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

407 views16 pages
14 Feb 2019
School
Department
Course
Professor
MATH 32BH
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 16 pages and 3 million more documents.

Already have an account? Log in
MATH 32134
find more resources at oneclass.com
find more resources at oneclass.com
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 16 pages and 3 million more documents.

Already have an account? Log in
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
find more resources at oneclass.com
find more resources at oneclass.com
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 16 pages and 3 million more documents.

Already have an account? Log in
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
find more resources at oneclass.com
find more resources at oneclass.com
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 16 pages and 3 million more documents.

Already have an account? Log in

Get OneClass Grade+

Unlimited access to all notes and study guides.

Grade+All Inclusive
$10 USD/m
You will be charged $120 USD upfront and auto renewed at the end of each cycle. You may cancel anytime under Payment Settings. For more information, see our Terms and Privacy.
Payments are encrypted using 256-bit SSL. Powered by Stripe.