User:Tohline/Appendix/Ramblings/DirectionCosines
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Direction Cosines
Basic Definitions and Relations
Here we follow the notation of MF53.
<math> \gamma_{ni} = \frac{1}{h_n} \frac{\partial x_i}{\partial \xi_n} = h_n \frac{\partial\xi_n}{\partial x_i} . </math>
This means that the following inverse relationship applies in general:
<math> \frac{\partial x_i}{\partial \xi_n} = h_n^2 \frac{\partial\xi_n}{\partial x_i} . </math>
Let's define a delta function, <math>\delta_{mn}</math> such that <math>\delta_{mn} = 1</math> if <math>m = n</math> but <math>\delta_{mn}=0</math> if <math>m \ne n</math>. The coordinate system <math>(\xi_1, \xi_2, \xi_3)</math> is orthogonal if all the direction cosines obey the following relation:
<math>\sum_s \gamma_{ms}\gamma_{ns} = \delta_{mn} .</math>
Usage
Scale Factors
The above relations can be used to define the scale factors <math>(h_1, h_2, h_3)</math>. For example,
<math>
\sum_s \gamma_{1s}\gamma_{1s} = \sum_s \biggl( h_1 \frac{\partial\xi_1}{\partial x_s} \biggr)^2 = 1
</math>
<math> \Rightarrow ~~~~~ h_1^2 = \biggl[ \biggl(\frac{\partial\xi_1}{\partial x} \biggr)^2 + \biggl(\frac{\partial\xi_1}{\partial y} \biggr)^2 + \biggl(\frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1} ; </math>
or,
<math>
\sum_s \gamma_{1s}\gamma_{1s} = \sum_s \biggl( \frac{1}{h_1} \frac{\partial x_s}{\partial\xi_1} \biggr)^2 = 1
</math>
<math> \Rightarrow ~~~~~ h_1^2 = \biggl[ \biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 \biggr] . </math>
Orthogonality
How can we check to make sure that the coordinate <math>\xi_1</math> is everywhere orthogonal to the coordinate <math>\xi_2</math>? Here we'll illustrate how orthogonality can be checked for any axisymmetric coordinate system; that is, we'll examine behavior only in the <math>(\varpi,z)</math> plane. First, note that,
<math> \frac{\partial\varpi}{\partial x} = \frac{\partial}{\partial x} (x^2 + y^2)^{1/2} = \frac{x}{\varpi} , </math>
and,
<math> \frac{\partial\varpi}{\partial y} = \frac{\partial}{\partial x} (x^2 + y^2)^{1/2} = \frac{y}{\varpi} , </math>
Hence,
<math> \frac{\partial\xi_i}{\partial x} = \frac{\partial\xi_i}{\partial \varpi}\frac{\partial\varpi}{\partial x} = \biggl(\frac{x}{\varpi}\biggr) \frac{\partial\xi_i}{\partial \varpi} , </math>
and,
<math> \frac{\partial\xi_i}{\partial y} = \frac{\partial\xi_i}{\partial \varpi}\frac{\partial\varpi}{\partial y} = \biggl(\frac{y}{\varpi}\biggr) \frac{\partial\xi_i}{\partial \varpi} . </math>
Therefore also,
<math>
\biggl( \frac{\partial\xi_i}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_i}{\partial y } \biggr)^2 = \biggl( \frac{\partial\xi_i}{\partial\varpi} \biggr)^2
</math>
<math> \Rightarrow ~~~~~ h_i^2 = \biggl[ \biggl(\frac{\partial\xi_i}{\partial \varpi} \biggr)^2 + \biggl(\frac{\partial\xi_i}{\partial z} \biggr)^2 \biggr]^{-1} . </math>
Unit Vectors
Direction cosines can be used to switch between the basis vectors of different orthogonal coordinate systems. For example,
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<math> \hat\imath </math> |
<math> = </math> |
<math> \hat{e}_1 \gamma_{11} + \hat{e}_2 \gamma_{21} + \hat{e}_3 \gamma_{31} ; </math> |
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<math> \hat\jmath </math> |
<math> = </math> |
<math> \hat{e}_1 \gamma_{12} + \hat{e}_2 \gamma_{22} + \hat{e}_3 \gamma_{32} ; </math> |
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<math> \hat{k} </math> |
<math> = </math> |
<math> \hat{e}_1 \gamma_{13} + \hat{e}_2 \gamma_{23} + \hat{e}_3 \gamma_{33} . </math> |
Position Vector
And, employing these relations tells us that in general the position vector is,
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<math> \vec{x} </math> |
<math> = </math> |
<math> \hat\imath x + \hat\jmath y + \hat{k}z </math> |
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<math> = </math> |
<math> (\hat{e}_1 \gamma_{11} + \hat{e}_2 \gamma_{21} + \hat{e}_3 \gamma_{31}) x + (\hat{e}_1 \gamma_{12} + \hat{e}_2 \gamma_{22} + \hat{e}_3 \gamma_{32})y + (\hat{e}_1 \gamma_{13} + \hat{e}_2 \gamma_{23} + \hat{e}_3 \gamma_{33})z </math> |
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<math> = </math> |
<math> \hat{e}_1(x\gamma_{11} + y\gamma_{12} + z\gamma_{13} ) + \hat{e}_2(x\gamma_{21} + y\gamma_{22} + z\gamma_{23} ) + \hat{e}_3 (x\gamma_{31} + y\gamma_{32} + z \gamma_{33}) . </math> |
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© 2014 - 2021 by Joel E. Tohline |