Syllabus for Math 42

Winter 2002

This is subject to change, so please check regularly.

Also, the chapters are pretty short in our text, so don't freak out over the reading assignments.

 

Week 1

Date

Class Topics

Reading (due next class)

Problems

Fri., Jan. 4

What's a manifold?

Geometry/Topology (sphere theorem and Hadamard's Theorem)

A little topology (open, closed, connected, compact, etc.)

Vector Fields & Integral Curves

Chpt. 1-3

HW 1 DUE January 11

Sat., Jan. 5

WE'RE NOT MEETING FOR THE SPECIAL DAY OF CLASSES

Week 2

Date

Class Topics

Reading (due next class)

Problems

Mon., Jan. 7

Vector Fields & Integral Curves

Chp. 4

Tues. Jan 8

Integral Curves

Tangent Spaces

Chp. 5

Wed., Jan 9

Tangent Spaces

Implicit Function Thm.

Surfaces

chp. 6

Fri., Jan 11

Functions on Surfaces & The method of Lagrange Multipliers

HW 1 DUE TODAY

Chp. 1-6

HW 2 Due Jan 18

Week 3

Date

Class Topics

Reading (due next class)

Problems

Mon., Jan. 14

Lagrange Multipliers

Vector Fields & Integral Curves on Surfaces

Chp. 8

Wed., Jan. 16

Orientation

Möbius Strip & Real Projective Space

Covariant Differentiation

Chp. 7

Fri., Jan. 18

Geodesics

HW 2 Due Today

Chp. 6 and Review

Week 4

Date

Class Topics

Reading (due next class)

Problems

Mon., Jan. 21

MLK Day (Class Moved to X-hour)

Tues., Jan 22

Geodesics

HW 2 & re-write of HW1 Due Today

Chp 9

Wed., Jan. 23

Parallelism on n-surfaces

Chp. 9

Fri., Jan, 25

Parallel Transport

More on Differentiating vector fields

Chp. 10 & 12

Week 5

Date

Class Topics

Reading (due next class)

Problems

Mon., Jan. 28

No Class

Wed., Jan. 30

Covariant Differentiation

The Shape Operator (a.k.a. Weingarten Map)

Guest Lecturer: Prof. Chernov

Chp. 11

Fri., Feb. 1

Curvature of plane Curves

HW 3 Due Today

Chp. 1-12

HW 4 Due Feb. 6

Week 6

Date

Class Topics

Reading (due next class)

Problems

Mon., Feb. 4

Curvature of Surfaces: Quadratic Forms, First & Second Fundamental Form, Normal Curvature

Chp. 1-12

Tues., Feb. 5

Gauss-Kronecker Curvature, Mean Curvature

Interpreting/Calculating Gaussian curvature

Chp. 1- 12

Wed., Feb. 6

Interpreting/Calculating Gauss-Kronecker Curvature

HW 4 Due Today

Review & Chp. 14

Revisions Due Monday

Fri., Feb. 8

Winter Carnival - No Class

Review & Chp. 14

Revisions Due Monday

Week 7

Date

Class Topics

Reading (due next class)

Problems

Mon., Feb. 11

Comments on Exercises 2.10 (Flows) and 8.7 (Holonomy)

Revisions Due Today

Tues. Feb. 12

Review Session

Wed., Feb. 13

Parametrized Surfaces

Chp. 15

Th., Feb. 14

Valentine's Day

Fri., Feb. 15

More on Parametrized Surfaces

Midterm 4-6pm

104 Bradley

Chp. 17

Week 8

Date

Class Topics

Reading (due next class)

Problems

Mon., Feb. 18

locally n-surfaces are parametrized n-surfaces.

Coordinate Charts and Atlases

Tues. Feb. 19

Stereographic Projection

Locally param. n-surfaces are n-surfaces

Revisiting smooth functions on n-surfaces

Chp. 22

HW 5 due Feb. 25

Oral Exam Topic Due March 4

Wed., Feb. 20

Inverse Func. Thm. for Surfaces

Smooth Structures on n-surfaces.

Fri., Feb. 22

New definition of Smooth functions using local param. (a.k.a., coordinate charts)

Final Call for re-writes of HW 3 & 4*

Chp. 23

* If you don't have a final grade for HW 3 or 4 you should hand in your first draft along with any revisions you'd like to make. The revisions are optional, but clearly doing them can improve your grade and, more importantly, your understanding of the material.

Week 9

Date

Class Topics

Reading (due next class)

Problems

Mon., Feb. 25

Rigid Motions

HW 5 due today

Tues. Feb. 26

Congruence preserves geometric properties

Chp. 23

HW 6 Due March 5

Wed., Feb. 27

More on Congruence

Isometries

Fri., March 1

Isometries

Chp. 24

Week 10

Date

Class Topics

Reading (due next class)

Problems

Mon., March 4

Isometries

Riemannian metrics

Oral Exam/Presentation Today

Tues. March 5

Riemannian Metrics

HW 6 Due Today

Wed., March 6

Riemannian metrics & Summing up the quarter