I would like to understand the idea of a 'curve of intersection' in $\mathbb{R}^{3}$. Say we are given a surface $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and a plane $y = x$. Then the curve of intersection is obtained by substituting $y=x$ into $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$, thus we get $z = x^{\frac{2}{3}}$. As I understand, this curve is in $\mathbb{R}^{3}$, so would the curve of intersection be given as the following set $$\bigg\{ (x,y,z): z = x^{\frac{1}{3}}y^{\frac{1}{3}},~ y = x ~\text{ and } ~z = x^{\frac{2}{3}} \bigg\} $$ or can we simply note that it is the set of points which satisfy $z = x^{\frac{2}{3}}$ and hence immediately consider it as a projection on the z-x plane, which is the curve $z = x^{\frac{2}{3}}$ in the z-x plane?
Are both of these interpretations valid?
Thanks.
Neither interpretation is correct. The curve of intersection is: $\mathcal{C} = \left\{(x,y,z): z = (xy)^{1/3}, x = y\right\} = \left\{(t,t,t^{2/3}): t \in \mathbb{R}\right\}$