Metodos matematicos para fisicos - arfken

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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

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

This book is printed on acid-free paper. ∞ Copyright c 2003, 2001, 1995, 1985, 1970, 1966 by Harcourt/Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy,recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, CA 92101-4495, USAhttp://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK Harcourt/Academic Press 200 Wheeler Road, Burlington, MA 01803 http://www.harcourt-ap.com International Standard Book Number: 0-12-059877-9 PRINTED IN THE UNITED STATES OF AMERICA 03 04 05 06 07 Q 9 8 7 6 5 4 3 2

1

Contents

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