I have these 2 surfaces, given in their Cartesian form, and want to find the resulting surface of intersection.
My approach is to find parametric representations. But I have two approaches, and not sure if both approaches are proper, or if one or the other is only proper.
Here are the equations:
$$\tag{1}z = 1 - x^2$$
$$\tag{2}x = y^3$$
Approach 1:
Substitute (2) into (1), we get the cartesian form of the intersection surface: $z = 1 - x^6$, then try to parameterize this, but I run into $2$ independent parameters with this approach.
Approach 2: Direct parameterization:
Just looking at both equations I see that '$x$' is the common variable so I set $x= t$. Then just plugging in '$t$' into the equations, and put them together in $(x,y,z)$, I get: $$(x,y,z) = (t, t^{1/3}, 1-t^2)$$
So not sure if either of these 2 approaches above give the correct parameterization.
Not sure if there is a systematic procedure to do parameterization of surfaces of intersection in $\mathbb R^3$, or if this is a creative type of process, and hence there is no systematic procedure to follow.
Hope someone can explain how to think about this kind of problem.
Two quick comments. First, you do not want to introduce $t^{1/3}$, as then the parametrization will not be differentiable — let alone smooth — at the origin.
Second, when you eliminate a variable as you did in approach 1, you are not done; rather, you've obtained the projection of the intersection into the $xz$-plane in this case. You still need to give the $y$ value for each point on that curve in the $xz$-plane. To convince yourself, consider the system of linear equations $x+y=0$, $2x-y=0$. If you eliminate $y$, you get $3x=0$, so $x=0$. Clearly the solution set is not the entire $y$-axis.
Oh, and glad you enjoy the book. There are also lectures on YouTube, linked in my profile.