- #1

- 232

- 1

## Homework Statement

Consider the following two basis sets (or triads) in [tex]{R}^3[/tex]:

[tex]

\{\vec{e}_1, \vec{e}_2, \vec{e}_3\} := \{(1, 0, 0), (0,1, 0), (0, 0, 1)\}

[/tex]

[tex]

\{\widehat{\vec{e}_1}, \widehat{\vec{e}_2}, \widehat{\vec{e}_3}\} := \{(1, 0, 0), (1,1, 0), (1, 1, 1)\}.

[/tex]

Let a covariant vector [tex]\tilde{u}[/tex] be defined by [tex]\tilde{u}(\alpha^i{\vec{e}_i} ):= \alpha^3-\alpha^2[/tex]. Obtain explicitly the components of [tex]\tilde{u}[/tex] relative to the corresponding bases [tex]\{\vec{e}_1, \vec{e}_2, \vec{e}_3\}[/tex] and [tex]\{\widehat{\vec{e}_1}, \widehat{\vec{e}_2}, \widehat{\vec{e}_3}\}[/tex].

## Homework Equations

[tex]

\tilde{a}(\vec{b})=\tilde{a}(\beta^j\vec{e}_j):=\alpha_j\beta^j

[/tex]

## The Attempt at a Solution

My attempt at a solution is just me running around in a bunch of directions. I really don't have a clear understanding of how to approach this.