What, physically, are you calculating when you're solving a contour integral?

I was going to post a snarky reply but I stopped when I couldn't actually get an answer. The way I understand it is that contour integration is just one of (infinitely) many ways to compute the closed integral between two points where regular integration only allows one.

>>7715733
I see. How would you visualize it?

fun fact: In german there is no difference between line-, curve-, or contourintegral

basically you're always integrating along a contour in space (real or complex)

File: cint.png (14KB, 748x572px) Image search: [Google] [Yandex] [Bing]

14KB, 748x572px

I always imagined it like the area under the curve (like a normal integral), only that the base is itself a curve in 2D space, instead of the x axis. I might be wrong, though...

https://en.wikipedia.org/wiki/Line_integral#/media/File:Line_integral_of_scalar_field.gif

this explains it quite well

>>7716201
Love that gif

>>7716201
>https://en.wikipedia.org/wiki/Line_integral#/media/File:Line_integral_of_scalar_field.gif
>>7717270
>>7715812
how do you know what sign the integral has? If you invert the boundaries, the integral should have the opposite sign, but how do you decide which one.

>>7716159
No, that is the correct way to think about, except that in general the curve isnt bound to being a subset of the plane. This >>7716201 gif explains it well.

>>7716201
how would one make such a gif? animate with tikz or something like that?

>>7715714
Scalar conour integral can be seen as area under the contour. Vector contour integral is usually work done by a force field when you move a test particle around the contour.

>>7716159
>>7716201
That's a little bit too basic for my taste. Is there a more physical interpretation, e.g. using the residue theorem and equating poles with charged particles? if you know what I mean ...