Math361. Lecture 2.
Inverse and implicit functions
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
University of Pennsylvania
by Elaine So
1. Week 1: Multivariable Derivatives
Denition 1.1.1. X = Rn is the nth-dimensional Euclidean vector space. Elements are vectors
= . .
Math361 Spring 2016. Lecture 4. Integration
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
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
The simplest example of a smooth submanif
Lecture 1. Dierentiable maps of Euclidean
Math361. Spring 2016
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 |
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
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) =
6 x 2y
= xdy dz + ydz dx + 3zdx dy
is a 2-form over R3 . Compute the integral of ov
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
Math 361: Homework 4
1. Pugh #22: If f : [a, b] [c, d] R is continuous, show that F (y) =
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