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

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

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

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

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

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

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

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

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

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