Metodos matematicos para fisicos - arfken
A = Ax x + Ayy + Azz, ˆ ˆ ˆ
2 A2 = Ax + A2 + A2 , y z
A · B = Ax Bx + Ay By + Az Bz
A×B =
Ay By Ax
Az Bz Ay By Cy
x− ˆ Az
Ax Bx
Az Bz
y+ ˆ Az Bz
Ax Bx
Ay By
z ˆ
A · (B × C) = Bx Cx
Ay Bz = C x By Cz
k
− Cy
Ax Bx
Az Bz
+ Cz
Ax Bx
Ay By
A × (B × C) = B A · C − C A · B,
εijk ε pqk = δip δ jq − δiq δ jp
VectorCalculus
F = −∇V (r) = − r dV dV = −ˆ r , r dr dr ∇ · (r f (r)) = 3 f (r) + r df , dr
∇ · (rr n−1 ) = (n + 2)r n−1 ∇(A · B) = (A · ∇)B + (B · ∇)A + A × (∇ × B) + B × (∇ × A) ∇ · (SA) = ∇S · A + S∇ · A, ∇ · (∇ × A) = 0, ∇×r = 0 ∇ × (A × B) = A ∇ · B − B ∇ · A + (B · ∇)A − (A · ∇)B, ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A ∇ · B d3 r =
V S
∇ · (A × B) = B · (∇ × A) − A · (∇ × B) ∇ × (r f (r)) = 0,
∇ ×(SA) = ∇S × A + S∇ × A,
B · da,
(Gauss),
S
(∇ × A) · da =
A · dl,
(Stokes) (Green)
(φ∇2 ψ − ψ∇2 φ)d3r =
V S
(φ∇ψ − ψ∇φ) · da,
∇2
1 = −4π δ(r), r
δ(ax) =
1 δ(x), |a| 1 2π
δ( f (x)) =
∞ −∞
δ(x − xi ) , | f (xi )| i, f (x )=0, f (x )=0
i i
δ(t − x) =
eiω(t−x) dω,
δ(r) = δ(x − t) =
d3 k −ik·r e , (2π )3
∞ n=0 ∗ ϕn (x)ϕn(t)
Curved OrthogonalCoordinates
Cylinder Coordinates q1 = ρ, q2 = ϕ, q3 = z; h1 = hρ = 1, h2 = hϕ = ρ, h3 = hz = 1, r = xρ cos ϕ + yρ sin ϕ + z z ˆ ˆ ˆ Spherical Polar Coordinates q1 = r, q2 = θ, q3 = ϕ; h1 = hr = 1, h2 = hθ = r, h3 = hϕ = r sin θ, r = xr sin θ cos ϕ + yr sin θ sin ϕ + z r cos θ ˆ ˆ ˆ dr =
i
hi dqi qi , ˆ
A=
i
A i qi , ˆ
A·B=
i
q1 ˆ A1 A i Bi, A × B = B1 F · dr =
L i
q2 ˆ A2 B2q3 ˆ A3 B3
f d3 r =
V
f (q1 , q2 , q3 )h1 h 2 h 3 dq1 dq2 dq3 B1 h 2 h 3 dq2 dq3 + qi ˆ
i
Fi hi dqi
B · da =
S
B 2 h1 h 3 dq1 dq3 +
B 3 h1 h 2 dq1 dq2 ,
∇V = ∇·F = ∇2 V =
1 ∂V , hi ∂qi
1 ∂ ∂ ∂ (F1 h 2 h 3 ) + (F2 h1 h 3 ) + (F3 h1 h 2 ) h1 h 2 h 3 ∂q1 ∂q2 ∂q3 ∂ h 2h 3 ∂ V 1 h1 h 2 h 3 ∂q1 h1 ∂q1 h1 q1 ˆ h 2 q2 ˆ
∂ ∂q2 ∂ ∂q1
+ h 3 q3 ˆ
∂ ∂q3
∂ h1 h 3 ∂ V ∂q2h 2 ∂q2
+
∂ h 2 h1 ∂ V ∂q3 h 3 ∂q3
1 ∇×F = h1 h 2 h 3
h1 F1
h 2 F2
h 3 F3
Mathematical Constants e = 2.718281828,
◦
π = 3.14159265,
◦
ln 10 = 2.302585093,
1 rad = 57.29577951 , 1 = 0.0174532925 rad, 1 1 1 γ = lim 1 + + + · · · + − ln(n + 1) = 0.577215661901532 n→∞ 2 3 n (Euler-Mascheroni number) 1 1 1 1 B1 = − , B2 = , B4 = B8 = − , B6 = , . . . (Bernoulli numbers)2 6 30 42
Essential Mathematical Methods for Physicists
Essential Mathematical Methods for Physicists
Hans J. Weber
University of Virginia Charlottesville, VA
George B. Arfken
Miami University Oxford, Ohio
Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
Sponsoring Editor Production Editor Editorial Assistant Marketing ManagerCover Design Printer and Binder
Barbara Holland Angela Dooley Karen Frost Marianne Rutter Richard Hannus Quebecor
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1
Contents
Preface 1 1.1 1.2 1.3 1.4 VECTOR ANALYSIS Elementary Approach Vectors and...
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