Click here to load reader
Upload
yahyakatar
View
45
Download
8
Embed Size (px)
Citation preview
Produits scalaire et vectoriel
~a ·~b = ‖~a‖ × ‖~b‖ × cos“
~a,~b”
‖~a ∧~b‖ = ‖~a‖ × ‖~b‖ ײ
˛
˛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)
Systemes de coordonnees orthogonaux
y
z
x
b
b
b
z
y
x
~ey
~ez
~ex
~eθ
~er
r
ρ
r sin θ ~eϕ
~ez
~eρ
~eϕ
ϕ
θ
M
m
O
~ex
~ey
ϕ
ϕ
~eρ
~eϕ
~ez
~eρ
θ
θ
~er
~eθ
ρ = Om > 0 0 6 ϕ < 2π
r = OM > 0 0 6 θ 6 π
~eρ = ~ex cos ϕ + ~ey sin ϕ, ~eϕ = −~ex sin ϕ + ~ey cos ϕ~er = ~ez cos θ + ~eρ sin θ, ~eθ = −~ez sin θ + ~eρ cos θx = ρ cos ϕ, y = ρ sin ϕ, z = r cos θ, ρ = r sin θ
d~r = dx~ex + dy~ey + dz~ez ; dτ = dx × dy × dzd~r = dρ~eρ + ρdϕ~eϕ + dz~ez ; dτ = ρdρ × dϕ × dzd~r = dr~er + rdθ~eθ + r sin θdϕ~eϕ ; dτ = r2 dr × sin θ dθ × dϕ
Operateurs differentiels
dF =−−→grad F · d~r ;
−−→grad F = ~∇F =
∂F
∂x~ex +
∂F
∂y~ey +
∂F
∂z~ez
I
S
~V · d~S =
Z
V
div ~V dτ (Ostrogradski ; S est fermee et delimite
le volume interieur V) ; div ~V = ~∇ · ~V =∂Vx
∂x+
∂Vy
∂y+
∂Vz
∂zI
Γ
~V · d~r =
Z
Σ
−→rot ~V d~S (Stokes ; Γ est fermee et constitue le bord
oriente de Σ) ;−→rot ~V = ~∇∧ ~V =
»
∂Vz
∂y− ∂Vy
∂z
–
~ex + . . .
∆F = div−−→grad F ; ∆F = ∇2F = ∂2F
∂x2 + ∂2F∂y2 + ∂2F
∂z2
−→rot−→rot ~V =−−→grad div ~V − ∆~V ; ∆~V = ∇2~V = ∆Vx~ex + . . .
d~V =“
d~r · −−→grad”
~V ;“
~a · −−→grad”
~V =“
~a · ~∇”
~V = ax∂~V∂x
+ . . .
Coordonnees cylindro-polaires
−−→grad F =
∂F
∂ρ~eρ +
1
ρ
∂F
∂ϕ~eϕ +
∂F
∂z~ez
div ~V =1
ρ
∂
∂ρ(ρVρ) +
∂Vϕ
∂ϕ
ff
+∂Vz
∂z
−→rot ~V =
1
ρ
∂Vz
∂ϕ− ∂Vϕ
∂z
ff
~eρ +
∂Vρ
∂z− ∂Vz
∂ρ
ff
~eϕ + . . .
. . . +1
ρ
∂
∂ρ(ρVϕ) − ∂Vρ
∂ϕ
ff
~ez
∆F =1
ρ
∂
∂ρ
„
ρ∂F
∂ρ
«
+1
ρ2
∂2F
∂ϕ2+
∂2F
∂z2
∆F (ρ) = 0 ⇒ F (ρ) = A ln ρ ⇒ ~V =−−→grad F = A~eρ/ρ
Coordonnees spheriques
−−→grad F =
∂F
∂r~er +
1
r
∂F
∂θ~eθ +
1
r sin θ
∂F
∂ϕ~eϕ
div ~V =1
r2
∂
∂r
`
r2Vr
´
+1
r sin θ
∂
∂θ(sin θVθ) +
∂Vϕ
∂ϕ
ff
−→rot ~V =1
r sin θ
∂
∂θ(sin θVϕ) − ∂Vθ
∂ϕ
ff
~er + . . .
. . . +1
r
1
sin θ
∂Vr
∂ϕ− ∂
∂r(rVϕ)
ff
~eθ +1
r
∂
∂r(rVθ) − ∂Vr
∂θ
ff
~eϕ
∆F =1
r2
∂
∂r
„
r2 ∂F
∂r
«
+1
r2 sin θ
∂
∂θ
„
sin θ∂F
∂θ
«
+1
r2 sin2 θ
∂2F
∂ϕ2
∆F (r) = 0 ⇒ F (r) = −A/r ⇒ ~V =−−→grad F = A~er/r2
Proprietes generales
−−→grad
“−−→Cte · ~r
”
=−−→Cte ;
−→rot“−−→Cte ∧ ~r
”
= 2 ×−−→Cte ; div~r = 3
−→rot“−−→grad F
”
= 0 ; −→rot ~y = 0 ⇒ ∃x /~y =−−→grad x
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∧−→rot ~V +~V ∧−→rot ~U+“
~V · −−→grad”
~U+“
~U · −−→grad”
~V
−→rot“
~U ∧ ~V”
=“
div ~V”
~U −“
div ~U”
~V −“
~U · −−→grad”
~V + . . .
. . . +“
~V · −−→grad”
~U
Theoremes integraux
Γ est fermee et constitue le bord oriente de Σ.
Stokes :
I
Γ
~V · d~r =
Z
Σ
−→rot ~V d~S
Kelvin :
I
Γ
F d~r =
Z
Σ
d~S ∧ −−→grad F
S est fermee et delimite le volume interieur V.
Ostrogradski :
I
S
~V · d~S =
Z
V
div ~V dτ
Gradient :
I
S
F d~S =
Z
V
−−→grad Fdτ
Primitives usuelles
Fonction Primitive
(x − a)n, n6=−1 1
n + 1(x − a)n+1
1
x − aln |x − a|
exp(ax)1
aexp(ax)
ln x x ln x − xcos x sin xsin x − cos xtan x − ln | cos x|
1
tan xln | sin x|
1/ cos2 x tanx
1/ sin2 x − 1
tanx
1/ cos x ln˛
˛
˛tan“x
2+
π
4
”˛
˛
˛
1/ sin x ln˛
˛
˛tan
x
2
˛
˛
˛
ch x sh xsh x ch x1/ ch2 x thx
1/ sh2 x − 1
thx1/ chx 2 arctan (exp(x))
1/ sh x ln˛
˛
˛th
x
2
˛
˛
˛
1
a2 + x2
1
aarctan
x
a1
a2 − x2
1
2aln
˛
˛
˛
˛
a + x
a − x
˛
˛
˛
˛
=1
aargth
x
a
1√a2 + x2
ln
x
a+
r
x2
a2+ 1
!
= argshx
a1√
a2 − x2arcsin
x
a`
1 ± x2´−3/2 x√
1 ± x2
Fonctions de Bessel
Equation de Bessel : x2y′′ + xy′ + (x2 − ν2)y = 0
Solution generale : y(x) = αJν(x) + βYν(x)
Jν(x) ∼x→0xν
2νν!; Yν(x) ∼x→0 −2ν(ν − 1)!
πxν
Jν+1(x) =2ν
xJν(x) − Jν−1(x), Yν+1(x) =
2ν
xYν(x) − Yν−1(x)
dJν
dx=
Jν+1(x) − Jν−1(x)
2,
dYν
dx=
Yν+1(x) − Yν−1(x)
2
Jν(x) =1
π
Z π
0
cos (νθ − x sin θ) dθ
sin (x sin θ) = 2∞X
n=1
J2n−1(x) sin ([2n − 1]θ)
cos (x sin θ) = J0(x) + 2
∞X
n=1
J2n(x) cos (2nθ)
xb b b b b b b b
b
0
1
Jν(x
)
ν = 0ν = 1
ν = 2
x
b
b b
1,2
2
0
1
4J2 1(π
x)
(πx)2
Moments d’inertie de solides pleins
J=
MR
2
2
J=
M“
R2 4
+h2
12
”
hR
b
R
J = 25MR2
a
b
c
J = M12
`
b2 + c2´
Spectre electromagnetique
400
nm
750
nm
UV
10
nm
X
1pm
γ IR
1m
m
µO
10
cm
radio
400
nm
750
nm
Fonctions de l’Optique
sinc(x) =sin x
x,
Z ∞
−∞
sinc(x) dx =
Z ∞
−∞
sinc2(x) dx = π
x
π
b
b
b b b0
1
sincx
sinc2
x
4 × 10−2
RN (x) =
˛
˛
˛
˛
˛
N−1X
k=0
exp (ikx)
˛
˛
˛
˛
˛
2
=sin2 Nx/2
sin2 x/2
x
πb b b
b
0
1
RN
(x)/
N2
N=
6N
=10
x = π
x = 2πN
˛
˛
˛
˛
˛
∞X
k=0
ρk exp (ikx)
˛
˛
˛
˛
˛
2
=1
(1 − ρ)2Fm(x) si ρ < 1
Fm(x) =1
1 + m sin2(x/2)avec m =
4ρ
(1 − ρ)2
x
πb b b
b
0
1
F m(x
)
m=
10
m=
100
Classification periodique des elements
metaux
non-metaux
semi-conducteurs
lanthanides
actinides
H He
Li Be B C N O F Ne
Na Mg Al Si P S Cl Ar
K Ca 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
+ Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No
Rayonnement thermique
du
dν=
8πhν3
c3
1
exp“
hνkBT
”
− 1;
dϕ
dν=
c
4
du
dν
Z ∞
0
x3dx
exp(x) − 1=
π4
15, σ =
2π5k4B
15c2h3
λmaxT = CW = 0, 201hc
kB= 2, 90 × 10−3 m · K
0, 98σT 4
dϕ
dλ
dϕ
dλ=
2πhc2
λ5
1
exp“
hcλkBT
”
− 1
Constantes fondamentales
c = 3, 00 × 108 m · s−1 me = 9, 11 × 10−31 kge = 1, 60 × 10−19 C mp ≃ mn ≃ 1, 67 × 10−27 kgǫ0 = 8, 85×10−12 F·m−1 µ0 = 4 × π × 10−7 H · m−1
F = 96 500 C · mol−1 NA = 6, 02 × 1023 mol−1
R = 8, 31 J · K−1 · mol−1 G = 6, 67 × 10−11 m3 · kg−1 ·h = 6, 63 × 10−34 J · s σ = 5, 67 × 10−8 W · m−2 · KkB = 1, 38×10−23 J·K−1 TT = 273, 16 K
Donnees astronomiques
M⊙ = 1, 99 × 1030 kg R⊙ = 6, 96 × 108 m1UA = 1, 50 × 1011 m 1AL = 9, 46 × 1015 m1pc = 3, 09 × 1016 m 1 j (solaire) = 86 400 s1 an = 365, 25 j (solaire) 1 j (sideral) = 86 164 s
Terre Lune
M = 5, 98 × 1024 kg M = 7, 35 × 1022 kgR = 6, 38 × 106 m R = 1, 74 × 106 md⊙ = 1UA dTerre = 3, 84 × 108 me = 0, 017 e = 0, 055T = 1an T = 27, 3 j (solaire)
http://physique.fauriel.org/classe/mp+/cours/help.pdf