formulaire Optique

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Produits scalaire et vectoriela b = a b cos a, b a b = a b sin a, b a b c = b (c a) = c a b = det a, b, c = vol a, b, c a (b c) = b (a c) c (a b) z z

2 2 2 F = div grad F ; F = 2 F = F + F + F x2 y 2 z 2 V = div V V ; V = 2 V = V e + . . . rot rot grad x x dV = dr grad V ; a grad V = a V = ax V + . . . x

Coordonnes cylindro-polaires e F 1 F F grad F = e + e + ez z V Vz 1 (V ) + + div V = z V = 1 Vz V e + V Vz e + . . . rot z z V 1 (V ) ez ... + 2 2 F 1 F F 1 + 2 + F = 2 z 2 F () = 0 F () = A ln V = grad F = Ae / Coordonnes sphriques e e 1 F 1 F F er + e + e grad F = r r r sin V 1 1 ` 2 (sin V ) + r Vr + div V = 2 r r r sin V V = 1 rot (sin V ) er + . . . r sin 1 Vr 1 Vr 1 (rV ) e + (rV ) e ... + r sin r r r 2 F 1 F 1 1 F r2 + 2 sin + 2 2 F = 2 r r r r sin r sin 2 F (r) = 0 F (r) = A/r V = grad F = Aer /r 2 Proprits gnrales e e e e grad Cte r = Cte ; rot Cte r = 2 Cte ; div r = 3

Thor`mes intgraux e e e est ferme et constitue le bord orient de . e Ie Z Stokes : V dr = rot V dS I Z Kelvin : F dr = dS grad F

Syst`mes de coordonnes orthogonaux e ee ey

S est ferme etIdlimite le volume intrieur V. e e e Z Ostrogradski : V dS = div V d S V I Z grad F d Gradient : F dS =S V

Primitives usuellesFonction (x a) 1 xa exp(ax)n, n=1

ex ez

e

rs in

er M e

er

ez e e ex x x

r

e

ey O ez

y y

ln x cos x sin x tan x 1 tan x 1/ cos2 x 1/ sin x 1/ cos x 1/ sin x ch x sh x 1/ ch2 x 1/ sh2 x 1/ ch x 1/ sh x 1 a2 + x2 1 a2 x2 1 a2 + x2 1 2 x2 ` a 2 3/2 1x2

Primitive 1 (x a)n+1 n+1 ln |x a| 1 exp(ax) a x ln x x sin x cos x ln | cos x| ln | sin x| tan x 1 tan x x ln tan + 2 4 x ln tan 2 sh x ch x th x 1 th x 2 arctan(exp(x)) x ln th 2 x 1 arctan a a a + x 1 = 1 argth x ln 2a a x a! a r x x2 x + + 1 = argsh ln a a2 a x arcsin a x 1 x2

e e

m 0 < 2 = Om 0 r = OM 0 0 e = ex cos + ey sin , e = ex sin + ey cos er = ez cos + e sin , e = ez sin + e cos x = cos , y = sin , z = r cos , = r sin

dr = dxex + dyey + dzez ; d = dx dy dz dr = de + de + dzez ; d = d d dz dr = drer + rde + r sin de ; d = r 2 dr sin d d

Oprateurs direntiels e eF F F dF = grad F dr ; grad F = F = ex + ey + ez x y z I Z V dS = div V d (Ostrogradski ; S est ferme et dlimite e e Vy Vz Vx + + le volume intrieur V) ; div V = V = e x y z I Z V dr = rot V dS (Stokes ; est ferme et constitue le bord e Vy Vz ex + . . . orient de ) ; rot V = V = e y zS V

F = 0 ; y = 0 x / y = x rot grad rot grad div rot V = 0 ; div y = 0 x / y = rot x

grad (F G) = F grad G + G grad F div F V = F div V + V grad F div U V = V rot U U rot V rot F V = F rot V + grad F V grad U V = U V + V U + V grad U + U grad V rot rot rot U V = div V U div U V U grad V + . . . . . . + V grad U

Fonctions de Bessel Equation de Bessel : x y + xy + (x )y = 0 Solution gnrale : y(x) = J (x) + Y (x) e e x 2 ( 1)! J (x) x0 ; Y (x) x0 2 ! x 2 2 J+1 (x) = J (x) J1 (x), Y+1 (x) = Y (x) Y1 (x) x x dJ J+1 (x) J1 (x) dY Y+1 (x) Y1 (x) = , = dx dx 2 Z 2 1 J (x) = cos ( x sin ) d 0 X J2n1 (x) sin ([2n 1]) sin (x sin ) = 22 2 2 n=1

Fonctions de lOptiquesinc(x) = 1 sincx sin x , x Z

Rayonnement thermiqueZ

sinc(x) dx =

sinc2 (x) dx =

4 102

0

x

du 8h 3 1 d c du = ; = d c3 exp h 1 d 4 d kB T Z 4 4 2 5 kB x3 dx = ,= 2 h3 exp(x) 1 15 15c 0 hc max T = CW = 0, 201 = 2, 90 103 m K kB d d

cos (x sin ) = J0 (x) + 2 1 J (x) =0 =1

RN (x)/N 2

x=

2 N

N = 10

n=1

X

J2n (x) cos (2n)

N1 2 X sin2 N x/2 RN (x) = exp (ikx) = sin2 x/2 k=0 1

sinc2 x

2hc2 1 d = hc d 5 exp 1 kB T 0, 98T 4

=2 0

x

1 1, 222 4J1 (x) (x)2

0 2 X 1 k exp (ikx) = Fm (x) si < 1 (1 )2 k=0 1 4 Fm (x) = avec m = (1 )2 1 + m sin2 (x/2) 1 m = 10 m = 100

N =6

x=

x

Constantes fondamentalesc = 3, 00 108 m s1 e = 1, 60 1019 C 0 = 8, 851012 Fm1 F = 96 500 C mol1 R = 8, 31 J K1 mol1 h = 6, 63 1034 J s kB = 1, 381023 JK1

0

x Fm (x)

Moments dinertie de solides pleinsh2 12

J = 2 M R2 5

J=

M R2 2

+

M 12

R2 4

a

` 2 b + c2

J=

h

J =M

R

c

0

x

me = 9, 11 1031 kg mp mn 1, 67 1027 kg 0 = 4 107 H m1 NA = 6, 02 1023 mol1 G = 6, 67 1011 m3 kg1 = 5, 67 108 W m2 K TT = 273, 16 K

Donnes astronomiques enon-mtaux He e semi-conducteurs mtaux e B C N O F Ne Al Si P S Cl Ar

b

Classication priodique des lments e eeH Li Be Na Mg K Ca

M = 1, 99 1030 kg 1 UA = 1, 50 1011 m 1 pc = 3, 09 1016 m 1 an = 365, 25 j (solaire) Terre M = 5, 98 1024 kg R = 6, 38 106 m d = 1 UA e = 0, 017 T = 1 an

R = 6, 96 108 m 1 AL = 9, 46 1015 m 1 j (solaire) = 86 400 s 1 j (sidral) = 86 164 s e Lune M = 7, 35 1022 kg R = 1, 74 106 m dTerre = 3, 84 108 m e = 0, 055 T = 27, 3 j (solaire)

Spectre lectromagntique e e400 nm 1 pm O 1 mm 750 nm

R

X 10 nm

UV 400 nm 750 nm

IR

radio 10 cm

Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba * Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra + Lr Rf Ha Sg Ns Hs Mt * La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb lanthanides + Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No actinides

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