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Let the $\vec{F}=\langle x,y,z\rangle$ be a force vector field in $\mathbb{R}^3$, the curve $C$ is the counterclockwise oriented ellipse given by parametric equation $\vec{r}(t)=\langle \cos t, \sin t,\cos t\rangle , t\in[0,2\pi]$, find the work required to move an object on $C$.

A

$1$

B

$0$

C

$\pi$

D

$-1$

E

$-\pi$

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