# A tensor problem

1. Sep 12, 2008

### atwood

1. The problem statement, all variables and given/known data
A two dimensional space has a metric tensor field
$$g=dq^1 \otimes dq^1 + 2 dq^1 \otimes dq^2 + 2dq^2 \otimes dq^1 + 3dq^2 \otimes dq^2$$.

With the help of g, find the covariant components of the contravariant vector field
$$\textbf{v}=3\partial _1 -4\partial _2$$

3. The attempt at a solution
Not much here as I don't really understand how this works.

g11=1, g12=2, g21=2, g22=3

$$\begin{center}G=\left(\begin{array}{ll} 1 & 2 \\ 2 & 3 \end{array}\right) \Rightarrow G^{-1}=\left(\begin{array}{ll} {-3} & \ 2 \\ \ 2 & {-1} \end{array}\right)\end{center}$$

g11=-3, g12=2, g21=2, g22=-1

Now the covariant components ai of v are given from the contravariant components ai like this:
a1=a1g11+a1g12=3*(-3)+3*2=-3
a2=a2g21+a2g22=(-4)*2+(-4)*(-1)=-4

So $$\textbf{v}=-3dq^1 -4dq^2$$

Does this make any sense at all?

2. Sep 12, 2008

### Dick

$a^i=a_k g^{ik}$, yes? So don't you mean $a^1=a_1 g^{11}+a_2 g^{12}$?

3. Sep 13, 2008

### atwood

Right, of course. I should never ask anything when tired.

Thanks!