Let be the angle between and , the angle between and , and the angle between and . Then the direction cosines are equivalent to the coordinates of a unit vector ,
(1)
| |||
(2)
| |||
(3)
|
From these definitions, it follows that
(4)
|
To find the Jacobian when performing integrals over direction cosines, use
(5)
| |||
(6)
| |||
(7)
|
The Jacobian is
(8)
|
Using
(9)
| |||
(10)
|
(11)
| |||
(12)
|
so
(13)
| |||
(14)
| |||
(15)
| |||
(16)
|
Direction cosines can also be defined between two sets of Cartesian coordinates,
(17)
|
(18)
|
(19)
|
(20)
|
(21)
|
(22)
|
(23)
|
(24)
|
(25)
|
Projections of the unprimed coordinates onto the primed coordinates yield
(26)
| |||
(27)
| |||
(28)
| |||
(29)
| |||
(30)
| |||
(31)
|
and
(32)
| |||
(33)
| |||
(34)
| |||
(35)
| |||
(36)
| |||
(37)
|
Projections of the primed coordinates onto the unprimed coordinates yield
(38)
| |||
(39)
| |||
(40)
| |||
(41)
| |||
(42)
| |||
(43)
|
and
(44)
| |||
(45)
| |||
(46)
|
Using the orthogonality of the coordinate system, it must be true that
(47)
|
(48)
|
giving the identities
(49)
|
for and , and
(50)
|
for . These two identities may be combined into the single identity
(51)
|
where is the Kronecker delta.