The given vector field is $X(x,y,z) = (y,-x,1)$ on $\mathbb{R}^3$. Calculate $\int_{x_1,C}^{x_2} (X|dx)$ for the following curves.
a. $\mu_1(t) = (\cos(t), -\sin(t), \frac{t}{2 \pi}), t \in [ 0, 2\pi]$
b. $\mu_2(t) = (\cos(t), \sin(t), \frac {t}{2\pi}), t \in [ 0, 2\pi]$
c. $\mu_3(t) = (\cos(2\pi-t), \sin(2\pi-t), 1-\frac{t}{2\pi}),t \in [ 0, 2\pi]$
In all the case, we start the points of integration with $\mu(0)$ and end with $\mu(2\pi)$
And draw a qualitative picture of X and the three curves and explain the result of computation.
I got for a. $1-2\pi$ and for b. $2\pi -1 $ and for c.$1+2\pi$