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.
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
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 |