I have quoted the question. This assignment is past due and I have questions about the solution:
Computer the path integral of $\int_C{f} \, ds $ where $f(x,y,z)= x^2$ and the path C is the intersection of the sphere $x^2+y^2+z^2=1$ and the plane $x+y+z = 0$.
So the way I see it is that the intersection of the sphere and curve gives us a circle on the xy plane with radius one. So i thought the parametrization is as simple as $x = cos(t)$ and $y=sin(t)$ but it's not. Here is the correct solution.
How is it that they are parametrizing using vectors. What is the reasoning/logic behind it?