10 - UFR Sciences de la Terre de l'Environnement et...
Transcript of 10 - UFR Sciences de la Terre de l'Environnement et...
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° w bJlpfikllCheh I jw a`cea\M\Uqih»eh J himjb fidmpheh K lC\m`phlCxh/f6mpkahi7h/\mjw bJUlf6llChjw `e\M\Mq/h»h6mjbofidmphengxak\m M P\gb7ea\fOM = [x, y, z] avec x = r sin θ cosλ; y = r sin θ sin λ; z = r cos θ ½POaQO¿
! "$#"$%& '()+*, ° wq|nUbºm\?½ OaQO¿¹xh/nmPlph`pqlCnUe`hPh/\ r, θ, λ ³
r =√
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x½PO ¿
avec 0 ≤ r ≤ ∞; 0 ≤ θ ≤ π; 0 ≤ λ ≤ 2π ½ Oa Æ ¿° b`h/lCmp`kf6mpka\¤½PO Æ ¿¹blplpnM`hPjw nM\Mf6mpq.eh>jb7f6a``h/lpx\UeMbd\f6h]h/\|mp`h (x, y, z) him (r, θ, λ) ° hlhi\Ulphi,JMjhleWwSq/|nUbºm\Ul7½PO'O¿n ½PO ¿]eMq6U\MllCh/\mnM\¤fcyUbd\U^ahi7h/\meh7f6a`cea\U\Mqihli ° bx|lCmpka\?eWw nM\xak\m M xhinm¹6mp`hlpxqf6UqlC¶mDxbd` (x, y, z) lChlDf6`e\M\Mq/h/lBfiba`Cmq/lph/\M\MhlBlC¶mDxUba` (r, θ, λ) lph/lDfi|`eMa\M\Mq/h/llpxMyMqi`nUh/l/@BA)AC@ ¾¬¬DJUO]¬©N]N]¡TF ¦¡TJ£C®¯F§¬DJ¥M¬©ªD¬©NLBVCF° hl)lCnU`Cuvbf6h/l r = cste ½vfinUjkjh¿Á θ = cste ½ÀxUba`bajjkzijkh¿©h6m λ = cste ½À7qi`eh/\¿lphfianMxhi\|mfcyUbf6nM\Mh>eh/nr.eh/nslCnMkºbd\|m¹ehl¹fianM`!Jh/lDbaxMxh/jq/h/l !+-21+D3!* 036 3N2 ° h/l¹f6nM`Jh/l©ehPfi|`eMa\M\Mq/h/l r, θ, λlpa\|m¹bd\bdjka^anUh/l©bansbºh/l¹ehPfi|`eMa\M\Mq/h/l x, y, z eWw nM\»`hixzi`h]`h/fÁmcbd\M^nMjbdk`ph ° hl©lpnM`Cuvbf6hl©ehf6a`cea\U\MqihlDlChf6nMxhi\|mhi\uÀa`sbd\|meh/l)ba\M^ajkh/l)e`aml³jkh.lplCmpz/7heMh.f6a`cea\U\Mqihl[lpxMyMq/`p|nMh/l[hlXm`CmyMa^a\Ubaj
@BA)A ¾FQ¥F¯JU§¯]N]£X¥¡ OLBNP§VXFa¥ ¢FOF QS¬¬DJUO]¬©N]N]¡TF§|¦¡TJU£p®¯F·¶m)jb,Jbalphf/bd`pmpqlCkhi\M\Uh`CmyMa\Ua`q/hPeh>h/f6mphinU`lnM\U¶mcbdk`phl ex
eyh6m ez xUbd`pmpk`)eh/lh/f6mphinU`l[\M`psbdnMtbdntlpnM`puvbaf6hleh>f6a`cea\U\Mqihlia\xh/nm]eqiU\Mk`)nM\Mh JUbalph 3* $+-%*%Q Er
Eθhim Eλ
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Er = ∂−−→OM∂r
Eθ = ∂−−→OM∂θ
Eλ = ∂−−→OM∂λ
;¹h6mpmph JUblChhlXm)a`pmpyUa^a\Ubdjkh7bakl\Ua\ \Ma`7qih7³P\ \Mdmhhr =
√
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Eλ.Eλ
° hl)|nUbd\|mpmpql hr hθh6m hλ
liw bdxUxh/jjkhi\|m)jhl[uvbaf6mphinU`l)eh>xU`px`Cm\M\Ubdjkmpq³hr = 1; hθ = r; hλ = r sin θ
P\ xh/nm]eq6\M`]nM\Mh JUbalphxMylC|nMha`pmpyM\Ma`7qih³
er = ∂−−→OM∂r
eθ = 1r
∂−−→OM∂θ
eλ = 1r sin θ
∂−−→OM∂λ
½PO ¿
P\gbbaj`l³
er =
sin θ cosλ
sin θ sinλ
cos θ
ex,ey ,ez
eθ =
cos θ cosλ
cos θ sin λ
− sin θ
ex,ey ,ez
eλ =
− sinλ
cosλ
0
ex,ey ,ez
½PO ¿
@BA)A _VC¡ ¢FN¹¥«O ÀL<JUQ'FW¥ OF©¬©Vp¯] ¢F·¶m.nM\¤xak\|m M ′ xM`fcyUh7eMn«x\|m&M eh»f6`e\M\Mq/h/l>fiba`Cmq/lph/\M\Mhl [x + dx, y + dy, z + dz] h6mGlpxMyMq6Y`k|nMhl [r + dr, θ + dθ, λ + dλ] MÂ]j`l[|nMhjhl©q/jq/h/\|mleh]jka\U^anMh/nM`l[eh −−−→MM ′ xM`d±Xhimpql¹lpnM` ex, ey, ez
lphi`a\|m[dx, dy, dz] UfihinxM`pa±Xh6mq/l)eMba\Uljhl)ek`phfÁmpka\l er, eθ, eλ
lphi`a\|m dr, rdθ, r sin θdλ
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−−−→MM ′ ³
ds2 = h2rdr2 + h2
θdθ2 + h2λdλ2 ⇒ ds2 = dr2 + r2dθ2 + r2 sin2 θdλ2 ½PO ¿
° w qijkqi7h/\m]eMhajknM7hxnM`)jh.lplXmzi7h>eh.f6`e\M\Mq/h/llCxUyMqi`k|nMhl[h/lCm]ea\M\Uqxbd`>³dV = hrhθhλdrdθdλ ⇒ dV = r2 sin θdrdθdλ ½ Oa ¿
;[bdjf6nUjh/`)xUbd`k\mqi^`bdmpka\Tjb7lCnM`puvbafihh6m)jh>ajknM7h>eWw nU\Mh.lCxUyMzi`h>eh`ba\ R
@BA)A ¤LBNªDF IFN©¥KOF QS¬¬DJUO]¬©N]N]¡TFP\txhinMmhixM`pk7hi`jkh/lf6a`ce\M\Mqihl©f/bd`pmpq/lpkhi\M\Mhl [Ax, Ay, Az] eWw nM\ ahfÁmph/nM` A hi\uÀa\fÁmpka\ eh>lChlf6a`pYe\M\Mqihl)lCxUyMqi`k|nMhl [Ar, Aθ, Aλ] him|f6hiYhi`clpbt³
A = Axex + Ayey + Azez ⇒ A = Arer + Aθeθ + Aλeλ
Pn hi\Ufia`hPh/\ k\|mp`enMlpba\m)nM\Uh>7bdmp`kfiheMh>xUballpba^ah³
Ax
Ay
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sin θ cosλ cos θ cosλ − sin λ
sin θ sin λ cos θ sinλ cosλ
cos θ − sin θ 0
Ar
Aθ
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Aλ
=
sin θ cosλ sin θ sinλ cos θ
cos θ cosλ cos θ sin λ − sin θ
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Ax
Ay
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a2+
y2
a2+
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c2= 1
¸©fi`pk`ph)fih6mCmhlCnU`Cuvbf6hhi\nmpkjkklbd\|mDjkh/lBf6`e\M\Mq/h/l<lCxUyMqi`k|nMhliP\h6xM`kh/`bPjkh)`ba\ r f67hnM\MhuÀ\UfÁm\eh θ λ P\ \Mamph a−ca = α jw bdxMjbºmkllphi7hi\|mehjw hijkjkxUlpaeh
·¶mnM\Uh`dmcbºm\ bdbdjkhaeh>mphlplph>bd\M^nMjkba`h Ω = [0, 0, Ω] ¸©fi`pk`phjhahfÁmph/nM`|mphlplph v = Ω ∧ r hi\¾fia`cea\M\Uqih/l]fibd`pmpqlCkhi\U\Mh/l]h6mhi\¾fia`cea\M\Uqih/l]lCxMyUqi`k|nMhli]q/`pUhi`bahf]jkh/l`hijbºm\Ul>½PO |¿¹h6mG½PO |¿Á,.- ?9 I0?0E4? F9CI63%8T=HGDF9CI6571 3@BA)AC@ H#JMLBN]OQSFTJUQWVXF ·nM`)nM\Mh.lpxMyMz/`ph>eh`cba\ R nM\g^a`cbd\Uef6h/`fijhh/lCmnM\_f6h/`fijheh>`cba\ R e\m)jkh.f6h/\m`phhlXmf6h/jnUehjkblCxMyUzi`h½vh6¾³Ujhlq/`pekhi\Ul)lpa\|mehl)^a`cbd\UeUl[fihi`cf6jkh/lc¿ ·nM`jkblCxMyUzi`haUjb an "!$# % ½ÀjkhxMjknUlf6nM`CmfcyMhi7k\ eWw nM\ xak\|mrtnM\ banmp`h¿h/lCmnM\^`ba\Ue f6hi`cf6jkh
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A
B
b
a
c
C90 90
jhl)f6²dmq/l>½ a, b, c ¿[eWw nM\ mp`bd\M^jh>lpxMyMqi`nUhl/w h6xM`7hi\|m)hi\ nM\MmpqeWw ba\M^ajkha jhlTbd\M^jhlTekz/e`h/l©½ A, B, C ¿eh/lTxMjkba\Ul©½|nMeqimph/`p7k\Mhi\|mjkh/lW^`ba\UeMlTfihi`cf6jkh/lTeMhB`ba\nM\Mmpq¿liw h6xM`kh/\|mhi\ nM\Mmpq.eWw bd\M^jh'& %( ) ³Mjkb7lpa77h>eh/l)ba\M^ajkh/l)eWw nM\ mp`bd\M^jh>lpxMyMqi`nUhe+*Szi`h.eh 180o
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AB
Cb
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Ox
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x1
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;<>=?<5@:ACBDE=F@GDHIEJ,K1LNMHOM2P(O, x1, x2, x3)
HQ@(O, x1X2, X3) R
;<>S1LNM:M:HTDH2UV E1<HXW UV LYE@:ACH2SZL[AGE<HTA:BY@CL[@:=?BY<D"V1LY<\YU?H
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x1 R;<^<B[@CHPU_La`aL[@:AC=?bQH7DH]S1LYMCM:LY\YHcDH
(O, x1X2, X3) → (O, x1, x2, x3)P
P =
1 0 00 cos c − sin c
0 sin c cos c
;<dbQL[U_beE1UFHcU?HIM bQBY`aSZB5M:LY<5@:HIMDEdfNHIbg@CHQEA −−→OC
DL[<ZM(O, x1, x2, x3)
S1E=?MDL[<1M(O, X1X2, X3)
P
AB
Cb
a
c
x
x
2
3
x1
O180−B
−−→OC(O,x1,x2,x3) =
sin a sin B
− sina cosB
cosa
O ! "$#"$%& '()+*,
AB
Cb
a
c
O
X
X
2
3
X1
A−−→OC(O,X1X2,X3) =
sin b sinA
sin b cosA
cos b
P\gb7jkb`hijbºmpka\¾³−−→OC(O,x1,x2,x3) = P .
−−→OC(O,X1X2,X3)
P\ 6Jmh/\m]baj`l[jkh.lClCmpz/h7³
sin b sinA = sin a sinB
− sina cosB = − cos b sin c + sin b cos c cosA
cosa = cos b cos c + sin b sin c cosA
½POaQO|¿ w NKjkh/luÀ`pnMjkh/l³ ( 7³
sin a
sin A=
sin b
sin B=
sin c
sin C½POaQO6O¿
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Paris :
latitude ϕo = 48o50′11.2”longitude λo = 2o20′13.8” Moscou :
latitude ϕ1 = 55o45′39.4”longitude λ1 = 37o39′53.1”
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Brest :
colatitude θo = 42o
longitude λo = −4o
Venezuela :
colatitude θ1 = 75o
longitude λ1 = −60o
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