My goal is to understand how work that is calculated be either a 2 or 3 dimensional line integral of a vector field can be visually represented as a area , similarly to single variable calculus as work being the area under a curve.
My problem is that I need 3 dimensions for a 2 dimensional line integral to see this otherwise the path can loop back on itself.
As for example if the path is a helix in a 3 dimensional vector field then clearly it goes in a circle so loops around a three dimensional space.
Now imagine a path with a 2 dimensional vector field. Can this path loop on itself and how can ANY path end up being the area under a curve ?