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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.

Advanced Calculus: Lecture 1 part 1: normed linear spaces 59:51 Here I give a very brief overview of linear algebra, for my students, I hope the first homework helps complete the review. Then I discuss normed linear spaces, metric spaces and inner product spaces. Again, the first homework will flesh out the discussion begun here. Finally, we define open sets and limits on a NLS and explore the norm-indendence of topology in the plane.

Lecture 1 part 2: squaring the circle 13:22

Lecture 2 part 1: limit laws in normed linear spaces 59:51

Lecture 2 part 2: limit laws for normed linear spaces 17:58

Lecture 3 part 1: Frechet Derivatives 59:51

Lecture 3 part 2: Frechet Derivatives 14:40

Lecture 4 part 1: product rule to rule them all (with sound) 22:11

Lecture 4 part 2: continuous differentiability (sound added) 24:52

Lecture 4 part 3: Jacobian Matrix and Linearization Example (with sound) 5:48

Lecture 4 part 1: continuous differentiability and product rules 59:51

Lecture 4 part 2: continuous differentiability and product rules 9:59

Lecture 5 part 1: continuous differentiabilty and chain rule 59:51

Lecture 5 part 2: continuous differentiabilty and chain rule 13:42

Lecture 6 part 1: inutition on inverse and implicit function theoremstutions 9 10 59:51

Lecture 6 part 2: inutition on inverse and implicit function theorems 17:14

Lecture 7 part 1: implicit differentiation 59:51

Lecture 7 part 2: implicit differentiation 16:58

Lecture 8 part 1: tangent and normal spaces 59:51

Lecture 8 part 2: tangent and normal spaces 8:49

Lecture 9 part 1: Lagrange multipiers 59:51

Lecture 9 part 2: Lagrange multipliers 12:21

Lecture 10 part 1: 2nd derivative test and quadratic forms 59:51

Lecture 10 part 2: 2nd derivative test and quadratic forms 17:04

Lecture 11 Part 1: differentiation under integral, variational calculus 59:51

Lecture 11 Part 2: differentiation under integral, variational calculus 13:11

Lecture 12 Part 1: examples of variational calculus 59:51

Lecture 12 Part 2: examples of variational calculus 17:11

Lecture 13 Part 1: Snell's Law, method of Lagrange multipliers 59:51

Lecture 13 Part 2: Central Force Problem 15:50

Lecture 14: Part 1 : review for Test 1 59:51

Lecture 14: Part 2 : review for Test 1 16:52

Lecture 15 Part 1: tensors and coordinate change 59:51

Lecture 15 Part 2: tensors and coordinate change 15:19

Lecture 16 Part 1: duals, double duals and isomorphisms of tensor products 59:51

Lecture 16 Part 2: duals, double duals and isomorphisms of tensor products 14:54

Lecture 17 Part 1: wedge products, flux and work form, determinants 59:51

Lecture 17 Part 2: wedge products, flux and work form, determinants 16:16

Lecture 18 Part 1: more on metrics and musical morphisms 59:51

Lecture 18 Part 2: more on metrics and musical morphisms 14:00

Lecture 18 Part 3: Hodge Duality 48:58

Lecture 19: manifolds and calculus, derivations and push-forwards 59:51

Lecture 19 part 2: differentials as a dual basis to coordinate derivations 16:42

Lecture 20 Part 1: exterior derivatives and pull-backs 59:51

Lecture 20 Part 2: properties of pull-backs 18:02

Lecture 21 Part 1: pull-backs, exact and closed forms, Poincare lemma 59:51

Lecture 21 Part 2: electromagnetism in differential forms 16:22

Lecture 22 Part 1: Homotopy and De Rahm Cohomology 59:51

Lecture 22 Part 2: convex sets and the convex hull 15:39

Lecture 23 Part 1: from polytopes to k-faces, simplicial Homology 59:51

Lecture 23 Part 2: Generalized Stokes Theorem 17:52

Lecture 24 Part 1: homology of S2, proof of GST 59:51

Lecture 24 Part 2: electrostatics in 4D at end 19:05

Lecture 25 Part 1: Hodge theory, coderivative and Laplacian 59:51

Lecture 25 Part 2: on solutions to DEqns and Frobenius 13:03

Lecture 26 Part 1: Frobenius Theorem about vectors, forms and foliations 59:51

Lecture 26 Part 2: statements of the Frobenius Theorem 13:04

Lecture 27 Part 1: calculations to illustrate Frobenius Theorem 59:51

Lecture 27 Part 2: on compatibility condition for PDE example 13:48

Lecture 28 part 1: a bit on jet spaces and DEqns 21:13

Lecture 28 part 2: a bit on jet spaces and DEqns 9:57

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