(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A two dimensional space has a metric tensor field

[tex]g=dq^1 \otimes dq^1 + 2 dq^1 \otimes dq^2 + 2dq^2 \otimes dq^1 + 3dq^2 \otimes dq^2[/tex].

With the help of g, find the covariant components of the contravariant vector field

[tex]\textbf{v}=3\partial _1 -4\partial _2[/tex]

3. The attempt at a solution

Not much here as I don't really understand how this works.

g_{11}=1, g_{12}=2, g_{21}=2, g_{22}=3

[tex]\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}[/tex]

g^{11}=-3, g^{12}=2, g^{21}=2, g^{22}=-1

Now the covariant components a^{i}ofvare given from the contravariant components a_{i}like this:

a^{1}=a_{1}g^{11}+a_{1}g^{12}=3*(-3)+3*2=-3

a^{2}=a_{2}g^{21}+a_{2}g^{22}=(-4)*2+(-4)*(-1)=-4

So [tex]\textbf{v}=-3dq^1 -4dq^2[/tex]

Does this make any sense at all?

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# Homework Help: A tensor problem

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