A 2D space has coordinates $x^1$ and $x^2$ with line element

$$\mathrm{d} l^{2}=\left(\mathrm{d} x^{1}\right)^{2}+\left(x^{1}\right)^{2}\left(\mathrm{~d} x^{2}\right)^{2}.$$

I'm looking to find the metric tensor and its dual metric. I understand the basics of summing over $g_{ij}$ for $n=2$ and using the Kronecker delta but the second non $dx^1$ term is tripping me up. I don't know how to use it in summing as its not a derivative.


Your question is a bit unclear but as I understand it you want to derive the metric tensor of this line element and its inverse. You have to remember that $dl^2 = g_{i j}dx^i dx^j$. So your metric is $\text{diag}(1,(x^1)^2)$. And so its inverse is $\text{diag}(1,(x^1)^{-2})$.

Edit for clarity :

\begin{equation} dl^2=g_{i j}dx^i dx^j=g_{11}(dx^1)^2+2g_{12}dx^1 dx^2+g_{22}(dx^2)^2 \end{equation}

So one has $g_{11}=1$, $g_{12}=0$ and $g_{22}=(x^1)^2$. Since the metric is diagonal finding its inverse is straightforward : $g^{11}=1$, $g^{12}=g^{21}=0$ and $g^{22}=(x^1)^{-2}$


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