Line integrals and reparametrization

Question

$C = (1,2,0)$, $B= (1,0,2)$.

part B: I have that my parametric equations are $x = 1$, $y = 2\cos(t)$, and $z = 2\sin (t)$.

part C: I don't know how to approach this. Do I start with the divergence of $G$ and then reparametrize with $\tau$?

Answer 1

Part B: Yes, this parametrization is correct. But also don't forget to state the range of the parameter $t$, i.e. $\ldots\le t\le\ldots$, so that the trajectory actually starts and stops where it's supposed to.

Part C: Yes, as you said simply find the divergence of $G$, and then integrate along the path $\tau$. This means that you will substitute the parametric expressions for $x$, $y$, and $z$ into the expression for $\operatorname{div}G$. But also make sure that you set up the line integral with respect to the arclength $ds$ (as it's given), where $ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}\,dt$.