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 that:
(a)
(b)
(c)
(d)
A is closed for every A Rd .
A = A
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 dicult to draw, like in 1(b), explain in words.
1. For eac
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].
(t+cos2 )
2
2 )(x2
2
2
2 )(x2
2
(b) (3x yx2 y +y ) dx+
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 on
A [i.e. U B open, a U such that for any z A U we have
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 there exists > 0 such that for any x U cfw_x0 with x x
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 g. First note that for a cube D S of edge
length we ha
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 several
variables. In this course we will rigorously develop
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, books, etc.
3. Turn OFF your cell phone and all other
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
v(Qi ) < .
i=1
Proposition.
i=1
(1) If B A and A has me
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 3i : ai cfw_0, 2 (this is the set of numbers in [0, 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 |xi |.
(b) V = Rn , x = (x1 , . . . , xn ), x1 = n |xi
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 is compact then there exist a unique point p X with
f (
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 standard bases of Rd and Rm :
(
)
x
xy
2e z
y (arbirary
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) = Df (a) + Dg (a).
(c) Df g (a) = f (a)Dg (a) + g(a)Df
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 b. That is, there exists a continuous
function : [0, 1
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 a norm on a vector space V over R. Denote by B(a, r) =
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 following ellipsoid, for parameters a, b, c > 0
x
2
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 partition P of Q such that U (f, P )
L(f, P ) < .
2. Dene
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner