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.
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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,
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<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,
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<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,
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<math>~v_y</math> |
<math>~=</math> |
<math>~v^2\cos\biggl(\frac{\pi}{2} - \alpha \biggr) = v^2 \sin\alpha</math> |
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<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,
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<math>~v_x</math> |
<math>~=</math> |
<math>~v^1 + v^2\cos\alpha = v_1 + \frac{v_y}{\tan\alpha}</math> |
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<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.)
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Covarient: The set of basis vectors, <math>~\hat{g}_1</math> and <math>~\hat{g}_2</math> (note the subscript indices), that are aligned with the coordinate directions, <math>~\Theta_1</math> and <math>~\Theta_2</math>, are generically referred to as covariant base vectors. Contravarient: A second set of vectors, which will be termed contravariant base vectors, <math>~\hat{g}^1</math> and <math>~\hat{g}^2</math> (denoted by superscript indices), will be aligned with a new set of coordinate directions, <math>~\Theta^1</math> and <math>~\Theta^2</math>. This new set of base vectors is defined as follows (see Fig. 1.15.5 of Kelly's Part III): the base vector <math>~\hat{g}^1</math> is perpendicular to <math>~\hat{g}_1</math> — that is, <math>~\hat{g}^1 \cdot \hat{g}_2 = 0</math> — and the base vector <math>~\hat{g}^2</math> is perpendicular to <math>~\hat{g}_2</math> — that is, <math>~\hat{g}_1 \cdot \hat{g}^2 = 0</math>. Further, we ensure that,
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Continuing with our 2D oblique coordinate system example and appreciating that Kelly has chosen to align the <math>~\hat{g}_1</math> basis vector with the <math>~\hat{e}_1</math> (Cartesian) basis vector, we see that the transformation between the two sets of covariant basis vectors is given by the relations,
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<math>~\hat{g}_1 = \hat{e}_1 \, ,</math> |
and, |
<math>~\hat{g}_2 = \hat{e}_1\cos\alpha + \hat{e}_2\sin\alpha \, .</math> |
These conditions lead to the following complementary set of contravariant basis vectors:
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<math>~\hat{g}^1 = \hat{e}_1 - \hat{e}_2 \biggl( \frac{1}{\tan\alpha}\biggr) \, ,</math> |
and, |
<math>~\hat{g}^2 = \hat{e}_2 \biggl( \frac{1}{\sin\alpha} \biggr) \, .</math> |
Note that, as defined herein, the magnitude (i.e., scalar lengths) of these contravariant basis vectors is not unity; they are, instead,
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<math>~|\hat{g}^1| \equiv \biggl[ \hat{g}^1 \cdot \hat{g}^1 \biggr]^{1 / 2}</math> |
<math>~=</math> |
<math>~\biggl\{ \biggl[\hat{e}_1 - \hat{e}_2 \biggl( \frac{1}{\tan\alpha}\biggr)\biggr] \cdot \biggl[\hat{e}_1 - \hat{e}_2 \biggl( \frac{1}{\tan\alpha}\biggr)\biggr] \biggr\}^{1 / 2} </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |