I'm working my way through "A Geometric Approach to Differential Forms" by Bachman and have a question about deciding the orientation of an integral of a differential form - basically whether you use a negative sign in where omega is the differential form. Bachman says, for the 1D case, "Just as there were for surfaces, for parametrized curves there is also a pictorial way to specifiy an orientation. All we have to do is place an arrowhead somewhere along the curve, and ask whether or not the derivative of the parameterization gives a tangent vector that points in the same direction."
C is the portion of the graph x=y^2 where 0 LE x LE 1. His arrowhead is drawn to point towards the origin. Integrate the 1-form omega = dx + dy over C with the indicated orientation.
The parametrization is phi(t) = (t^2,t) where 0 LE t LE 1. The derivative is d(phi)/dt = <2t,1>. At the point(0,0), the derivative is <0,1> and this points in the opposite direction to that of the arrowhead. Therefore, we have to use the negative sign in:
integral_C (omega) = - integral omega(<2t,1>) limits of integration are from 0 -> 1.
= -(2t+1) ^1 _0
My question is: If you are drawing the arrowhead pointing *towards* the origin, shouldn't the integral be from 1 -> 0 instead??? Because otherwise you just seem to be drawing an arrow and then comparing it to something you calculated and yet the direction of your arrow never enters into the calculation in some fashion????