Formulas Física

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Mathematical formulas 
668 
Appendix 1: Mathematical formulas 
A.1.1. Vector identities 
A, B, and C are vectors and a and b are scalars. 
 BACACBCBA (A.1.1) 
 BACCABCBA (A.1.2) 
 BABA (A.1.3) 
 baba (A.1.4) 
 BABA (A.1.5) 
 BBB aaa (A.1.6) 
 abbaab (A.1.7) 
 BBB aaa (A.1.8) 
 BAABBA (A.1.9) 
 ABBAABBABA (A.1.10) 
 BAABABBABA (A.1.11) 
 aa 2 (A.1.12) 
 0A (A.1.13) 
 0a (A.1.14) 
 AAA 2 (A.1.15) 
 
 
 
 
 
 
 
 
 
Mathematical formulas 
669 
A.1.2. Vector operations in the three coordinate systems 
Cartesian 
)19.1.A(
z
a
y
a
x
aa
)18.1.A(
y
A
x
A
x
A
z
A
z
A
y
A 
)17.1.A(
z
A
y
A
x
A
)16.1.A(
z
a
y
a
x
aa
2
2
2
2
2
2
2
xyzxyz
zyx
zyx
zyx
uuuA
A
uuu
Cylindrical 
)23.1.A(
z
aa
r
1
r
r
ar
r
1a
)22.1.A(A
r
A r
r
1
r
A
z
A
z
AA
r
1 
)21.1.A(
z
AA
r
1
r
rA
r
1
)20.1.A(
z
aa
r
1
r
aa
2
2
2
2
2
2
rzr
r
z
zr
z
zr
uuuA
A
uuu
Spherical 
)27.1.A(a
 sin 
1
a sin
 sin 
1
a
1a
)26.1.A(
AA 1
AA
 sin
11AA sin
 sin
1
)25.1.A(
A
 sin 
1 sin A
 sin 
1A1
)24.1.A(a
 sin 
1a1aa
2
2
2 22
2
2
2
2
2
u
uuA
A
uuu
 
Mathematical formulas 
670 
A.1.3. Summary of the transformations between coordinate 
systems 
Cartesian-cylindrical 
 
zz
sinry
cosrx
 
zz
x
y tan
yx r
1-
22
 (A.1.28) 
 ru u zu 
Xu cos sin 0 
Yu sin cos 0 
 Zu 0 0 1 
 (A.1.29) 
Cartesian-spherical 
 
cosz
sinsiny
cossinx
 
x
y tan
z
yx
tan
zyx
1 - 
22
1
222
 (A.1.30) 
 u u u 
Xu cossin coscos sin 
Yu sinsin sincos cos 
Zu cos sin 0 
 (A1.31) 
 Distance R=|r2-r1| in the three coordinate systems. 
1. Cartesian: 2/1212
2
12
2
12 )zz()yy()xx(R 
2. cylindrical: 
2/12
121212
2
1
2
2 zzcosrr2rrR 
3. spherical: 2/112121212
2
1
2
2 cossinsincoscos2R 
Mathematical formulas 
671 
A.1.4. Integral relations 
 
 dsAA
V
dv [divergence theorem] (A.1.32) 
 dlAdsA
S
 [Stokes’s theorem] (A.1.33) 
 dsAA
V
dv (A.1.34) 
 dsa adv
V
 (A.1.35) 
 dlds aa
S
 (A.1.36) 
 
Coordinate system Cartesian 
(x, y, z) 
Cylindrical 
(r, , z) 
Spherical 
( , , ) 
Unit vectors ux uy uz ur u uz u u u 
Differential length dl dx ux 
dy uy 
dz uz 
dr ur 
r d u 
dz uz 
d u 
 sin d u 
 d u 
Differential surface area ds dy dz ux 
dx dz uy 
dx dy uz 
r d dz ur 
dr dz u 
r dr d uz 
2 sin d d u 
 sin d d u 
 d d u 
Differential volume dv dx dy dz r dr d dz 2 sin d d d