Metodos matematicos para fisicos - arfken

Vector Identities
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

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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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Contents

Preface 1 1.1 1.2 1.3 1.4 VECTOR ANALYSIS Elementary Approach Vectors and...