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
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Outline
1
Integration
Basic Denitions
Measure Ze
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 Smooth Manifolds and
John Milnor, Topology from the Diere
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 are continuous.
A function u : Rm Rn can be written as u
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. Show that M is locally Euclidean and second
countable, b
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) Given F : M N a continuous map, dene F : C (N ) C (M ) by
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 coordinates.
k
l
1. (a) Suppose that T is a tensor eld of ty
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 which is dieomorphic to S k . (Hint. Mimic the
construct
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 are linearly dependent i
1 k = 0.
Lee 12-4 If the cfw_
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 = y x +
y
(b) W = x x + 2y y
(c) X = x x y y
(d) Y = x y
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 and Ak (M ), Al (M ) are closed forms, show that the coho
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 any collection cfw_O of open sets,
O is an open set.
N
3
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 the rest to lie on the round 3-sphere S3(2) of radius
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 definition of the integral of a real-valued function
f: A
R def
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 point x0
Rm if there is a linear
map A: Rm
Rn such that
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 the tangent space to M 3 at the matrix
1
0
0
0
Problem
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
1 / 46
Outline
1
Dierentiation
Chain Rule
Partial Deriv
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
Outline
1
Dierentiable Manifolds
k-Dimensional Smooth Mani
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 Theorem
Ryan Blair (U Penn)
Math 600 Day 5: Sards Theorem
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 / 21
Outline
1
Transition to abstract smooth manifolds
Pa
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 / 19
Outline
1
Tangent Bundles and Derivatives of maps
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
Vector elds on smooth manifolds.
Integral curves of vector
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 Derivatives
Ryan Blair (U Penn)
Math 600 Day 9: Lee Derivative
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, 2010
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Outline
1
Lie Bracket
Ryan Blair (U Penn)
Ma
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
Multilinear Algebra
Ryan Blair (U Penn)
Math 600 Day 11:
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
Denition
If V is an n-dimensional vector space, dene
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 dierential forms.
Now let f : Rm Rn be a dierentiable ma
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 Cohomology
Thursday October 28, 2010
1/9
Dierential forms on man
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 November 9, 2010
1 / 17
Outline
1
Integration on Chains
In Euc
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
Integration on Manifolds
Integration on Manifolds
The goal of
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 each point has an open neighborhood homeomorphic
to Rn and