Math 600 Day 2: Review of advanced Calculus
Ryan Blair
University of Pennsylvania
Tuesday September 14, 2010
Ryan Blair (U Penn)
Math 600 Day 2: Review of advanced Calculus Tuesday September 14, 2010
MATH 600 Notes
Andrew A. Cooper
These are my notes for MATH 600: Geometric Analysis and Topology, taught Fall 2012 at the
University of Pennsylvania. The course texts were John Lee, Introduction to Sm
MATH 600 Calculus Cheatsheet
This cheatsheet is not, of course, complete.
Given a natural number k , a function u : Rm R is called C k if u has all possible k th derivatives and the
k th derivatives a
MATH 600 Homework 1
Due 14 September 2012
Lee 1-1 Let X be the set of all points (x, y ) R2 , such that y = 1, and let M be the quotient of X by
the equivalence relation (x, 1) (x, 1) for all x = 0. S
MATH 600 Homework 2
Due 28 September 2012
Lee 2-6 If M is a topological space dene C (M ) as the space of continuous maps M R.
(a) Note that pointwise operations make C (M ) an algebra over R.
(b) Giv
MATH 600 Homework 3.5
Not due, but do!
This assignment is intended to get you comfortable with operations involve tensors. Almost all of it comes
down using the transformation law for tensors in coord
MATH 600 Homework 3
Due 12 October 2012
1. If V is a vector space, we dene the sphere of V by S (V ) = (V \ cfw_0) / , where v w if v = w for
some > 0.
(a) Show that S (Rk+1 ) has a smooth structure w
MATH 600 Homework 4
Due 2 November 2012
1. Let V be a nite-dimensional vector space, 1 , . . . , k , 1 , . . . , k V
Lee 12-3 Let V be a nite dimensional vector space. Show that the cfw_ i |i=1, ,k a
MATH 600 Homework 5
Due 19 November 2012
Lee 17-2 Compute the ows of the following vector elds on R2 . Recall that a ow is a pair D, , where
D R M and : D M is a local group action of R on M .
(a) V =
MATH 600 Homework 6
Not due, but do!
Lee 14-1 Consider T2 = S 1 S 1 R4 , dened by w2 + x2 = y 2 + z 2 = 1, and with the product orientation.
Compute T2 xyzdw dy .
Lee 15-3 If M is a smooth manifold an
MATH 600 Topology Cheatsheet
This cheatsheet is not, of course, complete.
A topological space is a pair (X, O), where X is a set and O is a collection of open subsets of X so that:
1. X O, O
2. For an
Math 600 - Geometry Analysis and Topology
Herman Gluck
Tuesday August 30, 2016
2 x 2 MATRICES
We view 2 x 2 matrices as points in Euclidean 4-space R4 , ignore the zero matrix
at the origin, and scale
Math 600 - Geometric Analysis and Topology
Herman Gluck
Tuesday September 3, 2013
2. REVIEW OF ADVANCED CALCULUS
INTEGRATION (following Spivak's "Calculus on Manifolds")
Basic Definitions
The definiti
Math 600 - Geometric Analysis and Topology
Herman Gluck
Tuesday September 3, 2013
1. REVIEW OF ADVANCED CALCULUS
- DIFFERENTIATION Rn is said to be
Definition. A function f : Rm
differentiable at the
1. Midterm 1
Due: In Lecture 10-21
Problem 1. Identify the set of real 2 2 matrices with R4 , as in an earlier
homework problem, and let M 3 denote the 3-diml submanifold of matrices of rank
one. Find
Math 600 Day 1: Review of advanced Calculus
Ryan Blair
University of Pennsylvania
Thursday September 8, 2010
Ryan Blair (U Penn)
Math 600 Day 1: Review of advanced Calculus Thursday September 8, 2010
Math 600 Day 4: Dierentiable Manifolds
Ryan Blair
University of Pennsylvania
Tuesday September 21, 2010
Ryan Blair (U Penn)
Math 600 Day 4: Dierentiable Manifolds
Tuesday September 21, 2010
1 / 15
Out
Math 600 Day 5: Sards Theorem
Ryan Blair
University of Pennsylvania
Thursday September 23, 2010
Ryan Blair (U Penn)
Math 600 Day 5: Sards Theorem
Thursday September 23, 2010
1 / 18
Outline
1
Sards The
Math 600 Day 6: Abstract Smooth Manifolds
Ryan Blair
University of Pennsylvania
Tuesday September 28, 2010
Ryan Blair (U Penn)
Math 600 Day 6: Abstract Smooth Manifolds
Tuesday September 28, 2010
1 /
Math 600 Day 7: Whitney Embedding Theorem
Ryan Blair
University of Pennsylvania
Thursday September 30, 2010
Ryan Blair (U Penn)
Math 600 Day 7: Whitney Embedding Theorem
Thursday September 30, 2010
1
Math 600 Day 8: Vector Fields
Ryan Blair
University of Pennsylvania
Tuesday October 5, 2010
Ryan Blair (U Penn)
Math 600 Day 8: Vector Fields
Tuesday October 5, 2010
1 / 21
Outline
1
Vector Fields
Vec
Math 600 Day 9: Lee Derivatives
Ryan Blair
University of Pennsylvania
Thursday October 7, 2010
Ryan Blair (U Penn)
Math 600 Day 9: Lee Derivatives
Thursday October 7, 2010
1 / 17
Outline
1
Lee Derivat
Math 600 Day 10: Lee Brackets of Vector Fields
Ryan Blair
University of Pennsylvania
Thursday October 14, 2010
Ryan Blair (U Penn)
Math 600 Day 10: Lee Brackets of Vector Fields Thursday October 14, 2
Math 600 Day 11: Multilinear Algebra
Ryan Blair
University of Pennsylvania
Tuesday October 19, 2010
Ryan Blair (U Penn)
Math 600 Day 11: Multilinear Algebra
Tuesday October 19, 2010
1 / 14
Outline
1
M
Math 600 Day 12: More Multilinear Algebra
Ryan Blair
University of Pennsylvania
Thursday October 21, 2010
Ryan Blair (U Penn)
Math 600 Day 12: More Multilinear Algebra
Thursday October 21, 2010
1 / 14
Math 600 Day 12: Dierential Forms
Ryan Blair
University of Pennsylvania
Tuesday October 26, 2010
Ryan Blair (U Penn)
Math 600 Day 13: Dierential Forms
Tuesday October 26, 2010
1 / 14
Induced maps on d
Math 600 Day 14: Homotopy Invariance of de Rham
Cohomology
Ryan Blair
University of Pennsylvania
Thursday October 28, 2010
Ryan Blair (U Penn)
Math 600 Day 14: Homotopy Invariance of de Rham Cohomolog
Math 600: Integration on Chains and Stokes Theorem
Ryan Blair
University of Pennsylvania
Tuesday November 9, 2010
Ryan Blair (U Penn)
Math 600: Integration on Chains and Stokes Theorem
Tuesday Novembe
Integration on Manifolds
Outline
1
Integration on Manifolds
Stokes Theorem on Manifolds
Ryan Blair (U Penn)
Math 600: Integration on Chains and Stokes Theorem November 11, 2010
Thursday
1 / 14
Integra
Math 600 - Geometric Analysis and Topology
Herman Gluck
Thursday September 1, 2016
3. TOPOLOGICAL and DIFFERENTIABLE MANIFOLDS
A topological manifold of dimension n is a topological space
in which eac