a tuples bear numbers
each Lis an element of R (xi ER)
R
se Use ase lengths distances
x LASER
is called the or lengti
Distance between Li is
(from abe)
a tuples bear numbers each Lis an element of R (xi ER) R se Use ase lengths distances x LASER is called the or lengti Distance between Li is (from abe)MAT冈
イーxeyellxllllaplcase
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一unions
Up,
zedleif. xe4ーーーーーーーーーーー
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Sat Differences Ae
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MAT 冈 イー  xeyellxlllla  plcase ーー Irialtex  4e 보 t4n  tylellelltlly .Ery . こ 46 2 叫 に 415. s  U4tua  Ilz / ltlliAll Cass 4 : 444 ) is 24. 以川 ーーーー ) : 4k  axame xt3 ) +  A ( ema ) rexa uxVE.4 ^ ニ 2j  t  k 트 14.  3 , a a subsetof R_b4 Piving.conditions 一 unions Up , zedle  if . xe4 ー ー ー ー ー ー ー ー ー ー ー χchahif.ed  and  zee Sat Differences Ae AAA c AUR2MAT
EN
0m2.
Ex/ A
UA
nktion a Line:
p.NER fixed
Exy Find the line belman... I.3.2). and 4.
Plane TnR
t,s ER
tut SV
pin, R are fixed
In IDL calculus, ha can
deine onesidad detvrdives.
In dimensions, this is a proAem,sone awoid it.
To say Whitwer a function red on a
set ucR is
LhR want u to "surround" ev
Det A soi NCR is a neghbourhood a point zeR if
also say that x is an i
N
non)
MAT EN 0m2. Ex/ A UA nktion a Line: p.NER fixed Exy Find the line belman... I.3.2). and 4. Plane TnR t,s ER tut SV pin, R are fixed In IDL calculus, ha can deine onesidad detvrdives. In dimensions, this is a proAem,sone awoid it. To say Whitwer a function red on a set ucR is LhR want u to "surround" ev Det A soi NCR is a neghbourhood a point zeR if also say that x is an i N non)MAT
A subset UCRn
s open if it is a neighbourhood each.
That rs is open if a neigh
THM Open bolls are open
Ex The open interval
as b
Lab
as xs b
Des. A swbeat ACRA Te cloegd iff its com,plement is open
ACR closed. RAA open
Ex Compeanads open bolts are closc
VE
Ex Closed intervals ahe closed. A
Car bl CR
Suppose xEAT the either xEtoo ay o
xe cb,
en A 133 closed
u V CR are open then the union is
LEMMA
open the viewsartion is spen
THM If A, R are closed, then AUR, 2 A AB are closed
MAT A subset UCRn s open if it is a neighbourhood each. That rs is open if a neigh THM Open bolls are open Ex The open interval as b Lab as xs b Des. A swbeat ACRA Te cloegd iff its com,plement is open ACR closed. RAA open Ex Compeanads open bolts are closc VE Ex Closed intervals ahe closed. A Car bl CR Suppose xEAT the either xEtoo ay o xe cb, en A 133 closed u V CR are open then the union is LEMMA open the viewsartion is spen THM If A, R are closed, then AUR, 2 A AB are closedMAT
The
Di uCR is pen a neighbourhood of each of rts points
its complement TisoRn
Lt. If u, V CBN are open, then uUV is pen,and unvisafen.
On HMW, shaA that. nute unions of open set
awe s not a
LP. If A BCR are desed AUB dosed AnB ceeeed
i. closad for. LEN closed.
Ex/ The amptM sg s the sd which contains no elenents,
R is closed (Rty is closed
They are open closed!
XER is a limit point a set ACR if every nechbourimod
N20, leek at P
THM ACR is clused it curioins all its limit points
limit has to be in A
bourhood 6f some point z EA
s a hivat po
A A
Ex
is a linn pant not i
I ACR. tine claswe st A is smallest closed set that
MAT The Di uCR is pen a neighbourhood of each of rts points its complement TisoRn Lt. If u, V CBN are open, then uUV is pen,and unvisafen. On HMW, shaA that. nute unions of open set awe s not a LP. If A BCR are desed AUB dosed AnB ceeeed i. closad for. LEN closed. Ex/ The amptM sg s the sd which contains no elenents, R is closed (Rty is closed They are open closed! XER is a limit point a set ACR if every nechbourimod N20, leek at P THM ACR is clused it curioins all its limit points limit has to be in A bourhood 6f some point z EA s a hivat po A A Ex is a linn pant not i I ACR. tine claswe st A is smallest closed set thatMAT SBT
THM
A s the set if ou limit
poi
is not in a closed F2
is not a limit A
Ex. The closure of an open boll is a bale
Take
o consider ex ER where os
take c desa enough to thenStc ESE
Exy
R take
A is
largest open set contorned in A
MM int A is the set all inte Nor points st A
this cant be
A
if nt sa limit A ond a limit point
The bosundaMA st A demoted DAY, Ts the set stall
boww dory pirds of A
MAT SBT THM A s the set if ou limit poi is not in a closed F2 is not a limit A Ex. The closure of an open boll is a bale Take o consider ex ER where os take c desa enough to thenStc ESE Exy R take A is largest open set contorned in A MM int A is the set all inte Nor points st A this cant be A if nt sa limit A ond a limit point The bosundaMA st A demoted DAY, Ts the set stall boww dory pirds of AMAT sl
sequence ausians to each natural numbers
n an object an.
Exp A.
an
hihai doab t mean for a sequence unverges a At L
n R
For every E20, there is NEN
st dan E for n N
For D, let N
than an L
LEO
Con
O, O E
Assume At A.
Naut to show B
closed
MAT sl sequence ausians to each natural numbers n an object an. Exp A. an hihai doab t mean for a sequence unverges a At L n R For every E20, there is NEN st dan E for n N For D, let N than an L LEO Con O, O E Assume At A. Naut to show B closedfunction as elements one set to elemanis
CO,DO)
Functions f
L nonlinen
XE
Gions.
LER is the
limit f as x approaches as 3.13
Ve o
orin
SIWAA
function as elements one set to elemanis CO,DO) Functions f L nonlinen XE Gions. LER is the limit f as x approaches as 3.13 Ve o orin SIWAAMAT
IHM Let Sc R
S R then
Ex
Pf Suppose
want to show. Li
Lee S.
Non say xe S and o lixalls S
io,of
Fot am
Let
Core)
is condi
May 17
where
MAT IHM Let Sc R S R then Ex Pf Suppose want to show. Li Lee S. Non say xe S and o lixalls S io,of Fot am Let Core) is condi May 17 whereMAT
loll
alls
THM Let SCR f S R the llowing are eguvalent
in verging to a
li
Ca
prove the contrapostiive (not (not
D Suppose f not cont a
For some o and
all
Then, a
Le is the oncle radius N
MAT loll alls THM Let SCR f S R the llowing are eguvalent in verging to a li Ca prove the contrapostiive (not (not D Suppose f not cont a For some o and all Then, a Le is the oncle radius NMAT
CRA 21 Am AC 2. the innade of A under
Ce e')
exp (ti,
exp (R)s lo
oo).
under f is the set IA)
xeu
Exl
THM IRA, the fo
OWN
are equivalent
bood of f
ca CN
exercise, si
R following
au
is conti wuows.
the Deimage a
anu closed szt under f closed
Some O
MAT CRA 21 Am AC 2. the innade of A under Ce e') exp (ti, exp (R)s lo oo). under f is the set IA) xeu Exl THM IRA, the fo OWN are equivalent bood of f ca CN exercise, si R following au is conti wuows. the Deimage a anu closed szt under f closed Some OMAT S31
exercise.
LS Continuous
deine forsy
Dot
is upper bound for
co,
S La.... BLIS L, So is
Loco
is the least upper bound
US
Aare
Ext is an uppe bound
pose y is an upper bound and 4s
L
then
and
so y is an upper
R has bound has a upper bound.
CR is bounded if FM
ER is bounded
ux is bounded
closed balls ane bawodod
ACR is compact if it is both closed bounded
Ex
Co, is compact
In general, any desod iniwwal the form Ca,bl
bER impac
Exp Cersed bolls are mpa
MAT S31 exercise. LS Continuous deine forsy Dot is upper bound for co, S La.... BLIS L, So is Loco is the least upper bound US Aare Ext is an uppe bound pose y is an upper bound and 4s L then and so y is an upper R has bound has a upper bound. CR is bounded if FM ER is bounded ux is bounded closed balls ane bawodod ACR is compact if it is both closed bounded Ex Co, is compact In general, any desod iniwwal the form Ca,bl bER impac Exp Cersed bolls are mpaMAT
Amazing IHM H k SAn is compact, and f k ig conf.
Conn
Ex, A
path a
The imoage any such path also campaa.
Comrade
Carde.
Ramads. Neither boundedness nor closedness is eneugh
K is boundsd
THM LA non compact subset kaR Aasa maximum
element.
That is, yek, st. xsy Y DLE
NOTE This is not true for noncompact sets
(0,l) has co mar, element..
k CR compact K. is Goda k has an upper
baud.
Show. supK a limit.
uppet bound
z z is an This commadiat the fact thai z is the least upper
k closed
f KCR ns com
then f has a max. on K. Tha
MAT Amazing IHM H k SAn is compact, and f k ig conf. Conn Ex, A path a The imoage any such path also campaa. Comrade Carde. Ramads. Neither boundedness nor closedness is eneugh K is boundsd THM LA non compact subset kaR Aasa maximum element. That is, yek, st. xsy Y DLE NOTE This is not true for noncompact sets (0,l) has co mar, element.. k CR compact K. is Goda k has an upper baud. Show. supK a limit. uppet bound z z is an This commadiat the fact thai z is the least upper k closed f KCR ns com then f has a max. on K. ThakCR is compact, empty f: k R is continuous
roo
has a minimum maximum on k
a has maximum
n the Ne Is C o st.
WIS2
f A is coorooted if it bas no discuorections.
Check Ulu
up PCB
A
check unvnA
Suppose. E unv.
mpossible2
ExeNasa SCQ.Ts dwsconozatod if it has air least two points
Ex is connected.
such connection cant exist comecled..
sat
Imagina a disconnection Cu,v)
(u,v
s not a das Coonaution.
A ixi is connected
kCR is compact, empty f: k R is continuous roo has a minimum maximum on k a has maximum n the Ne Is C o st. WIS2 f A is coorooted if it bas no discuorections. Check Ulu up PCB A check unvnA Suppose. E unv. mpossible2 ExeNasa SCQ.Ts dwsconozatod if it has air least two points Ex is connected. such connection cant exist comecled.. sat Imagina a disconnection Cu,v) (u,v s not a das Coonaution. A ixi is connectedMAT
Conn
Suppose. Curv) is a disconnection
EV
un I closed, since unIL VOI is the intersection
two closed
n I is bounded, as it is: a Subsze 5 bounded set I
and Con
empty asumpfion)
so these is a maximum zeunI.
EV
If
EV
so fats
tro
some openand
z Eu., But z is the maximum.
of
No swch dusconection can exist.
I IS con
the contrapositiv
Assume fIA) is duscoonaated and ove that A
for f
et (uiV) be a
is continuous, u is open
Cu ,V) is a disconnection A
Check.
disconnection
Ulin Vin A
so t
MAT Conn Suppose. Curv) is a disconnection EV un I closed, since unIL VOI is the intersection two closed n I is bounded, as it is: a Subsze 5 bounded set I and Con empty asumpfion) so these is a maximum zeunI. EV If EV so fats tro some openand z Eu., But z is the maximum. of No swch dusconection can exist. I IS con the contrapositiv Assume fIA) is duscoonaated and ove that A for f et (uiV) be a is continuous, u is open Cu ,V) is a disconnection A Check. disconnection Ulin Vin A so tMAT 37
Similarly, VLMA
be ve shown (u V) is a disconnection for A.
Exp Paths the fom Lo have conceded
Exeroke Show thas the textbook
connected
and de are
any two points
x u EA there is a continuous
function A
THM If ACRA is e connected. it is connected
Assume A is atenuise conneoRd, but is disconnetect
that is it has a disconnection (uv
There
is a path A st
V ane open
(u,v) is a disconnection of (to,
s coone tool
Exi Open balls are arc wise cono
Rad
from y to z.
The same function to show that closed balls
RT is a civis
nn
and are the only
clopen closed i open)
upsets of
Suppose
Than (A, A) is a disconnection
is en since A is closed
A AC ART
Value THM
ALCA is connectaci r: A R is continuous
so for some ae A fc 20 fo
some beA
MAT 37 Similarly, VLMA be ve shown (u V) is a disconnection for A. Exp Paths the fom Lo have conceded Exeroke Show thas the textbook connected and de are any two points x u EA there is a continuous function A THM If ACRA is e connected. it is connected Assume A is atenuise conneoRd, but is disconnetect that is it has a disconnection (uv There is a path A st V ane open (u,v) is a disconnection of (to, s coone tool Exi Open balls are arc wise cono Rad from y to z. The same function to show that closed balls RT is a civis nn and are the only clopen closed i open) upsets of Suppose Than (A, A) is a disconnection is en since A is closed A AC ART Value THM ALCA is connectaci r: A R is continuous so for some ae A fc 20 fo some beAMAT 3
Assume the hypotheses hold but fx o for xEA.
Then, f CA) is duscoone
(a) f(A) nu
EXA Show that the function f: R
xy kas a zero
is connected
is comtiwnows.
TKerefoNe, but f has a zero.
B are conne
EAnb then
u P, is
If Ai connected EI, and REAL VieI
UAE is connectRd...
SA SA SA. D and each Ai is
both compose and conceatok
connect&d
f R R differentiable at aER if the limit
DeO If T C
is an open interval, IR is differendiade
ael if th
limit
vector the point
Ct
The linit in each comporoud is a
T goes through 3D space
tEr
In the last example, elo
MAT 3 Assume the hypotheses hold but fx o for xEA. Then, f CA) is duscoone (a) f(A) nu EXA Show that the function f: R xy kas a zero is connected is comtiwnows. TKerefoNe, but f has a zero. B are conne EAnb then u P, is If Ai connected EI, and REAL VieI UAE is connectRd... SA SA SA. D and each Ai is both compose and conceatok connect&d f R R differentiable at aER if the limit DeO If T C is an open interval, IR is differendiade ael if th limit vector the point Ct The linit in each comporoud is a T goes through 3D space tEr In the last example, eloMAT
Cap
What about R R
Car differen Tate one variable
a time
open, u R the jth
at a (ai, az, an is the
limit
Also
derotod
and even
Exl CompuR the Martial derivatives
f R R
Xe
..Ex/ Portal daHvohves may exist, even when the function
is not continuous
exist (exerowse)
Ott
At Koio
O O
What is the total" dewiodive of a function?
nonsense, cart divide by XER
RER
CON
CR be econ f nuR and ae u
there is CER
c the gradiant of
V V
MAT Cap What about R R Car differen Tate one variable a time open, u R the jth at a (ai, az, an is the limit Also derotod and even Exl CompuR the Martial derivatives f R R Xe ..Ex/ Portal daHvohves may exist, even when the function is not continuous exist (exerowse) Ott At Koio O O What is the total" dewiodive of a function? nonsense, cart divide by XER RER CON CR be econ f nuR and ae u there is CER c the gradiant of V VwewAld
icatio se +c.ex.
is sort, at
a
in the u is
COO
3
Sa f is height tertan at
This mans if were at a Cs,L and walking in
the headma us (Est) mg'd be nalking down hill at
and a fan
ca)
attu
asCtu.
Cetu
at two
tw
you can see
TO, O
(a)
an)
Ca ,a
o)
an) Ca
laura
aitt
attei
e, whete cs cm) then
(c
So, if fis diff. at a then the gradewit is avon by
wewAld icatio se +c.ex. is sort, at a in the u is COO 3 Sa f is height tertan at This mans if were at a Cs,L and walking in the headma us (Est) mg'd be nalking down hill at and a fan ca) attu asCtu. Cetu at two tw you can see TO, O (a) an) Ca ,a o) an) Ca laura aitt attei e, whete cs cm) then (c So, if fis diff. at a then the gradewit is avon bydel
ecland.continuousinaei3kloounxxcl
Just paoveforua
Without loss taeneralit4rde(Gd)
CothaAdshXLr.consider tw.tu
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PartialDarius
Total Deriv
Partial Daris
 what aboutBysly?
Ideas approxiotannlinear function bu a linoaroe
Dog A lino at ap is a funation T: R^→R"st.f EX4ER
Recall
linear map TprRn ean bo hop esnted bu
anmonnatrix 4
denato thespace ofalllineat mnapsーーーーーー
et yj"ED
Recau. If Te L (RnD") and Sel RaN) and is
RocalRensTel L the and SelR:00andin
epresented.b4thmxnmatixA,Sb4 the lem
the ken
matrix P, than SeTE / (R" R) .isrepesenad bl
并
del ecland.continuousin  a  ei3kloounxxcl Just paove  for  ua Without loss taeneralit4r  de ( Gd ) CothaAdshXLr.consider tw.tu CuLfcut a ) )  Nant to  show  fis d ft . heknon ,  卉  蓊 ane.coof . all llvlS8 hill show Nghethat re , 0 ) le % yte ) ( Cz 土 5s ー レ ' t eLexl Similarly ( ztu ) s 4 ( uvilt ilvl ) : Ellvil ーーナ Tiviī tftV ) _ftoh  evkE Partial  Darius Total Deriv Partial Daris  what about  By  sly ? Ideas approxiot  ann  linear function bu a linoaroe Dog A lino at ap is a funation T : R ^ → R " st.f EX  4ER Recall linear map Tpr  Rn ean bo hop esnted bu an  mon  natrix 4 denato the  space ofall  lineat mnaps ーーーーーー et yj " ED Recau . If Te L ( RnD " ) and Sel RaN ) and is RocalRensTel L the and SelR : 00  andin epresented.b4th  mxn  matix  A , Sb4 the lem the ken matrix P , than SeTE / ( R " R ) .isrepesenad bl 并MAT
T is called the EMchet desauttve of f ot a E Ts
at a
then d
May
THMa 2dcR open, f R and the al desvatives
R cominuow, then f is differentiate and
Assume A is not compact
a) Shoon Ef: st. fCA) rs not bounded
FLA
continmeus
f is diff and vfizy
If an whit vector, st f is diff a a, then
same di Nection
Det. Let nicR be open, feuR. Then f is
T2dfa is FNéchet deutvative... Tofa is thematrix)
If c Ts the arodeut of fat a than EN R)
so Tedfa is the
FNéchet de Nivative.
MAT T is called the EMchet desauttve of f ot a E Ts at a then d May THMa 2dcR open, f R and the al desvatives R cominuow, then f is differentiate and Assume A is not compact a) Shoon Ef: st. fCA) rs not bounded FLA continmeus f is diff and vfizy If an whit vector, st f is diff a a, then same di Nection Det. Let nicR be open, feuR. Then f is T2dfa is FNéchet deutvative... Tofa is thematrix) If c Ts the arodeut of fat a than EN R) so Tedfa is the FNéchet de Nivative.MAT
before..... Then
L Suppose
usR.6pen, f Rm, and f has a Frichet.
Tm) so
Tux)
TTHM ueR is, and the partial ONe continuous then
f is daffoNeviobe at a and a is ven by (A
any dimension
MAT before..... Then L Suppose usR.6pen, f Rm, and f has a Frichet. Tm) so Tux) TTHM ueR is, and the partial ONe continuous then f is daffoNeviobe at a and a is ven by (A any dimensionMAT257
2142,zzay )
区→N (LA VIAD → (earc":")
22._一語
u.vwDEP(x.421.
01f 2.
之一O
R") is a linear map the non T is

Danaark : uThListha maximum q.condanaous
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MAT257 2142 , zzay ) 区 → N ( LA VIAD → ( ear  c " : " ) 22._ 一 語  u.vwDEP ( x.421 . 01f 2 . 之一 O R " ) is a linear map the no  n T is  Danaark : uThListha maximum q.condanaous ーーー 让 lTal_00.the compact  sek . o  Sel  thou . ITI : 5 Avi + 35y , " t 7w nd isacuatvector ーシー T ( Tax ) 7 囟干 χ_ Exorcise : SES  Jthat.l_l . that l  l is indeed a parm on LSLR ) . LA  N ) , e in Paticular  ll St Tilsl SltILTIL THM . ( the Chain  Rule 14 UcBy , .VeNAN open  f : 1d 크 V  3 三 V_R._ diff.nt aev , then at a and 并 ーーー without  les . of 0 , sod SeLLER " , st . → 壑 ST Titfit ) setto 4js  diff . at o , soaTe LATN )  st qgup  Tyl  a we  wast , toshcw_1o S is the Br aka LivwitT5xll sdal4 fa 止 巧 alalife  TSzl Show Otudhto 2010MATS히
.3baalO to ndsto
bounded
Doesn'tLagrk a 6Afcall be zero
for al
Since·ll fcoSlo) llooll 0% ll 011
Then,M
c . ItlS1,then
LetE>0 sinco
盔STdēilaa1_Tyll:0
co o
S20 stallfull_nforllzllsS
Asume Ssiartoビル主11 < S
119(f(x)エfu)
thatthelinet is zero.
If2dsR
isdiff at 7(t)e
or :R→Disdi
of tand
ー并ー. DefencDfreeDr
Ex Lut
Letret)。 ( QtLt2e)
(Tw)。(sint (sti) costI)
(sine (近ー! ) cost,l)(→ st.2t )
골을 뚫 등 H붚솔
ttel
Donut Skopcw: entl4
LAW l b cf flour to make 30, donuis Everytime the
priceofflourincreasingb4194bthemiseriyiramer
ePrice of dA
andthou sell Do less
How is their
buthePrico flour
fuantatut 千donaisseldーーー
MATS 히 .3baal  O to nds  to bounded Does n't  Lagrk a 6  Afcall be zero for al Since · ll fco  Slo ) l  lo  oll 0 % ll 011 Then , M c . ItlS1 , then Let  E > 0 sinco 盔 STdēilaa1_Tyll : 0 co o S20 sta  llfull_n  for  llzllsS Asume Ssiar  to ビル 主 11 < S 119 ( f ( x )  エ fu ) that  the  linet is zero . If2dsR isdiff at 7 ( t ) e or : R → Disdi of tand ー 并ー . DefencDfree  Dr Ex Lut Let  ret ) 。 ( Qt  Lt2e ) ( Tw ) 。 ( sint  ( st  i ) cost  I ) ( sine ( 近 ー ! ) cost , l )  ( → st.2t ) 골 을 뚫 등 H 붚솔 ttel Donut Skop  cw : entl4 LAW l b cf flour to make 30 , donuis Everytime the priceofflourincreasingb4194b  themiseriyiramer ePrice of dA andthou sell Do less How is their buthePrico flour fuantatut 千 donaisseld ーーーMAT
costs
So
eteer 55LOOa
Ex Perceived Hummdax
Humidex H a
o a Ca
T h ose functions cloud covet c suniokt
LLO
How does
OT oc.
N
MAT costs So eteer 55LOOa Ex Perceived Hummdax Humidex H a o a Ca T h ose functions cloud covet c suniokt LLO How does OT oc. Nel set st f for the
ev
It c so
function
Can wate y as unction a
Look at vative
o only the the level rs bad
ction least locally
We Can wrfte as functi
2 say that locally, the level sze f th
Note
A uction
f
hrte as FC
to,by
a and
el set st f for the ev It c so function Can wate y as unction a Look at vative o only the the level rs bad ction least locally We Can wrfte as functi 2 say that locally, the level sze f th Note A uction f hrte as FC to,by a andMAT
20 st.
Fix Such an x, then
so, F is st
cheating
(L, loth)
Intact, f Rs (see textbook
Bu the chain
Rule
What about Solv
a system of equations imp citu?
Det SCR s tive f is the set
graph
and th
Smt m
a,b
a SR SR a and a ci function
ERM
v aka.
MAT 20 st. Fix Such an x, then so, F is st cheating (L, loth) Intact, f Rs (see textbook Bu the chain Rule What about Solv a system of equations imp citu? Det SCR s tive f is the set graph and th Smt m a,b a SR SR a and a ci function ERM v aka.MAT
an
s to be
3atO
and
inveHabe and the TFT Phe
4 B
220
f Ca,b R is catinuous on la, bl and diffeNeut Table.
on a,b) then 3CEla,by st
ba
then
xel)
Then Y
is differentiable on Ca,b) and
Then
Y Co, R is cont. and diff
on Co,
by the Chain Rule
By the ID MT
Cay
(ba)
SO
MAT an s to be 3atO and inveHabe and the TFT Phe 4 B 220 f Ca,b R is catinuous on la, bl and diffeNeut Table. on a,b) then 3CEla,by st ba then xel) Then Y is differentiable on Ca,b) and Then Y Co, R is cont. and diff on Co, by the Chain Rule By the ID MT Cay (ba) SOMAT
between a Buppose SSR is a set
except Rossibly at the endpoints a Then,
As is cost. on Coil and duft on Co
then
MAT between a Buppose SSR is a set except Rossibly at the endpoints a Then, As is cost. on Coil and duft on Co thenMAT
4
Le abERA and let L be the line szoment betwzen them
Suppose SER is a set contai
L and f S IR
immens on L and diff on
ports.) Then 3 CEL st
Ex Let
Then tot every... VER sboru that 3 CERN
st
is diff. Since sin and
zH llxl
Take VER. If
then guv
Sin
Then, by the MVT, 3 c bzkween a st
v o)
CO
O Sino
Then, 3 c betwann a o b at
Look at n
Cordlany If abER, L is the
line.
gen Athena
s diff on L
u is a set containing L f ure
and M on L than
M Mball
Co
MAT 4 Le abERA and let L be the line szoment betwzen them Suppose SER is a set contai L and f S IR immens on L and diff on ports.) Then 3 CEL st Ex Let Then tot every... VER sboru that 3 CERN st is diff. Since sin and zH llxl Take VER. If then guv Sin Then, by the MVT, 3 c bzkween a st v o) CO O Sino Then, 3 c betwann a o b at Look at n Cordlany If abER, L is the line. gen Athena s diff on L u is a set containing L f ure and M on L than M Mball CoMAT
A, BCR are convex, then An Lis comex
hMT is convex for a o, b x
very Convex set is aron ise connected
ected 3 connected
s diff and vfo20 Vx Then f rs constant on u
lab
To do HNS riaprowdy, well use MVT.
Apply MNT, 3 Ste Ca art) st
att
Qtt)
Ca
lab)
Switch ra the roles
MAT A, BCR are convex, then An Lis comex hMT is convex for a o, b x very Convex set is aron ise connected ected 3 connected s diff and vfo20 Vx Then f rs constant on u lab To do HNS riaprowdy, well use MVT. Apply MNT, 3 Ste Ca art) st att Qtt) Ca lab) Switch ra the rolesn, x reru are all infinitely
sin, cos
exist and are
Como)
Ex
lst order
Ond ordet
3rd order
ade) If all of its
partial to
Higher Order Partial DeNvatives and the Chain Rale
e fuay) inhere xiy are functions sit.
TOS
Hee
TOS
WTA
Ext Palos Coordinotes.
t sine
TOM
sine.
cose.
sin0
n, x reru are all infinitely sin, cos exist and are Como) Ex lst order Ond ordet 3rd order ade) If all of its partial to Higher Order Partial DeNvatives and the Chain Rale e fuay) inhere xiy are functions sit. TOS Hee TOS WTA Ext Palos Coordinotes. t sine TOM sine. cose. sin0MAT ST
Macan:
Multiindex Nato tion
outtiindex is an ntuple con
nk
is
MAT ST Macan: Multiindex Nato tion outtiindex is an ntuple con nk isMAT
June.
non neoptiva integers,
ER
If f aRR is Ch then Comm
Can also have constants.
ER ,they
Cc,, c
Consider
Apply the mWFromval THM
Lamma suppose abeN and Lis the
is CT
Then, fo to
IR is Ck and
dti
Let c b
a
whoe mejo
Claim
this Induction
hdds.
Buppose
MAT June. non neoptiva integers, ER If f aRR is Ch then Comm Can also have constants. ER ,they Cc,, c Consider Apply the mWFromval THM Lamma suppose abeN and Lis the is CT Then, fo to IR is Ck and dti Let c b a whoe mejo Claim this Induction hdds. BupposeMAT
chain Mule
ey induction, we proved the dam
b a
at b
Land a, XEL, then 3 cel
a st
Xras
Exl Applu Toutor's THM to exp.. on the interval. Do,
1
For each n
3 ca e lo st
expun" exp rand
xa)
l
Since Cielo
If IER is opan interval, f I
R s cs and ael is
fuar o s o then f has a local maximum
inuoub i
Apply Taylor's THM. Nith nei such
(C
a local
Let UCR be
If a, Eu are st. the line L between. at x is
contanodinu than 3. CEL
MAT chain Mule ey induction, we proved the dam b a at b Land a, XEL, then 3 cel a st Xras Exl Applu Toutor's THM to exp.. on the interval. Do, 1 For each n 3 ca e lo st expun" exp rand xa) l Since Cielo If IER is opan interval, f I R s cs and ael is fuar o s o then f has a local maximum inuoub i Apply Taylor's THM. Nith nei such (C a local Let UCR be If a, Eu are st. the line L between. at x is contanodinu than 3. CELshowed that
these m
Xmam
If we take Taylors THM nitth neo, 3esL st.
MVT
What if lol
Co, O
Let v Exay so
Also
Then
the
VtVi Oz
an an
where ai calls
m m motNx A
and v
IS. a column vector.
v.Av
Nun
showed that these m Xmam If we take Taylors THM nitth neo, 3esL st. MVT What if lol Co, O Let v Exay so Also Then the VtVi Oz an an where ai calls m m motNx A and v IS. a column vector. v.Av NunMAT
Det If.ucR is open, R is c and ae then the
a is the morax Hessat a
we showed
n is that. CEL.
a)
bliss
than
Then for
et be as in Tautors THM. Then st
for CEesia) we have that
This implie in particular, that
MAT Det If.ucR is open, R is c and ae then the a is the morax Hessat a we showed n is that. CEL. a) bliss than Then for et be as in Tautors THM. Then st for CEesia) we have that This implie in particular, thatMATS
Ex Coosider the curve SCA dejined bui
Find the line to S ugh the point
coo.
the S can be inritten with y
LOAD
line is given bu
the taseert plane to the
sphere as. radius 3
DZ
Near. the sphere is quenas a graph of a C
n
function f: R
2,2)
So the taraent plane is given the set
F R is C continuously
O
On
MATS Ex Coosider the curve SCA dejined bui Find the line to S ugh the point coo. the S can be inritten with y LOAD line is given bu the taseert plane to the sphere as. radius 3 DZ Near. the sphere is quenas a graph of a C n function f: R 2,2) So the taraent plane is given the set F R is C continuously O OnMAT
these is a neobbourhood CS st.
a)
As this holds fo all vectors urne have
a is a citi
Taylor's THM hithin L at a CHtical
Note The Hessian minix a
Lmxm matNx.
in modrix A is symmetric Then
it is diagonalizable and Nts engenvalues are real.
rthegonal SPT
where P is o
and
where the a Nethe egenvalnes
A
deAnte if all of its eiaenvalues ane
nagaliva.
non
000
nods it has at least one posieve and one oeaotive
MAT these is a neobbourhood CS st. a) As this holds fo all vectors urne have a is a citi Taylor's THM hithin L at a CHtical Note The Hessian minix a Lmxm matNx. in modrix A is symmetric Then it is diagonalizable and Nts engenvalues are real. rthegonal SPT where P is o and where the a Nethe egenvalnes A deAnte if all of its eiaenvalues ane nagaliva. non 000 nods it has at least one posieve and one oeaotiveMAT
Ex J o Lhas eigenvolna and 2. ositive
o 3 go the matrix is positive definite.
Eigenvalues
is posrtive detinxte
S o positive sem
(bit pot pasti dginite)
O O
nte Card
5 O
O TO 3
Note
A is an
A is an
eigen
veR ,vro, alt.
A
Ts pasitve detincte iff. A is negative nite
A Ts pas.
A is Sam
DCE
P is invertible v o
et Pax then as
Anvr
Nav.
bloe 20 and Aivit o fot ton
SSWAN
is as eigenvolve A Tlus means
zigenvalves are
positive
MAT Ex J o Lhas eigenvolna and 2. ositive o 3 go the matrix is positive definite. Eigenvalues is posrtive detinxte S o positive sem (bit pot pasti dginite) O O nte Card 5 O O TO 3 Note A is an A is an eigen veR ,vro, alt. A Ts pasitve detincte iff. A is negative nite A Ts pas. A is Sam DCE P is invertible v o et Pax then as Anvr Nav. bloe 20 and Aivit o fot ton SSWAN is as eigenvolve A Tlus means zigenvalves are positiveMAT
3 O
S
open, f t is
Hessa Ts postive
te, has a local man at a
its e
ane POSitive.
o
A summitic nxn matrix A Dail is n2aa definite
Hess af is posttive de inite, f has a locol mio. ai a
Exercise Check for Noactive ddinite motices that IE 3
inita ness is an open
MAT 3 O S open, f t is Hessa Ts postive te, has a local man at a its e ane POSitive. o A summitic nxn matrix A Dail is n2aa definite Hess af is posttive de inite, f has a locol mio. ai a Exercise Check for Noactive ddinite motices that IE 3 inita ness is an openMAT 37
also
I.OO
Note that the determinant a is a continuas
fundren the cells of the matix (ID fact, rtis a
Mnomal....)
O
Oka ark
these is Ek o st.
O
Let E min
SE for n then
O
end continuously on.
(O
AAte
WTaylor TH
Here,
hei has a local minimum a.
a
Hess
Hessa
Ex
Also a
So CL1) is the only critical
point.
RSS
MAT 37 also I.OO Note that the determinant a is a continuas fundren the cells of the matix (ID fact, rtis a Mnomal....) O Oka ark these is Ek o st. O Let E min SE for n then O end continuously on. (O AAte WTaylor TH Here, hei has a local minimum a. a Hess Hessa Ex Also a So CL1) is the only critical point. RSSMAT BI
Only cal part at
Hess
(A)
inite.
O f has neither a min a max. at Coy.
COLO)
rt which is rather a min
called saddle
THAM TiusR open,
a c
and Hassi
does have a
Let A20 be an
VERA (Hess, V. Av
hithoki loss generobru IVI
v ninth
O
3 t o st. v. Hess 20.
tw
cannot ha
MAT BI Only cal part at Hess (A) inite. O f has neither a min a max. at Coy. COLO) rt which is rather a min called saddle THAM TiusR open, a c and Hassi does have a Let A20 be an VERA (Hess, V. Av hithoki loss generobru IVI v ninth O 3 t o st. v. Hess 20. tw cannot haMATJ3T
4G
Recall.... A summetic n n matrixA is posative semidefinite if
ali its eigenvolve are nonneoptre.
A
semi
of its eigenvalue abe non pastiva.
In words,the THM abae states iff is C and has a
local max. at a then Vf (a) o and Hesse is. Megafive
a rogative egenvaluwe
not have a local min. ata.
In other nds if f is ci and has a local min ae a,then
Hessat Ts.poSfive nite.
For f is cat a CHfical
Hoss pas.
local min pas...sem
max. Hess
point and is indefinite, then f has a saddle.
part at a
D it has a postive eigenvalue.
it has a
so b
dozs not ha Local min. at a
does not has a local max..at a
it has a saddle point
and det (Hessaf) so. Then are the eigenvalue
Hessif, SO, so one has to be regodive and the other
Positive
Hessf is ind inter and has a saddle it at a
Mas a saddle pot at
Hessa i is sevi
extNemwm ot a saddle point
Ex
ve CHtical point, i
e a ertical
pant Hess
MATJ3T 4G Recall.... A summetic n n matrixA is posative semidefinite if ali its eigenvolve are nonneoptre. A semi of its eigenvalue abe non pastiva. In words,the THM abae states iff is C and has a local max. at a then Vf (a) o and Hesse is. Megafive a rogative egenvaluwe not have a local min. ata. In other nds if f is ci and has a local min ae a,then Hessat Ts.poSfive nite. For f is cat a CHfical Hoss pas. local min pas...sem max. Hess point and is indefinite, then f has a saddle. part at a D it has a postive eigenvalue. it has a so b dozs not ha Local min. at a does not has a local max..at a it has a saddle point and det (Hessaf) so. Then are the eigenvalue Hessif, SO, so one has to be regodive and the other Positive Hessf is ind inter and has a saddle it at a Mas a saddle pot at Hessa i is sevi extNemwm ot a saddle point Ex ve CHtical point, i e a ertical pant HessMAT
on t
Com
since k
in the interiot, say ot aEK "Kt then
is a loca max, it nill be
NOTE ak
Attoa iter, the max. of f On
k has to occurr at
one the point looy. LO), (011
the max
of f occurs act (LO) C,o).
H ne look functions on sets which are not bounded
may mst have aldol miniona ot maxima
Let SCR be unbounded, f 3 R...
E. o, so that ln E. SLI.
HM I SCR closed, unbounded and f S is
CODinuaub and linn oo... has a
global min., but
no
max
MAT on t Com s
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