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

**. One property of these unit base vectors is the Kronecker delta where**

*orthonormal basis*\[ \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij} \]

According to Pahl and Damrath [], the

__, \( \mathbb{R}^n \), is written as \( \mathbf{b}_1, \ldots, \mathbf{b}_n \). This basis is__*basis vectors of a real vector space*__orthogonal__if the__are__*basis vectors*__pairwise____orthogonal__and__orthonormal__if the__have a__*basis vectors***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

**and**

*raised***.**

*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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