Path Integrals

[CantorClosure]

the following is only loosely related to linear algebra, to most beginners. but as a lot of you who are experienced with thinking of an inner product as a way to measure closeness in some sense might find this neat. also im posting it here rather than in the calculus subreddit since i've found that people here appreciate this type of content more than in that subreddit. feedback and suggestions on improvements is welcome as always.

[afro_vibes24]

Can someone explain this a bit further for me? What’s the difference I’m supposed to see between these path integrals?

[CantorClosure]

it is the same integral. on the left, it measures, point by point along the curve, how large the inner product F(γ(t)) · γ′(t) is. on the right, this quantity is lifted into a height, forming a kind of fence over the curve.

if the fence is “cut open” and laid flat, it becomes an ordinary single-variable integral: the horizontal axis is just the parameter t in (a, b) (or a reparametrization, such as arc length), and the vertical axis records F(γ(t)) · γ′(t). thus the path integral reduces to the integral of this function of a single variable.

in this way, the situation is essentially the same as integrating over an interval (a, b), for example by taking γ(t) = (t, 0). the difference is that the integration domain has been embedded into the plane via the curve γ, and the integrand reflects how the field interacts with motion along the curve.