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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
= . .
Math 361: Homework 3
Monday, February 17
1. Use the contraction principle to show that the equation x3 + x2 6x + 1 = 0 has a unique real solution
over the interval [1, 1]
Proof. Consider the map f : [1, 1] R dened by f (x) = 1 (x3 + x2 + 1). We intend to
Math 361- Quiz 5
March 18, 2014
1. We let f be a zero-form and be a k-form, i.e. f 0 (E) and k (E) for some
open set E Rn .
(a) (Worth 5) Show that
d(f ) = df + f d
You may assume WLOG that = gI dxI for I = cfw_i1 , , ik for this part.
Proof. We have
1. True or false: if is a k-form and k is odd, then = 0. What if k is even and k 2?
I claim that this statement is true. Assume that =
represents a specic iu1 , ., iuk Then, we have
aI (x)dxI , where I = I1 , ., In and each Iu
aIc (x)aId (x)dxIc
Math361 Homework 08
April 24, 2014
1. Claim: If m (A) = 0 for some A R, then m (A B) = m (B) for any subset B in R.
Proof. Since B A B, and then by countable sub additivity, we have m (B) m (A B)
m (B) + m (A) = m (B)
2. Claim: Any subset A R consisting
Math 361- Quiz 7
April 10, 2014
1. (Worth 10) Recall Stokes Theorem: If is a k-chain of class C in open set V Rm
and k1 (V ) is of C class, then
For this problem we will consider k = 1 and m = 1 and will be an ane oriented
= [p0 ,
Math 361- Quiz 4
March 6, 2014
1. Consider the following region E = cfw_(x, y) R2 : 0 x 1, 0 y 1. And consider
the map T : E R2 dened as
T (x, y) = (x, (1 x)y) = (u, v)
We will want to implement the Change of Variables Theorem in this problem
(a) (worth 2
Math 361- Quiz 8
April 17, 2014
1. (Worth 7) Recall an algebra A is a nonempty collection of subsets of R such that
(a) If A A = A A.
(b) If A, B A = A B A
Show that any algebra has the empty set using only the conditions above.
Proof. We know that A is n