Solving a Tensor Problem in 2D Space

[atwood]

Homework Statement

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
$$\textbf{v}=3\partial _1 -4\partial _2$$

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 a_i 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?

 

[Dick]

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

 

[atwood]

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

Thanks!