NRL_FORMULARY_09-1

CONTENTS

Numerical and Algebraic . . . . . . . . . . . . . . . . . . . .

.

3

Vector Identities . . . . . . . . . . . . . . . . . . . . . . . .

.

4

Di erential Operators in Curvilinear Coordinates . . . . . . . . . .

.

6

Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . 10

International System (SI) Nomenclature . . . . . . . . . . . . . .

. 13

Metric Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Physical Constants (SI) . . . . . . . . . . . . . . . . . . . . .

.

14

Physical Constants (cgs)

. . . . . . . . . . . . . . . . . . . .

.

16

Formula Conversion

. . . . . . . . . . . . . . . . . . . . . .

.

18

Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . .

.

19

Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . 20

Electromagnetic Frequency/Wavelength Bands . . . . . . . . . . .

. 21

AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . .

.

22

Dimensionless Numbers of Fluid Mechanics . . . . . . . . . . . .

. 23

Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

26

Fundamental Plasma Parameters . . . . . . . . . . . . . . . . .

. 28

Plasma Dispersion Function . . . . . . . . . . . . . . . . . . .

.

30

Collisions and Transport

. . . . . . . . . . . . . . . . . . . .

.

31

Approximate Magnitudes in Some Typical Plasmas . . . . . . . . .

. 40

Ionospheric Parameters . . . . . . . . . . . . . . . . . . . . .

.

42

Solar Physics Parameters . . . . . . . . . . . . . . . . . . . .

.

43

Thermonuclear Fusion

. . . . . . . . . . . . . . . . . . . . .

.

44

Relativistic Electron Beams . . . . . . . . . . . . . . . . . . .

.

46

Beam Instabilities . . . . . . . . . . . . . . . . . . . . . . .

.

48

Lasers

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

51

Atomic Physics and Radiation . . . . . . . . . . . . . . . . . .

.

53

Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . .

.

59

Complex (Dusty) Plasmas . . . . . . . . . . . . . . . . . . . .

.

62

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

71

Euler-Mascheroni constant1 γ = 0.57722

Gamma Function (x + 1) = x (x):

(1/6) = 5.5663

(3/5) = 1.4892

(1/5) = 4.5908

(2/3) = 1.3541

(1/4) = 3.6256

(3/4) = 1.2254

(1/3) = 2.6789

(4/5) = 1.1642

(2/5) = 2.2182

(5/6) = 1.1288

(1/2) = 1.7725 = √

(1)

= 1.0

π

Binomial Theorem (good for | x |< 1 or α = positive integer):

X

α

∞

α

k

α(α − 1) 2

α(α − 1)(α − 2)

3

(1 + x)

=

k

x ≡ 1 + αx +

2!

x +

3!

x + . . . .

n

X

x

x + kz

y

y + (n − k)z

k=0

x + kz

k

y + (n − k)z

n − k

x + y

y + nz

=

x + n

.

x + y + nz

nX

∞

(−1)nJα−γn(z)Jβ+γn(z)

=

π

J

(z)J

β−γµ

(z).

n + µ

sin µπ

α+γµ

(27)

dV f =

dSf

V

S

(28)

ZV

dV · A = ZS dS · A

(29)

ZV

dV ·T = ZS dS ·T

(30)

ZV

dV × A =

ZS dS × A

(31)

ZV dV (f 2g − g 2f ) = ZS dS · (f g − g f )

Z

(34)

ZS dS · × A = IC dl · A

(35)

ZS (dS × ) × A = IC dl × A

(36)

ZS dS · ( f × g) = IC f dg = − IC gdf

DIFFERENTIAL OPERATORS IN

CURVILINEAR COORDINATES5

Cylindrical Coordinates

Divergence

1 ∂

1 ∂Aφ

∂Az

· A =

(rAr ) +

+

r

∂r

r

∂φ

∂z

Gradient

( f )r =

∂f

1 ∂f

∂f

; ( f )φ =

; ( f )z =

∂r

r

∂φ

∂z

Curl

( × A)r =

1 ∂Az

∂Aφ

−

r

∂φ

∂z

( × A)φ =

∂Ar

−

∂Az

∂z

∂r

( × A)z =

1 ∂

(rAφ ) −

1 ∂Ar

r ∂r

r

∂φ

Laplacian

r

2f =

1 ∂

∂f

+

1 ∂2f

+

∂2f

r

∂r

∂r

r2

∂φ2

∂z2

Laplacian of a vector

( 2A)r = 2Ar −

2 ∂Aφ

−

Ar

r2

∂φ

r2

( 2A)φ = 2Aφ +

2 ∂Ar

−

Aφ

r2

∂φ

r2

( 2A)z = 2Az

Components of (A · )B

(A · B)r = Ar

∂Br

+

Aφ ∂Br

+ Az

∂Br

−

AφBφ

∂r

r ∂φ

∂z

r

(A · B)φ = Ar

∂Bφ

+

Aφ ∂Bφ

+ Az

∂Bφ

+

Aφ Br

∂r

r ∂φ

∂z

r

(A · B)z = Ar

∂Bz

+

Aφ ∂Bz

+ Az

∂Bz

∂r

r ∂φ

∂z

Divergence of a tensor

( · T )r =

1 ∂

(rTrr ) +

1 ∂Tφr

+

∂Tzr

−

Tφφ

r ∂r

r

∂φ

∂z

r

( · T )φ =

1 ∂

(rTrφ ) +

1 ∂Tφφ

+

∂Tzφ

+

Tφr

r ∂r

r

∂φ

∂z

r

( · T )z =

1 ∂

(rTrz ) +

1 ∂Tφz

+

∂Tzz

r ∂r

r

∂φ

∂z

Spherical Coordinates

Divergence

· A =

1

∂

(r2Ar ) +

1

∂

1

∂Aφ

(sin θAθ ) +

r2

∂r

r sin θ

∂θ

r sin θ

∂φ

Gradient

( f )r =

∂f

1 ∂f

1

∂f

;

( f )θ =

;

( f )φ =

∂r

r

∂θ

r sin θ

∂φ

Curl

( × A)r =

1

∂

(sin θAφ ) −

1

∂Aθ

r sin θ ∂θ

r sin θ

∂φ

( × A)θ =

1

∂Ar

−

1 ∂

(rAφ )

r sin θ

∂φ

r

∂r

( × A)φ =

1 ∂

(rAθ ) −

1 ∂Ar

r ∂r

r

∂θ

Laplacian

1 ∂

∂f

1

∂

∂f

1

∂2f

2f =

r2

+

sin θ

+

r2

∂r

∂r

r2 sin θ

∂θ

∂θ

r2 sin2 θ ∂φ2

Laplacian of a vector

2

2

2Ar

2 ∂Aθ

2 cot θAθ

2

∂Aφ

(

A)r

=

Ar −

−

−

−

r2

r2

∂θ

r2

r2 sin θ

∂φ

2

2

2

∂Ar

Aθ

2 cos θ

∂Aφ

(

A)θ

=

Aθ +

−

−

r2

∂θ

r2 sin2 θ

r2 sin2 θ

∂φ

2

2

Aφ

2

∂Ar

2 cos θ

∂Aθ

(

A)φ

=

Aφ

−

+

+

r2 sin2 θ

r2 sin θ

∂φ

r2 sin2 θ

∂φ

Components of (A · )B

(A · B)r = Ar

∂Br

+

Aθ ∂Br

+

Aφ

∂Br

−

Aθ Bθ + AφBφ

∂r

r

∂θ

r sin θ

∂φ

r

(A · B)θ = Ar

∂Bθ

+

Aθ ∂Bθ

+

Aφ

∂Bθ

+

Aθ Br

−

cot θAφBφ

∂r

r

∂θ

r sin θ

∂φ

r

r

(A · B)φ = Ar

∂Bφ

+

Aθ ∂Bφ

+

Aφ

∂Bφ

+

Aφ Br

+

cot θAφBθ

∂r

r

∂θ

r sin θ

∂φ

r

r

Divergence of a tensor

( · T )r =

1 ∂

(r2Trr ) +

1

∂

(sin θTθr )

r2 ∂r

r sin θ

∂θ

1

∂Tφr

Tθθ + Tφφ

+

−

r sin θ

∂φ

r

( · T )θ =

1 ∂

(r2Trθ ) +

1

∂

(sin θTθθ )

r2 ∂r

r sin θ

∂θ

1