User:Tohline/Appendix/Ramblings/DirectionCosines
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Direction Cosines
Basic Definitions and Relations
Here we follow the notation of MF53.
This means that the following inverse relationship applies in general:
The coordinate system (ξ_{1},ξ_{2},ξ_{3}) is orthogonal if all the direction cosines obey the following relation:
DC.01 

General Orthogonality Condition 



where the Kronecker delta function, δ_{mn}, is defined such that δ_{mn} = 1 if m = n but δ_{mn} = 0 if .
Usage
Scale Factors
The above relations can be used to define the scale factors (h_{1},h_{2},h_{3}). For example,
or,
Unit Vectors
Direction cosines can be used to switch between the basis vectors of different orthogonal coordinate systems. The defining expressions are:
and,
More explicitly, this last expression(s) implies,

= 


= 


= 

notice that we have liberally used the idea that, for orthogonal systems, γ_{nm} = γ_{mn}.
Orthogonality
How can we check to make sure that the coordinate ξ_{1} is everywhere orthogonal to the coordinate ξ_{2}? Well, for an orthogonal system, the unit vectors should be everywhere perpendicular to one another, that is, the dot product of two (different) unit vectors should be zero at all coordinate positions. Drawing on the above unitvector transformation expressions, this means that, for ,
This is precisely the condition enforced on direction cosines in conjunction with their definition, shown above as Equation DC.01. Notice as well that, when m = n, Equation DC.01 is equivalent to the statement, .
Here we'll illustrate how orthogonality can be checked for any axisymmetric coordinate system; that is, we'll examine behavior only in the plane. First, note that,
and,
Hence,
and,
Therefore also,
The relationship between the direction cosines when gives a key orthogonality condition. Take, for example, m = 1 and n = 2:
∑  γ_{1s}γ_{2s} = 0. 
s 
This means that if ξ_{1} is orthogonal to ξ_{2},
Hence,
DC.02 

An Example Orthogonality Condition 


Position Vector
Employing the unitvector transformation relations tells us that in general the position vector is,

= 


= 


= 

© 2014  2020 by Joel E. Tohline 