S1 is the circular cylinder of radius 2 with the y-axis as its central axis.
S2 is the surface described by $y = x^2 – z^2$.
Curve C is the intersection of these two surfaces.
I want to calculate the circulation of some vector field around C. But to be able to do that I have to parameterize C first (e.g. $r(r)$). When I looked at one of the possible parameterization, I did not understand how to come up with it, could anyone please help? Thanks.
This is the parameterization: $r(t) = 〈cos(t), cos(2t), sin(t)〉, t ∈[0, 2𝜋] $
Best Answer
Let $t$ be angle around the $y$-axis. For any given value of $t$, you can construct a line $L_t$ lying on the cylinder, passing through the point $(2\cos t, 0, 2 \sin t)$. A parametric equation for this line is: $$ L_t(u) = (2\cos t, 0, 2 \sin t) + u(0,1,0) = (2\cos t, u, 2 \sin t) $$ At the point where this line intersects the surface $y=x^2-z^2$, we have $$ u = 4\cos^2 t - 4 \sin^2 t = 4 \cos 2t $$ and so the corresponding point is $(2\cos t, 4 \cos 2t, 2 \sin t)$.
If the cylinder actually has diameter = 2, rather than radius = 2, then we get $(\cos t, \cos 2t, \sin t)$ by a similar method.