Boundary orientation for a cylinder 
Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :(
 A: Unless your book/class is using some strange conventions, the orientations of the boundary components should match the little almost-circle drawn in Figure 21.8.  This means that on $C_0$ the orientation goes from left to right across the front of the picture and (therefore) from right to left across the back. That's because the little circle in the picture is oriented from left to right where it goes near $C_0$ on the front of the cylinder.  So $C_0$ would be oriented counterclockwise as seen from above.  $C_1$ would have the opposite orientation, clockwise as seen from the top.  (In this answer, I've assumed that you're working with reasonable definitions of boundary orientations and that the little circle in hte picture indicates the 2-dimensional orientation in a reasonable way.)
A: Take a point $p$ on $C_0$ and two tangent vectors $v_1$ and $v_2$ as shown in the picture below. The orientation given to the cylinder by the order pair $[v_1,v_2]$ at $p$ agrees with the one coming from the one on the cylinder(described by the loop). The orientation $[v_1,v_2]$ restrict to the orientation $[ v_2]$ on $C_0$ which is a contour-clock-wise on $C_0$. Doing the same for $C_1$ you will get a clock-wise orientation on $C^1$.
