MAT 237, Problem set 3
Due: Fri. Nov.5 at 1:30 pm
1. Here is an equivalent denition of disconnectedness (call it DISCON2):
Hints/solutions
a set S is said to be disconnected if there is a pair of open sets (U1 , U2 ) such that (S U1 ) and
(S U2 ) are disj
MAT 237

Problem set 1
solutions and hints
1. Let x and a be in Rn .
a) Prove the conjugate identity for the norm, that is :
(x a ) (x + a ) = x 2 a 2
b) Use part (a) to prove the owing fact from geometry: let AXB be a triangle inscribed in a circle
M31691
Weot 56 532135 ePFOblem Session
Ago MAC/TA ojjice hours TBA
_
'9'? is cu? inferior 93 S <=>EP>O,S.t BCrfXDC S,
33: g; boundary Point <=> \7 r>o .80 .3?) ns¢¢and 503530
Def= S is Callai open if if curtains none of ifs bowie/fag po'tniS.
Dgf: $13 cw
section 1.1

from R to Rn
required reading
Why Rn :
This course attempts at generalizing functions one variable like y = f (x) to functions of several variables;
this means functions that take more than one variables as their argument (or their input:) d
section 1.1
Dot product and Norm

Absolute value versus Norm
Recall that the convention of absolute value helped us dene the concept of magnitude of a number,
and the absolute value of the dierence of two real numbers r s was considered as the distance
section 1.1
inequality 1.3
2norm vs innity norm
Inequality 1.3 is a very important inequality which can be used as a legitimate tool (fact) about norm
of a ntuple in relation to the absolute value of the components of the ntuple. If we were to say the
MAT 237

Notes on 2.9
Extreme value problems
As much as the previous section, 2.8 was about the local extremes, working in a small open ball around
a critical point, this section deals with the (absolute) extreme points on a set. In 2.8 we worked with
a
MAT 237
Notes on 2.8
Critical points
Throughout this section the function f is considered to be dierentiable on an open set S , and in this
case a critical point is a point at which the gradient vanishes. It is then theorem 2.78 which suggests
for a dier
June 6'56
Chewing/«1&1
w¢f®<3
'XI
:\
78" E
m=ag+m+awdxn
d ax. okt 397.. oLE
' R
Chm ml; it +' JCC'J=f(Xu'>Xi,  was
&=&C+U"jt:"3bn>
w X it 9 =§C+*'+7b >
Xn n S0[:E')=j: C j ( 1))
3i : ?> Kn
3: EkaK
cutT; yFeKnaR
i=im i 5.321: "ir
g; ax. ag + *
WMYLWWWWWVMWMMWmumulpvysawrmmrg
rearm .. M.
M  mo, ., ..:.WW., , _ (mm, or», MAvagmyatmquvapawlw mowers».~_.u.«.m.~rrwmw.gw=.,a..r.~..,,¢.,w,,._.,w,,,,
Mmet»Mg.marus«awapwwmw
I 11.1 THE LEAST UP
MAT237 Supplementary readings for section 2.8
1. Signicance of f (a) when f (a) vanishes:
Taylors theorem for the function f at the point a, of second order, announce that
f (a + h) = f (a) + f (a)h + f (a)h2 + R
for a negligible remainder R. This means t
MAT 237
Notes on 2.6
Higher order partial derivative
The most important result is this section is the equality of mixed partial derivatives when the function is
C
2
(that is any partial derivatives of order two are continuous.) This is theorem 2.45. Note
MAT 237
Notes on 1.2
Logic and Topology
LOGIC, SETS and TOPOLOGY
Set notations and operations (see optional reading postings) can be thought of as a transliation of
symbolic logic into the language of sets. Symbolic logic is about properties like P which
MAT 237
Notes on 1.3
Continuity1
if
A function f : A Rm (where A is a subset of Rn ) is said to be Continuous at a point a point a A
such that x S x a  < = f (x f (a ) <
This statement can be translated to
such that x A x B (, a ) = f (x B ( , f (a
MAT 237
Notes on 1.3
Continuity2
How to decide if a function of several variables is continuous?
As in the case of functions of one variable, a function f : Rn Rm is continuous at a point a Rn if
lim f (x ) = f (a ). In the case of n = 2 this means a pair