Math361. Lecture 2.
Inverse and implicit functions
A.A.Kirillov
Spring 2014
1
Introduction
In this lecture we study the local structure of a smooth map f : Rn
Rm under condition that its derivative Df (x) has a constant rank k in a
neighborhood of a give
MATH 361
University of Pennsylvania
Spring 2016
Supplementary Lecture
by Elaine So
1. Week 1: Multivariable Derivatives
1.1. Preliminaries.
Denition 1.1.1. X = Rn is the nth-dimensional Euclidean vector space. Elements are vectors
1
x
.
1
n
1
n T
= . .
Math361 Spring 2016. Lecture 4. Integration
A.A.Kirillov
Spring 2014
In this lecture we collect main facts about the integration of functions
of several variables. The proofs can be found in Chapter 8 of the textbook.
You are supposed to know the formulat
Lecture 3. Smooth manifolds
A.A.Kirillov
Math361. Spring 2016
In this lecture the adjective smooth is used as a synonym for the
expression of class C k where k 1 depends on the context.
Smooth submanifolds of Rn
1
The simplest example of a smooth submanif
Lecture 1. Dierentiable maps of Euclidean
domains
A.A.Kirillov
Math361. Spring 2016
1
Introduction
f
An Euclidean domain is an open subset X of Rn . Consider a map X Rm .
To dene such a map we need m real-valued functions of n real variables.
Indeed, a po
Math 361- Quiz 1
January 28, 2013
1. (Worth 5) Prove that the following is indeed a norm on the vector space Rn .
|x| = max |xi |
1in
Proof. (a) If x = 0 = |x| = 0. If |x| = 0 = max1in |xi | = 0 =
|xi | = 0 i = x = 0. Now |x| 0 because it is a max of abso
Math 361- Quiz 2
Feb 6, 2014
1. (Worth 5) Show that if V, | |V and W, | |W are nite dimensional vector spaces
equipped with norms and T : V W is a linear map then T is continuous. Use the
( , ) proof giving the explicit value of as a function of and speci
Solutions
HW #6
MATH 361
April 5, 2014
1) Suppose S is the portion of the plane x + 2y + 2z = 6 that lies inside the rst octant
given by the mapping:
(x, y) =
x, y,
6 x 2y
2
and
= xdy dz + ydz dx + 3zdx dy
is a 2-form over R3 . Compute the integral of ov
Anonymous
Math 361: Homework 5
Rudin 10.12 Let Ik be the set of all u = (u1 , ., uk ) Rk with 0 ui 1 for all i; let Qk be the set of all
x = (x1 , ., xk ) Rk with xi 0, xi 1. (Ik is the unit cube; Qk is the standard simplex in Rk ). Dene
x = T (u) by
x1 =
Math 361: Homework 4
1. Pugh #22: If f : [a, b] [c, d] R is continuous, show that F (y) =
b
a
f (x, y)dx is continuous.
Proof. Given some , pick s.t. if |d(x1 , y1 ), (x2 , y2 )| < , then |f (x1 , y1 ) f (x2 , y2 )| <
is allowed by the uniform continuity
January 31, 2014
Math 361: Homework 1 Solutions
1. We say that two norms | | and | | on a vector space V are equivalent or comparable if the topology
they dene on V are the same, i.e., for any sequence of vectors cfw_xk and x in V ,
lim |xk x| = 0 if and
February 7, 2014
Math 361: Homework 2 Solutions
1. Let U be an open subset in Rn , f, g : U Rm be two dierentiable functions and a, b be any two real
numbers. Show that af + bg is again dierentiable and
D(af + bg) = aDf + bDg
Since f is dierentiable, we k