
Lecture 20 part 1: isometries of surfaces 26:31
Lecture 20 part 2: Gauss' Awesome Theorem 23:35
Lecture 21 part 1: orthogonal patches and Gaussian Curvature 43:03
Lecture 21 part 2: total Gaussian curvature 13:02
Lecture 22: congruence of surfaces 6:33
Lecture 23: metric on geometric surface 26:58
Lecture 24: curved abstract surfaces 42:03
Lecture 25: covariant derivatives again 41:17
Lecture 26: geodesics on geometric surfaces 29:57
Lecture 27 part 1: Gauss Bonnet Theorem 55:55
Lecture 27 part 2: euler characteristic of torus 3:43
Lecture 28: applicationa of Gauss Bonnet 18:18
Lecture 29: I, II and III form notation 17:08

Complex Analysis (Fall 2015) by James Cook at Liberty University

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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.

Complex Analysis: Lecture 1: complex numbers 28:39 Here we discuss a few possible models for the complex numbers. Our choice, for convenience, is that the complex numbers are the plane with the Gauss multiplication. That said, we use the notation a+ib in almost everything which follows. Complex conjugation is defined and connections to the real and imaginary part maps are given. We also discuss why every nonzero complex number has a multiplicative inverse (but, I stop short of mentioning that makes C a field, more on that next time). We are about half-way through section 1.1 of http://www.supermath.info/GuideToGame... also the course page is at http://www.supermath.info/Complex.html
Lecture 2: imaginary exponential and nth roots 54:07
Lecture 3: branch cuts, complex exponential 55:02
Lecture 4: more on square root and trig and hyperbolics 54:01
Lecture 5: sequential limits, open sets 45:38
Lecture 6: topological terms and complex differentiability 51:06
Lecture 7: Caratheodory applied and CR equations 52:35
Lecture 8: partial proof of Cauchy Riemann result 55:28
Lecture 9: CR equations and Inverse Function Theorem 51:13
Lecture 10: harmonic functions 48:54
Lecture 11: conformality and FLTs 49:55
Lecture 12: circles and lines and infinity fun with FLT 49:15
Lecture 13: solution to quiz 1 50:20
Lecture 14: exact forms and real integration 50:03
Lecture 15: closed and exact forms 53:41
Lecture 16: closed forms and harmonic conjugates 50:27
Lecture 17 : fluid flow, circulation and flux, Laplace equation 54:01
Lecture 18: Laplace Solns, Complex Integral Introduced 53:30
Lecture 19: basics of complex integration 53:46
Lecture 20: Cauchy's Theorem and Integral Formula 56:19
Lecture 21: examples of applying Cauchy's Integral Formula 50:02
Lecture 22: an example with partial fractions, Morera and Goursat proofs 53:26
Lecture 23: solution to Quiz 2 51:41
Lecture 24: review for Test 2 48:58
Lecture 25: sequences, convergence 48:29
Lecture 26: domain of power series 50:26
Lecture 27: power series calculation 49:36
Lecture 28: power series calculation, order of zeros 55:08
Lecture 29: analytic continuation elementary comments 51:12
Lecture 29.5: singularity theorems 38:57
Lecture 30: partial fractions, residues 53:42
Lecture 31: Cauchy's Residue Thm and the 4 rules 51:03
Lecture 32: integration via residue theory 49:05
Lecture 33: argument principal, Rouche's Theorem 51:35
Lecture 34 part 1: comment on non-Euclidean geometry, Mittag Leffler exhibited 11:29
Lecture 34 part 2: some infinite product calculations 37:12
Lecture 35: series calculation via residues 44:33

PDEs and Geometry (Spring 2017) by James Cook

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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.

Math 495: pdes and regular Sturm Lioville, 2-2-17, part 1 59:51 (I'm working on Chapter 11 of the 4th edition of Nagle, Saff and Snider, I might say things from Haberman or Logan in time... then, eventually, we'll get to some differential geometry)
pdes and regular Sturm Lioville, 2-2-17, part 2 27:02
regular Sturm Liouville, working toward Green's, 2-7-17, part 1 59:51
regular Sturm Liouville, working toward Green's, 2-7-17, part 2 5:01
on Green's Functions for PDEs, Laplace Fourier examples, 2-14-17, part 1 59:51
on Green's Functions for PDEs, Laplace Fourier examples, 2-14-17, part 2 44:37