17
1r2FTC for f t e f q ft dy indep of pets C zD F k 0 Co loop pre Etienne conservative F is a gradient Ferg 7 Dif Daf c a potential fr f Hi fu DiDg 9 D i 4 Ex 2 p 341 5 1122 0 F II it T.I Converse fails Fq t

e f ft of pets - University of Pennsylvania

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Page 1: e f ft of pets - University of Pennsylvania

1r2FTC for ft e

f q ft dy indep of petsCzD

F k 0Co looppre

Etienneconservative

F is a gradientFerg 7 Dif Dafc a potential fr

fHi fuDiDg 9 D i 4

Ex 2 p 341 5 1122 0

F II it T.IConverse fails

Fq t

Page 2: e f ft of pets - University of Pennsylvania

Everseholes

S Simply Conn

udins't

disjoint of open

pagepCohn

V f diff form

422 F If f iffy1

diff f n w e f dict h dyC curve parans

L a ST R2414 1 2157 I

L Ch LD 4 It X

Page 3: e f ft of pets - University of Pennsylvania

k HH 2

Say F 09 off 9fA

f dirt fadsdx t y dy

D 9212 FTC Oy

Fida E cel If11

Sif at t Ififfdt

sie o

S d at Sd9

TeX 2

Page 4: e f ft of pets - University of Pennsylvania

Ex 2 P 341on S IR O

vf F on S

1 on Sdaff Wf X doe not d fund

Cr Qr 707 Ftt

not AEYE Ii o

do well de awayfrom 0

More a Chapare Green's Thom

chap 1 Multiple S'sI die

J faadx I

Page 5: e f ft of pets - University of Pennsylvania

f x y f R 712

reE.ie DentGraph of Z f IX yVolume horde 9 Ove S

fix DdnAarea

dxdystep fins 4P

If i integrableSame here Th 11.5 P 358

F Fusinisthn

How to compute 9fftgS fffixidA

y 1 su t

gf.tl fixiidx dymsxtfeekfrstgefn Cross

Page 6: e f ft of pets - University of Pennsylvania

c c Sr Ipf seustriaSup info are

I is

11.8 worked out ex's

f cost fitbounded

out on closed regainbounded

small span feeunit cont

Samepf pp 313 364

Piecewise cat

IIlull p c iit

fix.nl p c cont exc

fu ManCg

res

graph of cont fu

I

Page 7: e f ft of pets - University of Pennsylvania

FI iiiiThin 11.7 11.8

In Ih Det ADA IdailsiuxSSfdAfSfdA

I Ig Max StatitSg Kinds

type HI 7,2

I

SSFDA Sd Inga a dyx 4 l

Page 8: e f ft of pets - University of Pennsylvania

S x y p

ETxadA

Exam ex

fi a Eli1

10 I Y examples

10 16 app l to physics

fix.FI

Page 9: e f ft of pets - University of Pennsylvania

Mass fix ydA 5

If Density 1 aHdXmess IDA SDA area of S

If d sity Varies

af densityM1

da

Center ofFda

IIIf d s.tn is const

centroid

G reais Thunopen

q s IRcont diff QprC in S

L C a 5 s

09 da O

x

Page 10: e f ft of pets - University of Pennsylvania

Hoa Efi toffsd9 dq dxt d2

P Ef a IfF 0y PEtQ2dcf P dxtQdy

EEE II S

Converse If 912 0t F pet Qrf Is Fa gradin

Not alwaysEx 2 P 341

notsimp 75 112 O iConn

w do

Converse Ok fr 5 C regains

0

Page 11: e f ft of pets - University of Pennsylvania

then PdxtQdy o

IIF PEtQz is a gradient

Reason Crais Thun

PdxtQ.dz TILDA

R notosedby C M

So if thnPHS o

i LHS o

S df ey.SI dx

dsldxdyD

Page 12: e f ft of pets - University of Pennsylvania

T iiii i udType I

i

Both types

Green's Th m

Q

Page 13: e f ft of pets - University of Pennsylvania

tQg S f 9,51DA

Ex C x 5 1

R X t y El

22 t e DX t 3 x cos es dy

Up OT

Sf 3 2 DA DA areaCRITR

Ex C OR

area R Sps d A

SS s d A

Page 14: e f ft of pets - University of Pennsylvania

1 09Y

Ex it Q X P O

2 Pi y Qo

3 P Ey Q Ex

area 427 xdy a 2dx

I xdy f yd a

Ellipse IEPf of Greens Thn

PdxtQd2 II IDA

Special case R I and II

Page 15: e f ft of pets - University of Pennsylvania

S

Ssp gdA erol asin Iget Pdx

Sf DA eval using II

get Q dy

On each piece

p 382 Ikon whole resin

of

ExSS DA

R dy dy

Page 16: e f ft of pets - University of Pennsylvania

w P dy t Q d 2 diff i fo rnO fo r

f Ix n dy dag iThe area ett Teydy rdx dx rdg

fa si IW P d Xt Q dyd w dPsd d Q n dy

DP dxtfIydyDQ 9 dxtdfg.dzd w IIydyndx t d dxndy

ffIat d

dx nd x O

dy Ady so

due 9 0 darn dy

Page 17: e f ft of pets - University of Pennsylvania

OrnW Pdx Q dy

w

e w

T Green stl

TPFTCa E

L Ca n CL Iat I

9 C TR x is Ldiff

d y 9 let 9117 cellsO din S

d C E It