We have been given the following phase portrait, and asked to guess whether the following lines will result in a positive, zero, or negative line integral when evaluated on the vector field:
- from (-1,-1) to (1,-1),
- from (-1,0) to (1,0).
If the line integral represents the work done to drag a particle through the fluid over the given lines, it feels like both should be negative since they go against how the vector field suggests particles should move. However, I was marked correct for choosing (1) as negative, but wrong for (2) as negative (the correct answer wasn't given to me).
Clearly I am thinking about this relationship wrong, so how should these questions be considered?