Components of a covariant vector

[roldy]

Homework Statement

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

<br /> \{\vec{e}_1, \vec{e}_2, \vec{e}_3\} := \{(1, 0, 0), (0,1, 0), (0, 0, 1)\}<br />

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

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

Homework Equations

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

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.

 

[ystael]

Start by computing \tilde{u}(\vec{e}_j) for j = 1,2,3. (These are the components of \tilde{u} relative to the basis \{\vec{e}_1, \vec{e}_2, \vec{e}_3\}, or more properly speaking, relative to the corresponding dual basis.)

Then expand the new basis vectors \widehat{\vec{e}_j} in terms of the \vec{e}_k, and use that to compute \tilde{u}(\widehat{\vec{e}_j}) for j = 1,2,3.

 

[roldy]

I'm actually confused at how to do the first part, calculating \tilde{u}(\vec{e}_j).
I know I need to use this relationship \tilde{u}(\alpha^i{\vec{e}_i} ):= \alpha^3-\alpha^2.

So expanding the relationship:

<br /> \tilde{u}(\alpha^1\vec{e}_1+\alpha^2\vec{e}_2+\alpha^3\vec{e}_3)=\alpha^3-\alpha^2<br />

What does the above actually mean?

I asked my professor about this problem and he helped a little. He actually worked through a portion of it. This is what came about.

<br /> \tilde{u}(\vec{e}_1)=\tilde(u)\left((1)\vec{e}_1+(0)\vec{e}_2+(0)\vec{e}_3\right)=0-0=0<br />

The thing I don't understand about this is why are there basis \vec{e}_1, \vec{e}_2, \vec{e}_3 in the parenthesis when you are trying to figure out \tilde{u}(\vec{e}_1).

I need a really simplified (dumbed down) explanation on this. I'm not used of working with anything so abstract.

Would you know of any books that are easy on the reader?

 

[roldy]

Does anyone else have insight on this problem?