So I am currently looking at how to calculate the integral of a complex function $f(z)$ within a contour $\gamma$. That is, an integral of the form $$ \int_{\gamma} f(z) \; dz $$ Where the contour is given as a function in terms of $t$.
Say that we have a contour given by the equation $$ \gamma (t) = r \; e^{2 \pi i t} \quad 0 \leq t \leq 1 $$
Then this contour is a circle or radius 2, once anti-clockwise about the origin.
Why is this the case? How would a function of this form be plotted, and how do we determine the direction of it?
Since $\gamma(t)=re^{2\pi i t}=r\cos(2\pi t)+ir\sin(2\pi t)$, we can plot the curve just as in calculus, in the plane with $(r\cos(2\pi t),r\sin(2\pi t))$. If you plot this, you'll see its a circle centered at the origin of radius $r$.
To figure the direction, you could take the derivative, $\gamma'(t)=2\pi i re^{2\pi i t}$. At $t=0$, the corresponding point on the circle is $(r,0)$. Moreover, $\gamma'(0)=2\pi i r$, which is vertical in the complex plane. Therefore, the direction of the curve is pointing upwards; the direction is counterclockwise around the circle.