释义 |
Direction CosineLet be the Angle between and , the Angle between and , and the Angle between and . Then the direction cosines are equivalent to the coordinatesof a Unit Vector ,
 | (1) |
 | (2) |
 | (3) |
From these definitions, it follows that
 | (4) |
To find the Jacobian when performing integrals over direction cosines, use
The Jacobian is
 | (8) |
Using
so
Direction cosines can also be defined between two sets of Cartesian Coordinates,
 | (13) |
 | (14) |
 | (15) |
 | (16) |
 | (17) |
 | (18) |
 | (19) |
 | (20) |
 | (21) |
Projections of the unprimed coordinates onto the primed coordinates yield
and
Projections of the primed coordinates onto the unprimed coordinates yield
and
 | (31) |
 | (32) |
 | (33) |
Using the orthogonality of the coordinate system, it must be true that
 | (34) |
 | (35) |
giving the identities
 | (36) |
for and , and
 | (37) |
for . These two identities may be combined into the single identity
 | (38) |
where is the Kronecker Delta.
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