INTRODUCTION TO MODERN DIFFERENTIAL GEOMETRY

MATH 453 2012 Jerzy Kocik

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Office: Jerzy Kocik, Neckers 377 A. E-mail: jkocik at siu.edu Poster for this class. The adjacent digital sculpture is created by Jos Leys. Here is its full-size full-size version. Is it a one-sided surface?

Announcements

• Consider participating in Symmetry and geometry: applications of group theory,
  a course that is proposed for Spring 2013. Here is the proposal.
• Interesting sites -- here we shall collect relevant sites. What you see now is a temporary rather unordered and random collection.
• Ciderella offers a version for free. Geogebra is also very user-friendly (and free).
• Maple -- The course will be an occasion to get acquainted with Maple -- a computer program that can spice the life of every student of mathematics. This will be done via projects, homework and some lab demonstrations. (the availability of the program will be explained during our first meeting). One may as well use different programs like Matlab or Mathematica.

Classes

week 1

[20 Aug]

Lecture 1: Introduction.
Lecture 2: Review linear spaces.
Lecture 3: Dual space.

week 2

[27 Aug]

Lecture 4: Some proofs.
Lecture 5: Drawing covectors.
Lecture 6: What is inner product, really.

week 3

[3 Sep]

Lecture 7: Tensors.
Lecture 8: Surfaces.

week 4

[10 Sep]

Lecture 9: Surgery of surfaces. Projective space.
Lecture 10: Fundamental group.
Lecture 11: Differential manifold.

week 5

[17 Sep]

Lecture 12: Examples of manifolds. Stereographic projection.
Lecture 13: Life on manifolds: curves and scalar functions.
Lecture 14: What is "vector", really.

week 6

[24 Sep]

Lecture 15: Vector sand vector fields
Lecture 16: Lie bracket
Lecture 17: Review

week 7

[1 Oct]

Lecture 18: Differential 1-forms
Lecture 19: Exterior differential forms
Lecture 20: Functions and Pfaff forms

week 8

[8 Oct]

Lecture 21: Exterior derivative
Lecture 22: Exterior derivative

week 9

[15 Oct]

Lecture 23: Exterior derivative
Lecture 24: Concantenation with vectors / vector fields
Lecture 25: Riemann manifolds

week 10

[22 Oct]

Lecture 26: Mandala of 3D Calc
Lecture 27: Mandala of 3D Calc
Lecture 28: Rank of a differential 1-form. Induced inner product

week 11

[29 Oct]

Lecture 29: The concept of length of a curve
Lecture 30: Poincare upper half-plane
Lecture 31: Back to 3D Mandala

week 12

[5 Nov]

Lecture 32: Induced maps
Lecture 33: Induced maps
Lecture 34: Induced maps, line integral.

week 13

[12 Nov]

Lecture 35: Integrals of exterior differential forms
Lecture 36: Chains and boundaries.
Lecture 37: Stokes theorem -- proof.

week 14

[19 Nov]

Lecture 38: Stokes theorem in 3D (Calculus revisited)
BREAK

week 15

[26 Nov]

Lecture 39: Stokes theorem (review)
Lecture 40: Electrostatics
Lecture 41: Maxwell equations in 4D