User:Tohline/Appendix/Mathematics/ScaleFactors

Scale Factors for Orthogonal Curvilinear Coordinate Systems

Here we lean heavily on the class notes and associated references that have been provided by P. A. Kelly in a collection titled, Mechanics Lecture Notes: An Introduction to Solid Mechanics, as they appeared online in early 2021. See especially the subsection of Part III in which the properties of Vectors and Tensors are discussed.

Getting Started

Following Kelly, we will use <math>~\hat{e}_i</math> and <math>~x_i</math> when referencing, respectively, the three (i = 1,3) basis vectors and coordinate "curves" of the Cartesian coordinate system; and we will use <math>~\hat{g}_i</math> and <math>~\Theta_i</math> when referencing, respectively, the three (i = 1,3) basis vectors and coordinate curves of some other, curvilinear coordinate system.

2D Oblique Coordinate System Example

Consider a vector, <math>~\vec{v}</math>, which in Cartesian coordinates is described by the expression,

<math>~\vec{v}</math>

<math>~=</math>

<math>~ \hat{e}_1 v_x + \hat{e}_2 v_y \, . </math>

Referencing Figure 1.16.4 of Kelly's Part III, we appreciate that in a two-dimensional (2D) oblique coordinate system where <math>~\alpha</math> is the (less than 90°) angle between the two basis vectors, the same vector will be represented by the expression,

<math>~\vec{v}</math>

<math>~=</math>

<math>~ \hat{g}_1 v^1 + \hat{g}_2 v^2 \, . </math>

The angle between <math>~\hat{g}_2</math> and <math>~\hat{e}_2</math> is, (π/2 - α), so we appreciate that,

<math>~v_y</math>	<math>~=</math>	<math>~v^2\cos\biggl(\frac{\pi}{2} - \alpha \biggr) = v^2 \sin\alpha</math>
<math>~\Rightarrow~~~v^2</math>	<math>~=</math>	<math>~\frac{v_y}{\sin\alpha} \, .</math>

Next, from a visual inspection of the figure, we appreciate that <math>~v_x</math> is longer than <math>~v^1</math> by the amount, <math>~v^2\cos\alpha</math>; that is,

<math>~v_x</math>	<math>~=</math>	<math>~v^1 + v^2\cos\alpha = v_1 + \frac{v_y}{\tan\alpha}</math>
<math>~\Rightarrow ~~~ v^1</math>	<math>~=</math>	<math>~v_x - \frac{v_y}{\tan\alpha} \, .</math>

(These are the same pair of transformation relations that appear as Eq. (1.16.3) of Kelly's Part III.)

See Also

© 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