liiee i - cuhk mathematics...sdfds signed area underthe graphof f over c propi if c is a piecewise...
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cont'd from previous notes
0Jats Ig
detSCH F 3 at3 3 drainrule
3 Fat 3 t
3 I II East IE3 3a
LI iEE I
det J F fo at p2ftTx
So Hak satisfy tee requirements and we have
Hok H Y AI ySTCXlXaEaa x
F
This completes case 1
Case 3 ps o
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Sure detJA IIT3 3 77 to atp5 I CPS to
Interchanging thevariables YE Yy
Then the new mapping EHI satisfies the conditionin case 1 Applying case 1 to F then interchanging
back to Xi Xie
step let by K k4 be a diffeomorphism
from region Ri to REKCRD Then foranyfunction
f Yi Yz on Rz
4 yddyidyz fffoklxyxdfdetJCKYdx.chz
SftCkcxyxyxDPaYfjYjldadx
Pt By additivity property of integrations and cutting Randcorrespondingly Rz KCR into small regions we
may assume Refa by Ecd L aEX Eb CExed
For any fixed yz X2
y KCXi Xa KCXy ya for aEX Eb
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can be regarded as a transformation of T variable
Note that 3 3 Det Ito 3 det JustoSteph
Note also that Rz is of special fam
keyed kca ya e g s kcb.gs if 3 o
or key Ed Hb y E y s kca ya if 3 so
By Fubini's Thm assuming 3 o the othercase issnuiliar
Giddydyes Saki.gl fcy yzsdyi dy
and changeofvariable famula intraceable wiplies
Smh fly ya dy Jabtch 943 DX
fabtckcx.ws xD 3 dx sniff ILY
Cy dydy fabflkcxyxgx.BY dxi dX2
fcdJabfCklxyXD XDldetJCk dxidxsee 3 o
SL Stk xD xD IdetJ IDXda
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step If the changeofvariables formula holdsfor F G then it holds fa FoG
Ef Easily by 5 FOG JCF JCG drainRule
IdetJffoG I facts HdetJEDI
Finalstep Combutg steps I 3 and using additivity
property of integration we've proved the Thun6
f a general change of variables formula
Actually this appliesto all dimensions
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VectorAnalysis
Notation Usually in textbooks vectors are denoted by
boldface i but hard to do it on screen
so my notation of vectors are
general vectors T E F Fdifferential
unit erectus E j pi nsoperator
Line integrals in 1123 Rn
path integrals3
DESI The line integral of afunction for a card
path lute C with parametrization
F Tea by 1133
1 7 Fft Hits yet zitsis fir Ids tufo f Fita 0so
where P is a partition of Ta b and
DSi J
xi5tCoyi72tCozi5Cie.dslengthelement ofthe cave t.ITECXiYi
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Remarks
4 If f 1 Sc ds arc length of d
2 The definition is well defined ie the RHS in thedefinition is independent of the parametrization fits
Def9 Formula for line integralNotations as in Reff then
Str ds fab fleets I Fttsldt
where Fft Nct yfts TD
Fft is FctitySince
TTose e I FEti oti I IF Eti Gti
if He carve a differentiable
osi JTEoyfzoj.IT5tfFi5tEfjotiI JxTttytj z Oti
IF Eti l OtiRemade as ds if ft Idt is usually refereed as
the arc length element where TitstxtetHtt TD
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and IF'cty J xEtD7CyEtD4GttD
Suppose the curried is parametrized by a new parameter I
t E t E is increasingis aTrash tab III 0 dd o
ds IF Idt ladIect Idtftp.fetf dt fadIefdE
by chainrules
ds and hence fCBds is Ident oftheparametrization of d
t3 If Ftt isonlytpiecewise differentiable
afthen the RHS of tz to bbecomessure ofeach piece
If TabJ Eto tou a Eth ti U w htk i.tkIta b
such that F is differentiable thenEtit to
Str ds EI iteratirttsldt
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egI fairy zj x 3g't Z
C line segment joining theorigin and 91,4
Find Safa yieldsSold i Parametrize d by
Fct t 1,1 1 Lt t t too B
ie XA _t yet t Eet't
Ftt 1 I l t bGEED
nifty BHence Safey ds So fet t t Bdt
Soft 3t7tBdt o checky
egI let C be cute in RZ ie Zito
and it has 2 parametrization
F Ct Cost suits t EEF 3
Facts ft t TE El I
Suppose f Fund Scfcxyids
wesimply omit the 7 variable as C is a planecame and
f is indeep of 2
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Sole Ci Fits Cost suits Eet EE
fix yids feast suit I cast suit dt
JIE cost IC suit ost dt
LETcost dt z check
Lii Fact Ft t let El
fixyids I FtJfdttFt2YttttD2dt.ndt z checkr
check
This verifiesthe fact that the line integral 5 aidep ofthe parametrization
a
graphof f overC
Sdfdssigned area under thegraphof f over C
PropI If C is a piecewise smooth Anne made byjoining Ci dy du end to end then
Scfds Sd fds
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Pf clear from the remarkof RefIRemark end to end mealies
end point of Ck initial end pointof Ck
9342 i let fCx y X 3y4z again
di de Cs are lise segments as in the figure
ai bC
E sy 90,0
X d2 aCl1,0
we already did foes o eg33
One can similarly do Sqogfds fast Sgfds
Iz Zz Cox
The observation is Sdfds of Squefds
even C Gods have the sameendpointsConclusion Line integral of a function depends not only
on the end points but also t.eepath