According to Betten [1], an orthonormal basis example includes the three-dimensional rectangular Cartesian coordinate system, \( x_i, \; i = 1, 2, 3 \), where a vector is (as also noted in a previous post)
\[ \mathbf{V} = \left( V_1, V_2, V_3 \right) = V_1 \mathbf{e}_1 + V_2 \mathbf{e}_2 + V_3 \mathbf{e}_3 \]
and the unit base vectors are \( \mathbf{e}_1 \), \( \mathbf{e}_2 \), \( \mathbf{e}_3 \). These unit base vectors make up the orthonormal basis. One property of these unit base vectors is the Kronecker delta where
\[ \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij} \]
According to Pahl and Damrath [], the basis vectors of a real vector space, \( \mathbb{R}^n \), is written as \( \mathbf{b}_1, \ldots, \mathbf{b}_n \). This basis is orthogonal if the basis vectors are pairwise orthogonal and orthonormal if the basis vectors have a magnitude of one and are pairwise orthogonal.
\[ \text{orthogonal basis:} \qquad i \ne m \quad \Rightarrow \quad \mathbf{b}_i \cdot \mathbf{b}_m = 0 \]
\[ \text{orthonormal basis:} \qquad i = m \quad \Rightarrow \quad \mathbf{b}_i \cdot \mathbf{b}_m = 1 \qquad i \ne m \quad \Rightarrow \quad \mathbf{b}_i \cdot \mathbf{b}_m = 0 \]
A canonical basis...
A covariant basis is written as [2]
\[ \mathbf{b}_1, \ldots, \mathbf{b}_n \]
where the index is based on subscripts. While a contravariant basis has an index as a superscript shown as [2]
\[ \mathbf{b}^1, \ldots, \mathbf{b}^n \]
A more general form of bases results in a more general coordinate system (such as cylindrical) known as a curvlinear coordinate system. It is sometimes more useful to work in such coordinate systems.
In such a convention, the rectangular Cartesian right-handed orthogonal coordinates, \( x_i \), define a three-dimensional Euclidean space [1]. Curvlinear coordinates can be expressed as \( \xi^i \) and the transformation between rectangular coordinates and curvlinear coordinates is
\[ x_i = x_i \left( \xi^p \right) \Leftrightarrow \xi_i = \xi_i \left( x^p \right) \]
Slattery [3] gives a curvilinear coordinate system where a spatial vector field can be written as a linear combination of the natural basis
\[ \mathbf{u} = u^i \mathbf{g}_i \]
or a linear combination of the dual basis
\[ \mathbf{u} = u_i \mathbf{g}^i \]
In the rectangular Cartesian coordinate system, covariant and contravariant components are unecessary since the natural and dual basis vectors are the same [3]. Thus,
\[ \mathbf{u} = u^i \mathbf{g}_i = u^i g_{ki} \mathbf{g}_i = u_k \mathbf{g}^k \]
From this one can separate into
\[ \left( u^i g_{ki} - u_k \right) \mathbf{g}^k = 0 \]
and
\[ u^i g_{ki} - u_k = 0 \]
which comes out to be
\[ u_k = g_{ki u^i} \]
Similarly,
\[i \mathbf{u} = u_i \mathbf{g}^i = u^i g^{ji} \mathbf{g}_j = u^j \mathbf{g}_k \]
and
\[ u^j = g^{ji}u_i \]
These relations allow indices to be raised and lowered.
References:
[1] J. Betten. Creep Mechanics, 3rd ed. Springer, Berlin, Germany. 2008.
[2] P. J. Pahl and R. Damrath. Mathematical Foundations of Computational Engineering: A Handbook Springer, Berlin, Germany. 2001.
[3] J. C. Slattery. Advanced Transport Phenomena, Cambridge University Press, Cambridge, UK. 1999.
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