Arfken - Essential Mathematical Methods for Physicists (Academic, 2003).pdf

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Vector Identities
A 2
A x +
A y +
A z ,
=
+
+
=
·
=
A x B x +
A y B y +
A
A x x
A y y
A z z ,
A
B
A z B z
A y
A z
A x
A z
A x
A y
A
×
B
=
x
−
y
+
z
B y
B z
B x
B z
B x
B y
−
+
A x
A y
A z
A y
A z
A x
A z
A x
A y
A
·
( B
×
C )
=
B x
B y
B z
=
C x
C y
C z
B y
B z
B x
B z
B x
B y
C x
C y
C z
A × ( B × C ) = BA · C − CA · B ,
k ε ijk ε pq k = δ ip δ jq − δ iq δ jp
Vector Calculus
r
r
dV
dr =− r dV
∇ · ( r f ( r )) = 3 f ( r ) + r df
F =− ∇ V ( r ) =−
dr ,
dr ,
∇ · ( r r n − 1 ) = ( n + 2) r n − 1
∇
( A
·
B )
=
( A
· ∇
) B
+
( B
· ∇
) A
+
A
×
(
∇ ×
B )
+
B
×
(
∇ ×
A )
∇ ·
( S A )
= ∇
S
·
A
+
S
∇ ·
A ,
∇ ·
( A
×
B )
=
B
·
(
∇ ×
A )
−
A
·
(
∇ ×
B )
∇ ·
(
∇ ×
A )
=
0,
∇ ×
( S A )
= ∇
S
×
A
+
S
∇ ×
A ,
∇ ×
( r f ( r ))
=
0,
∇ ×
r
=
0
∇ ×
( A
×
B )
=
A
∇ ·
B
−
B
∇ ·
A
+
( B
· ∇
) A
−
( A
· ∇
) B ,
2 A
∇ ×
∇ ×
= ∇
∇ ·
− ∇
(
A )
(
A )
B d 3 r
V ∇ ·
=
B
·
d a ,
(Gauss),
(
∇ ×
A )
·
d a
=
A
·
d l ,
(Stokes)
S
S
2
2
φ ) d 3 r =
( φ ∇
ψ − ψ ∇
( φ ∇ ψ − ψ ∇ φ ) · d a ,
(Green)
V
S
2 1
1
| a | δ ( x ),
δ ( x − x i )
| f ( x i ) |
∇
r =− 4 πδ ( r ),
δ ( ax ) =
δ ( f ( x )) =
,
i , f ( x i ) = 0, f ( x i ) = 0
∞
d 3 k
(2
1
2
e i ω ( t − x ) d ω
) 3 e − i k · r ,
δ
( t − x )
=
,
δ
( r )
=
π
π
−∞
∞
n = 0 ϕ n ( x ) ϕ n ( t )
δ ( x − t ) =
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Curved Orthogonal Coordinates
Cylinder Coordinates
q 1 = ρ
,
q 2 = ϕ
,
q 3 =
z ;
h 1 =
h ρ =
1, h 2 =
h ϕ = ρ
, h 3 =
h z =
1,
r
=
x
ρ
cos
ϕ +
y
ρ
sin
ϕ +
z z
Spherical Polar Coordinates
q 1 =
r , q 2 = θ
, q 3 = ϕ
; h 1 =
h r =
1, h 2 =
h θ =
r , h 3 =
h ϕ =
r sin
θ
,
r = x r sin θ cos ϕ + y r sin θ sin ϕ + z r cos θ
q 1
q 2
q 3
A 1
A 2
A 3
d r
=
h i dq i q i ,
A
=
A i q i ,
A
·
B
=
A i B i , A
×
B
=
B 1
B 2
B 3
i
i
i
fd 3 r
=
f ( q 1 , q 2 , q 3 ) h 1 h 2 h 3 dq 1 dq 2 dq 3
F
·
d r
=
F i h i dq i
V
L
i
B · d a =
B 1 h 2 h 3 dq 2 dq 3 +
B 2 h 1 h 3 dq 1 dq 3 +
B 3 h 1 h 2 dq 1 dq 2 ,
S
h i ∂ V
1
∇ V =
q i
,
∂ q i
i
∂
∂
q 3 ( F 3 h 1 h 2 )
∂
∂
∂
1
h 1 h 2 h 3
∇ ·
=
+
+
F
q 1 ( F 1 h 2 h 3 )
q 2 ( F 2 h 1 h 3 )
∂
∂
∂
h 2 h 3
h 1
h 1 h 3
h 2
h 2 h 1
h 3
1
h 1 h 2 h 3
∂
V
∂
∂
∂
V
∂
∂
V
2 V
∇
=
+
+
∂
∂
∂
∂
q 1
q 1
q 2
q 2
q 3
q 3
h 1 q 1
h 2 q 2
h 3 q 3
1
h 1 h 2 h 3
∂
∂ q 1
∂
∂ q 2
∂
∂ q 3
∇ × F =
h 1 F 1
h 2 F 2
h 3 F 3
Mathematical Constants
=
.
π =
.
=
.
e
2
718281828,
3
14159265,
ln 10
2
302585093,
29577951 ◦ ,
◦ =
1 rad
=
57
.
0
.
0174532925 rad,
1 +
n − ln( n + 1)
1
2 +
1
3 +···+
1
γ = lim
n →∞
= 0 . 577215661901532
(Euler-Mascheroni number)
1
2 , B 2 =
1
6 , B 4 = B 8 =−
1
30 , B 6 =
1
42 , ...
B 1 =−
(Bernoulli numbers)
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Essential Mathematical
Methods for Physicists
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