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MATH 32BH Study Guide - Winter 2019, Comprehensive Midterm Notes - Xz, Xtc, Xd-Picture Card


Department
Mathematics
Course Code
MATH 32BH
Professor
Arant, T.J.
Study Guide
Midterm

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MATH 32BH

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MATH 32134
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01/09/2019
Lemmas Lcf ,p)EUH ,p)
Definition LE Racompact interval ,Papartition of B,
than apartition P'of Iis arefinement if P's P
BER "
is abox ,
PePix Pax ---xPn apartition of B,than
P' =P,
'xBe'x---XPn '
apartition of Bis
arefinementofp-ifp.is
Pi for each i-4,2,
...,n
The point is each P-sub box is partitioned by P'
-subbox
p#P' MIEL
Lemme BER "
abox f:BRbounded ,P,P'partitions of B
P'arefinement of P.Then ,
Klf
,P)ELcf ,P')EUH .p')EUH ,p)
Hoot
:Begin by noting if J'EJ󲍻my Cft Emp HI
Fix aP-sub box J
msHIWUJt-m-HIEyol.LT')󲍻msHI volts ')E,I󲍻mjHI roll J')
Summing over all P-sub boxes J,
Uf .p)=Imy A)roll J)EE my 'HI roll J't)=L Lf,p
'I
JFsubbox JR subbox (EET
Argument for Uis similar I'P' -subbox
(T'ET 󲍻Ms Hts Me 'Htt
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Definition (Common Refinement )
BERN abox P=P,x.-xPn P'=P ,
'
x.-.XP '
npartitions
tph.IT#xti79Ipontu9snY
'
is III. 󲍻HE
Lemma_ BE Rn ,
f:BRas usual P,P'partition of B
Then Lff ,Pl
sUH ,P')
Boot Let P"be the common refinement of P,P'
Lff ,P)eLI f,P")§
UH ,P")EUH ,P')
first Lemma sectored Lemma
Definition
BER "box ,f:BRbounded
LfBf =sup ILl f,p):Papartition of B)
UfBf =int IUlf ,p):Papartition of
Bl'integrable
If Lf Bt -
-U!Bt ,then we say fis lRiemann )(over
B)
Notes :
Lf Bt always exists since any Ulf ,pl is an upper bound for
{Lff ,p):Papartition of B)
Similarly ,Uf Bt always exists
LIB fell fest
This follows from :
Lemme Let U,LER set .l
su for every IEL,
hell .Then ,sup
LE infill
Boot Each be Lis alower bound for It .So ,LE int th
for each be L,i. e.intel is an upper bound for L
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