For a while now I've been trying to find motivation and a good intuition behind the line integral for a vector field. This is the first time I'm learning this topic and I'm not interested in too much formal rigor but rather a strong geometric/mathematical intuition behind the concept.
- I've heard of the intuition for a line integral over a scalar field as being the "area of the fence between the curve and the surface/function." and I'm happy with this interpretation. Is there a similar idea for vector fields?
- I've seen the wikipedia GIF's for both scalar fields and vector fields.
- I've seen almost all the other related pages on MSE with a similar question but without finding a satisfying answer.
- I'm trying to find an explanation which doesn't rely on work from physics
While this vector field GIF was quite useful, I was still unable to understand the motivation behind the integral. I might not have understood the GIF completely, but from what I could tell it looked like the connection between the integral and an area being calculated. But wouldn't this be true of any integral? What connection does this integral and area have with the original setting of a field an a curve?
But what is the geometric significance of this? Why are we specifically taking the dot product of these two objects ($\mathbf F(\mathbf r)$ and $d\mathbf r$)? What is the motivation behind finding this integral? What is the significance/meaning of the output to this integral?
- I understand that in a physical setting one might be trying to find the work done by the force field. However just in a mathematical context what would be the use of finding this information provided by the integral?