Laplaceant...Lecturers vectoroperators in orthogonal curvilinear coordinates GradientFf Divergence F...
Transcript of 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