MAT322 - Analysis in Several Dimensions
Homework 2
Due in Class: February 11, 2014
Remark. When
nothing else is specied you may assume that the norm is the Euclidean
n
2
norm x =
i=1 xi .
1. Prove th
MAT322 - Analysis in Several Dimensions
Homework 11
Due in Class: Tuesday, April 29, 2014
Remark: For those of you how write in LaTex, leave some room for the drawings
and draw by hand. When it is dic
MAT322 - Analysis in Several Dimensions
Homework 12
Due in Class: Thursday, May 8, 2014
1. Compute the following line integrals:
( 4t3 )
(a) (x + y) dx + (x y) dy, where (t) = t2 +2t4 , and t [0, 1].
Lagrange Multipliers - Examples
Yaar Solomon
Theorem (Lagrange Multipliers). Suppose that B Rn open, f, g1 , . . . , gk : B R
C 1 , A = cfw_x B : i gi (x) = 0, and a A a local minimum (maximum) of f o
Limits of Functions
Yssr Solomon
Denition 0.1. Let U Rd be an open set, x0 U , and f : U cfw_x0 Rm . We say
that the limit of f as x approaches x0 is y, and denote limxx0 f (x) = y if for every
> 0
Measure Zero Sets Under Lipschitz Maps
Yaar Solomon
Proposition. Let S Rn be an open set, E S of measure zero, g : S Rn
Lipschitz, then g(E) has measure zero.
Proof. Let K be the Lipschitz constant of
MAT322 - ANALYSIS IN SEVERAL DIMENSIONS
1. Course Description
The notions of a continuous, dierentiable and integrable function of
one variable calculus are naturally generalized to functions of sever
MAT322 - Analysis in Several Dimensions
Midterm
Yaar Solomon
Tuesday, March 25, 2014
Name (First - Last):
Stony Brook ID:
Signature:
Instructions
1. Start when told to; stop when told to.
2. No notes,
Properties of Measure Zero Sets
Yaar Solomon
Denition. We say that a set E Rn has measure zero if for every > 0 there is a
countable collection of closed boxes Q1 , Q2 , . . . such that
()
E
Qi , and
MAT322 - Analysis in Several Dimensions
Homework 10
Due in Class: Tuesday, April 22, 2014
1. Let C be the standard Cantor set (that was dened in question 1 of ex.9).
cfw_ a
i
(a) Explain why C =
i=1
MAT322 - Analysis in Several Dimensions
Homework 1
Due in Class: February 4, 2014
1. Show that the following formulas dene a norm of the linear space V :
(a) V = Rn , x = (x1 , . . . , xn ), x = maxi
MAT322 - Analysis in Several Dimensions
Homework 3
Due in Class: Thursday, February 20, 2014
1. Let X Rd be closed, and f : X X satises f (x1 ) f (x2 ) < x1 x2 ,
for any x1 , x2 X.
(a) Show that if X
MAT322 - Analysis in Several Dimensions
Homework 4
Due in Class: Thursday, February 27, 2014
1. For the following functions f : Rd Rm write the martix representation of
Df (x0 ) with respect to the st
MAT322 - Analysis in Several Dimensions
Homework 5
Due in Class: Thursday, March 6, 2014
1. Let a U Rd , U Rd open, R, and f, g : U Rm dierentiable at a.
Show that:
(a) Df (a) = Df (a).
(b) Df +g (a)
MAT322 - Analysis in Several Dimensions
Homework 5 - Partial Solutions
Yaar Solomon
4. Denition. We say that a set A Rd is connected if for any two points a, b A
there exists a path in A between a and
MAT322 - Analysis in Several Dimensions
Homework 6
Due in Class: Tuesday, March 25, 2014
Remark. Questions 1 and 2 deal with facts that we used in the proof of the Inverse
Function Theorem:
1. Let be
MAT322 - Analysis in Several Dimensions
Homework 7
Due in Class: Tuesday, March 25, 2014
1. Find global minimum and maximum of f (x, y) = x + y 2 subject to the constrain
x2 + y 2 = 4.
2. Consider the
MAT322 - Analysis in Several Dimensions
Homework 8
Due in Class: Thursday, April 3, 2014
1. Let Q Rn be a rectangle, f : Q R bounded. Prove that f is integrable
i
on Q for every > 0 there exists a par
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