How to visualize a line integral

[Isaiah Gray]

Hi,

I am trying to get a useful heuristic picture of a line integral, like the area
under a curve for an ordinary integral. My current one is: if I place a particle
in a force field, then the line integral from point A to B is the change in kinetic energy
of the particle from A to B. This works for simple cases, but not
for something like the circulation of the magnetic field, which does no work.
Are there any more accurate ways to picture a line integral of a vector field?

 

[Edgardo]

You can visualize a line integral as a curtain. See for example this video by Salman Khan:
Introduction to the Line Integral

 

[HallsofIvy]

Perhaps for idio-syncratic reasons, I prefer to visualize line integrals as "force integrated along the path" giving "work done".

 

[chiro]

Hey Isaiah Gray and welcome to the forums.

When we have integrals involving dot or inner products, one way to visualize this is to think about them in terms of projections.

You are summing up infinitesimal projections with these kinds of integrals in exactly the same way that you are summing up changes in a function with an ordinary function based integral.

Now the projection in a physical case might be for example a force, but in another case it might mean something different, but if you understand what a projection means in the context of your problem, it will make a lot more sense.