Inspired by Wesley Clark’szombie calculus thread which was recently resurrected, I got to thinking about the arc length formula of which this is a typical explanation.
Being integrally challenged, as it were, I usually can’t actually apply this formula, but I do understand its derivation, which I find interesting in that it involves both integration and differentiation in a way that many proofs don’t.
Now for my question: I get how the derivative of a function is correlates to the rate of change, or to the slope of the tangent on a graph. Similarly with a definite integral and the area under the curve, or the total work performed against an opposing force. But what does arc length tell us, apart from the length of the graph itself? While conceding that arc length is interesting in and of itself, I wonder what wider physical applications it has.
There are a lot of applications where you want to know the total length across a function. For example, if you are calculating air flow over a wing and you want to know the total length that the air will travel as it goes across the wing’s surface.
I think a common classroom example is figuring out how many miles will be added to your odometer going over a mountain (with the mountain represented as a function). It’s obviously going to be a lot longer than just the distance from point A to point B on either side of the mountain.
Integrals have many applications that aren’t actually about geometric curves and areas. Are there any applications of arc length that aren’t geometric?
Exactly what I was trying to say, but better stated. Obviously there’s practical value in knowing the distance that has to be covered by air flowing over a wing, or the length of a trajectory as opposed to just the maximum altitude or terrestrial distance, but those examples do seem essentially geometric. Mapping work as a physical concept to the area under a curve makes sense intuitively, but I’m not seeing “work” as something inherently related to a geometric area.
I’m not a professional mathematician/physicist, but I think the answer is that only graphs with the same units on both axis, such as the geometric solution for distance, have arc lengths that are meaningful. My reasoning goes as follows:
Look at a simple graph such as distance vs time. The rate of increase for the graph equals speed, because it has units distance over time. The area under the graph doesn’t have meaning because it has units of distance multiplied by time. The arc length doesn’t have meaning because it can’t be solved when the units aren’t the same on both axis. With constant speed and a linear graph the length is sqrt(t^2 + (vt)^2).
Changing to a speed vs time graph the area under the graph becomes meaningful because speed times time is distance, but the arc length remains an expression with non-compatible units in each term.
This probably isn’t a bad way of looking at things for the most part. I had a chat a few years back at a math conference with a biologist who was imploring mathematicians to use units with everything. Following the units can make concepts easier to understand.
I disagree with your statement on the area under a time vs distance graph however. It has meaning, just not a conventional or easily interpreted one. There’s no reason why we can’t measure meter-seconds, it’s just that there’s not a common use for them, so the unit isn’t used AFAIK. The arc-length example fails because you’re trying to add quantities of incompatible dimension. The fundamental difference between the two examples is that for the area example, the measured quantity isn’t useful, whereas in the arc-length example, the measured quantity doesn’t even make scientific sense.
I agree that it’s a different argument than the arc length. But is there really a meaningful interpretation of the area under the time vs distance graph?
Time vs distance (or position) is a fundamental physical description. The time derivative is time vs velocity. The time derivative of that is acceleration. Next we have jerk.
Going the other way you get to “something” with units meter-seconds, but as you say theres no common use for this, and I question whether it’s even possible to give a meaningful interpretation. It’s something that increases even if there is no velocity. If we call it x, and s(t) = 4, then x(t) = 4x + c, while if s(t)=0, x(t) = c.
