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Components of a covariant vector

  • Thread starter roldy
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
352
0
Start by computing [tex]\tilde{u}(\vec{e}_j)[/tex] for [tex]j = 1,2,3[/tex]. (These are the components of [tex]\tilde{u}[/tex] relative to the basis [tex]\{\vec{e}_1, \vec{e}_2, \vec{e}_3\}[/tex], or more properly speaking, relative to the corresponding dual basis.)

Then expand the new basis vectors [tex]\widehat{\vec{e}_j}[/tex] in terms of the [tex]\vec{e}_k[/tex], and use that to compute [tex]\tilde{u}(\widehat{\vec{e}_j})[/tex] for [tex]j = 1,2,3[/tex].
 
  • #3
232
1
I'm actually confused at how to do the first part, calculating [tex] \tilde{u}(\vec{e}_j)[/tex].
I know I need to use this relationship [tex]\tilde{u}(\alpha^i{\vec{e}_i} ):= \alpha^3-\alpha^2[/tex].

So expanding the relationship:

[tex]
\tilde{u}(\alpha^1\vec{e}_1+\alpha^2\vec{e}_2+\alpha^3\vec{e}_3)=\alpha^3-\alpha^2
[/tex]

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.

[tex]
\tilde{u}(\vec{e}_1)=\tilde(u)\left((1)\vec{e}_1+(0)\vec{e}_2+(0)\vec{e}_3\right)=0-0=0
[/tex]

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

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?
 
  • #4
232
1
Does anyone else have insight on this problem?
 

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