laplaceant...lecturers vectoroperators in orthogonal curvilinear coordinates gradientff divergence f...

Lecturers

vectoroperators in orthogonalcurvilinear coordinates

Gradient FfDivergence F T

Cure Exp

LaplaceantGradient Consider afunction in 3D G RY Z

supposeyou want tofigureouthowthisfunctionchanges asyoutravel adistances in somedirection

Let'sstart at apoint no yo zo nowgoto apoint Iniy27whichis at

adistance5 in thedirectionof it

Cx no net yo j 1 Zo E SCILetThe a it by tie

N not sa

l L

we can replace y z7in ofwind Gotsa

time no a yob Zo e all areconstant

depends on thedistance s

Ifs IE Es i

Idn a t zag bt2 c

Init J tfer dI

DI Flo Ids

if ut is a unitvector maximum changeof4wouldbe in

thedirectionofTH

now let us define a surface ofconstant

201Is

O F lo Ti

there it is thetangentvector tothatsurface

Fol is perpendiculartosurface

Anotherwaytounderstandthis is thefollowing

suppose youchose of constant andnow choosingavector in thedirection

suchthat it remains on thesurface

Ko sitthisis also ontheplane

tangentvector

onlypossibledirectionleftis tangential

Example

2292122 4 O

Ff 2 ritzy It 2E I

n

µ

E IEt ten r

whatsthetangentrectors

Whatdoes dadsmean

M

E t.in E

dnonJ onto dyoyJ

Let'stalk aboutvectorfields

A T's uit y j

Ti 2

t

f f yB I 46T145

aiI T a 4

so it'seasyto see that divergence tells usarsonhowfast arectorfield

goesawayfromthesource Similarly onecan thinkaboutcurl

Trytovisualize thesein the contextofElectricandmagneticfield

Okso wealreadystudiedhowa functionchanges if wemove adistance

s in aparticulardirectionfddIg Fair

however I wasdefined usingCartesian coordinatesystem

Ianweconvertthis in curvilinearcoordinatesystem

if wego in r direction ds dr

dois thecomponentofFf in thatdirection

fords dv Floshouldbe glop

for O'direction des ordo I 9shouldbe 210rao

for 2direction des dz 89Shouldbe atJZ

If Era intfifty Iot2 Iz dream7

and we can even writegeneralconditions

Recall fromthe previous lecture

ds hfdnfihudnuehf.dk

dS h dntdS hudn I des hydx distance

na ki l NT direction

I74 in any curvilinear

coordinatesystem

ft t.E.net ii.tt.Enieif a iiFHowaboutDivergenceimlandlaplaria

F F Let J Viet viruses assuminganortnogonalcoordinatesystem

weumpon III f o.E.fi if t Enme 10 2 i 22 Nz threedifferentcases

Mittensthen If Ih he hz

men In Ino In In Ifm

F Fx Fxs Fxs I Fm Tx 808 3

f ofo

Vhhah v te hmm then hints

et att T 4 to 8 T

F h2h34 yet neigh Flhahzvi huh3mFeh

O heproved

2 h2h3Vi gun jd hih3V2 112 Chihiro2 3hihrhz.IM hihr.hr

Ex Let'sdo aquickenercise

findF it in cylindricalcoordinatesystem

Laplacian E.tvo v24

eol hiEitthen E to

hLIElhhfI

Ex findfor cylindrical coordinatesystem

hie hzez h3e3Ex

iv nd

too iEm eI i enI.chihteglhih.lt hsb

hi

E'xeI o Extort CExT Fx Folhi

became

F Ipl 0 I'xp hii Chava

Ex h3V3h3

II f'ss.im tenI.nIExih Icancherkheotherteems

TE yi ing

ei fia

I titty 2k

in I in

that if it was T y2 n2j YZ z2 o

zaki 2yI

Eidhdoesbniqnan

h.ly pify