User:Tohline/Appendix/Ramblings/DirectionCosines

Direction Cosines

Following [MF53], a generalized coordinate system consists of a threefold family of surfaces whose equations in terms of Cartesian coordinates are, <math>~\xi_1(x,y,z) = </math> constant, <math>~\xi_2(x,y,z) = </math> constant, and <math>~\xi_3(x,y,z) =</math> constant. The lines of intersection of these surfaces constitute three families of lines, in general curved. At any point <math>~(x, y, z)</math> or <math>~(\xi_1, \xi_2, \xi_3)</math> we can place three unit vectors — <math>~(\hat\imath, \hat\jmath, \hat{k})</math> or <math>~(\hat{e}_1, \hat{e}_2, \hat{e}_3)</math>, respectively — each tangent to the corresponding coordinate line of the curvilinear system which goes through the point.

The three angles measured between any one of these unit vectors, <math>~\hat{e}_n</math>, and the three unit vectors of the Cartesian coordinate system, <math>~\hat\imath, \hat\jmath, \hat{k}</math>, are referred to as the direction cosines of the unit vector, <math>~\hat{e}_n</math>. Specifically,

<math>~\gamma_{n1}</math>

<math>~\equiv</math>

<math>~\hat{e}_n \cdot \hat\imath \, ,</math>

<math>~\gamma_{n2}</math>

<math>~\equiv</math>

<math>~\hat{e}_n \cdot \hat\jmath \, ,</math>

and,

<math>~\gamma_{n3}</math>

<math>~\equiv</math>

<math>~\hat{e}_n \cdot \hat{k} \, .</math>

Basic Definitions and Relations

The three direction cosines that are associated with the unit vector, <math>~\hat{e}_n</math>, can be obtained from the defining functional relationship, <math>~\xi_n(x, y, z)</math>, and an associated "scale factor," <math>~h_n</math>, (discussed immediately below) via the expressions,

<math>~\gamma_{ni}</math>	<math>~=</math>	<math>~h_n \frac{\partial\xi_n}{\partial x_i} \, ;</math>	or,	<math>~\gamma_{ni}</math>	<math>~=</math>	<math>~h_n \frac{\partial\xi_n}{\partial x_i} \, ;</math>
[ MF53, §1.3, Eq. (1.3.5) ]

depending on whether the <math>~\xi</math>'s are given in terms of <math>~x, y, z</math> or visa versa. 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>

The coordinate system <math>~(\xi_1, \xi_2, \xi_3)</math> is orthogonal if all the direction cosines obey the following …

DC.01

General Orthogonality Condition
<math>~\sum_s \gamma_{ms}\gamma_{ns}</math> <math>~=</math> <math>~\sum_s \gamma_{sm}\gamma_{sn}</math> <math>~=</math> <math>~\delta_{mn} \, ,</math> [ MF53, §1.3, Eq. (1.3.1) ]

where the Kronecker delta function, <math>~\delta_{mn}</math>, is defined such that <math>~\delta_{mn} = 1</math> if <math>~m = n</math> but <math>~\delta_{mn}=0</math> if <math>~m \ne n</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}</math>	<math>~=</math>	<math>~\sum_s \biggl( h_1 \frac{\partial\xi_1}{\partial x_s} \biggr)^2 = 1</math>
<math>~\Rightarrow~~~ h_1^2</math>	<math>~=</math>	<math>~\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}</math>	<math>~=</math>	<math>~\sum_s \biggl( \frac{1}{h_1} \frac{\partial x_s}{\partial\xi_1} \biggr)^2 = 1</math>
<math>~\Rightarrow ~~~ h_1^2</math>	<math>~=</math>	<math>~\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>
[ MF53, §1.3, Eq. (1.3.4) ]

Unit Vectors

Direction cosines can be used to switch between the basis vectors of different orthogonal coordinate systems. The defining expressions are:

<math> \hat{e}_n = \hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k}\gamma_{n3} ; </math>

and,

<math> \hat\imath = \sum_{n=1,3}\hat{e}_n \gamma_{n1} ; ~~~~\mathrm{etc.} </math>

More explicitly, this last expression(s) implies,

<math> \hat\imath </math>	<math> = </math>	<math> \hat{e}_1 \gamma_{11} + \hat{e}_2 \gamma_{21} + \hat{e}_3 \gamma_{31} ; </math>
<math> \hat\jmath </math>	<math> = </math>	<math> \hat{e}_1 \gamma_{12} + \hat{e}_2 \gamma_{22} + \hat{e}_3 \gamma_{32} ; </math>
<math> \hat{k} </math>	<math> = </math>	<math> \hat{e}_1 \gamma_{13} + \hat{e}_2 \gamma_{23} + \hat{e}_3 \gamma_{33} ; </math>

notice that we have liberally used the idea that, for orthogonal systems, <math>~\gamma_{nm} = \gamma_{mn}</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>? 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 unit-vector transformation expressions, this means that, for <math>m \ne n</math>,

<math> \hat{e}_m \cdot \hat{e}_n = \biggl[ \hat\imath \gamma_{m1} + \hat\jmath \gamma_{m2} + \hat{k}\gamma_{m3} \biggr] \cdot \biggl[ \hat\imath \gamma_{n1} + \hat\jmath \gamma_{n2} + \hat{k}\gamma_{n3} \biggr] = \gamma_{m1}\gamma_{n1} + \gamma_{m2}\gamma_{n2} + \gamma_{m1}\gamma_{n2} = 0 </math>

<math> \Rightarrow ~~~~~ \sum_{s=1}^3 \gamma_{ms}\gamma_{ns} = 0 . </math>

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 <math>~m = n</math>, Equation DC.01 is equivalent to the statement, <math>~\hat{e}_m\cdot \hat{e}_m = 1</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>

The relationship between the direction cosines when <math>m \ne n</math> gives a key orthogonality condition. Take, for example, <math>~m=1</math> and <math>~n=2</math>:

<math>~\sum_s \gamma_{1s}\gamma_{2s} = 0 .</math>

This means that if <math>~\xi_1</math> is orthogonal to <math>~\xi_2</math>,

<math>~ h_1 \frac{\partial\xi_1}{\partial x} \cdot h_2 \frac{\partial\xi_2}{\partial x} + h_1 \frac{\partial\xi_1}{\partial y} \cdot h_2 \frac{\partial\xi_2}{\partial y} + h_1 \frac{\partial\xi_1}{\partial z} \cdot h_2 \frac{\partial\xi_2}{\partial z}= 0 </math>

<math> \Rightarrow ~~~~~ h_1 h_2\biggl[ \biggl( \frac{x^2}{\varpi^2} \biggr) \frac{\partial\xi_1}{\partial \varpi} \cdot \frac{\partial\xi_2}{\partial \varpi} + \biggl( \frac{y^2}{\varpi^2} \biggr) \frac{\partial\xi_1}{\partial \varpi} \cdot \frac{\partial\xi_2}{\partial \varpi} + \frac{\partial\xi_1}{\partial z} \cdot \frac{\partial\xi_2}{\partial z} \biggr] = 0 .

</math>

Hence,

DC.02

An Example Orthogonality Condition
<math> \frac{\partial\xi_1}{\partial \varpi} \cdot \frac{\partial\xi_2}{\partial \varpi} = - \frac{\partial\xi_1}{\partial z} \cdot \frac{\partial\xi_2}{\partial z} . </math>

Position Vector

Employing the unit-vector transformation relations tells us that in general the position vector is,

<math> \vec{x} </math>	<math> = </math>	<math> \hat\imath x + \hat\jmath y + \hat{k}z </math>
	<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>
	<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>

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation