Basis vector

Description

An arbitrary vector (often referred to as the radius vector) \(\vec{r}\) is

\[\vectorcompbasis,\]

where \(X^i\) is the contravariant component, while \(\vec{E}_i\) is the covariant basis vector defined as

\[\vec{E}_i \equiv \pder{\vec{r}}{X^i}.\]

Note that \(\vec{E}_i\) is not necessarily normalised; namely

\[\vec{E}_i \cdot \vec{E}_i\]

is not necessarily \(1\). The normalised version \(\vec{\hat{E}}_i\) is given by

\[\vec{\hat{E}}_i \equiv \frac{\vec{E}_i}{H_i},\]

where \(H_i\) is the scale factor elaborated later. Since the basis vectors are orthogonal to each other, I have

\[\orthogonal,\]

where \(\delta\) is the Kronecker delta.

The same vector can be written using the covariant component and the contravariant basis vector as well:

\[\vec{r} = \sum_i X_i \vec{E}^i,\]

where the covariant and the contravariant vectors satisfy (by definition)

\[\vec{E}_i \cdot \vec{E}^j \equiv \delta_i^j.\]

Although the covariant vectors are mostly used in this document, the contravariant vector will appear to define the nabla operator.

In summary, the three different basis vectors are related by

\[\vec{\hat{E}}_i = \frac{\vec{E}_i}{H_i} = H_i \vec{E}^i,\]

as well as the three components:

\[\hat{X}_i = H_i X^i = \frac{1}{H_i} X_i.\]

Note that, for the Cartesian coordinate, \(h_i \equiv 1\) and all the three representations are identical:

\[\vec{\hat{e}}_i \equiv \vec{e}_i \equiv \vec{e}^i,\]

\[\hat{x}_i \equiv x^i \equiv x_i.\]