General changeofconduiate formula in 1132

Suppose x gcu v

y haguedenoted by

guoplane

Xy plane

4cu v Cx y 4 G R

a

Idea Weneed to findArea KGWARTE

as Gk point

If 4 is a diffeomorphism I 1 onto a 4,4 EC

dis c

gcutouyvtovs gcu.US 12 out out

I hlutoyutous hcu.us IT out If out4 09 9Cut ou Hou ga v fudout If ou

ay oh HattonHob NYU Iheuoutshout

In matrixfam

1 1 1Iq IuH g a aa.gu.msCanton cutobutous µ'T

GAi l

Gre Ecups cutoun gcuiyhlu.vn appro.x.pdrallegnaas

In givenbytheapproxlinearmap

By lunar algebra

dAx A a9Y ftp.EIEHdAiuis

211

I tutlDeff Define the Jacobian STUN of the coordinates

transformation gcu v

y ehcu.us

BYyou notation 3 1 det III

with this notation we should have the formula

fix ysdxdig fftcgiyubhcu.us detfIgIuuIq dudry Luis

ffffxcu.us youas Jiu v dudu

fffcxasmycu.ua 3YgdudvegI

X rcesO

y rauiodistr os

Jer a 394 detf.FI Eq rkheok

and I fcx.ysdxdyfgffirosorsinospgfxfh.FI drda

fqffCrasqrsuiOrdr do

sameformula as before

Third Suppose to Y H Y is a diffeomorphism t t onto sit

Gand O C d mapping a region Gclosedandbounded

in the ut plane onto a region Rclosed and bounded

in the xy plane exceptpossibly on theboundary Suppose

fany is continuous on R then

I fixyidxdy fool 1344,4 Idudt

Notes i is foolcyuj flxcyubycu.us

ai ol is a diffeomorphism 1344 11 0

Tripleintegrals substitutions in triple integrals

lolYu w exyGOR Dc

cases

with gcuv wy hcu u w

tinto cont differentiable

2 ku youand inverse also cont differentiable

Retd Jacobian determent oftransformationin 1133

94 ywIetFu Fu Fu

Jiu v wH Y t

agu Ef

E outa gl I

Note chain rule

a aim Iihf IT IFsTtT istsaa.sixi.su i

u i

2 dim

out Ext

s din I zY

ThmI Undersimilar conditionsof 1hm6

FIXy dxdydzffqffocfcyywsfscyu.wsdududw

fqffFfgiuyw3hlyyw3klywD ffYY g

dudvdw

egI Jo JEE 2tz dxdy a4dis 7 ie

lower limit X 2x yO 3µg

p

upper limit X Yz11 2x y zt Z

Define u zx Ya

v yv 4

Then x EutEvy v

to y2 4 0 4 0 mo u zzx y z 4 2

Ifo soso.ITgdetfIyuIIIIJo4

detfo4 t

4i fzttkxzddxdy J.J.kz z dudv 2 check

egII Tze dxdy

x ys LDomain of integration 2

let u xTylo FI

this should simplifythe integration

V nThen x y 0 1boundary X ty 4 1curves a I

y 2 UU Z4 1 y

And theJacobian

39TH detf

Express Xy intermsof ur Iy uw

YYuYuy detf'T YI Checka

I S gYfe dxdy2

u

G I2 fJ ve dudu u z nl T

or 2 f Jfreakinglaude

I 1

let do Sis ve 1371 dudu ve dudu

fine du da check5,2 zue Z D du 2ece 2

eglf revisit VolumeofEllipsiod

D at E t En s I ca b co

Vol D 8J Job Joc dz dyDX

change of variablesa I

Hi2 CW3fYY w det fo b ab e fo

Vol SJ JF

dzdydx

s pffu.is dwdrdu

Saia D transforms to the solid unit ballUFftwk l

abc f to fit dudu duabc Vol solidunitball wi cu v w space

21T I µof of spherical

abc Sos So faint dpd4 do quiffftp.spuiaathe

abc

GI let D IN y EIR 1 111911124 El

Evaluate S cxtyt 4dV

can use symmetric Hay tx 9z

to reducehalf but not to the 1stoctant

since fauistanextytz XtyZ

under ay z Cxy

City174 is notsymmetric ui all reflectionwith

respect to thecoordinates lines

Sold If 7 0 atthen 1 111314

1 4 11ab x y It ri'trI I i yi y

i i I x iiinI t

l X l 1

Boundarysurfaces are givenby

x y Z 1 8 surfaces

let U XtytZ bandayplanes

xty Z a It only 60 11 cutoffw x y Z w I surfaces

meedtofindfamulafuotherzbandaysurfaces