Consider a surface with the equation $z=x^3 − 3xy^2$.

So I have parametrization $p(u,v) = \begin{bmatrix} u \\ v \\ u^3-3uv^2\end{bmatrix}~.$

I have found that the first fundamental form of this surface is $$G=\begin{bmatrix}1+9(u^2-v^2) & -18uv(u^2-v^2)\\-18uv(u^2-v^2) & 1+36u^2v^2 \end{bmatrix}~.$$

Now I need to calculate length of a curve $u-v=0$ for $u,v \in (0,1)$. I know length is equal to

$$\int_0^1 \|c'(t)\| dt=\int_0^1 \sqrt{(u'\,\, u')\begin{bmatrix}1 & 0\\0 & 1+36u^4 \end{bmatrix} (u'\,\,u')^T} du~,$$

however I am not able to find a suitable parametrization of the curve so that this integral is computable (by hand, non numerically).

Any recommendations? Thanks in advance!


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