There is One True Integral (in Vector Calculus)

When I took vector calculus, one of my biggest issues was keeping straight the various different integrals we have to do. The names might differ depending on your reference, but by my count there are five core integrals that come up:

• Path integrals: ∫ f(c(t)) ║c’(t)║ dt
• Line integrals: ∫ F(c(t)) ⋅ c’(t) dt
• Surface integrals: ∬ f(Φ(u,v)) ║∂Φ/∂u × ∂Φ/∂v║ du dv
• Flux integrals: ∬ F(Φ(u,v)) ⋅ (∂Φ/∂u × ∂Φ/∂v) du dv
• Curl integrals: ∬ (∇ × F(Φ)) ⋅ (∂Φ/∂u × ∂Φ/∂v) du dv

We’ve come a long way from merely looking at the area under curves! Thankfully, these integrals are largely the same thing, and they all find areas in their own right. After all, they are integrals, and at the end of the (possibly very long) day we are still doing the same thing we were doing in Calc 1.

The question we need to ask ourselves is what does the height of the curve represent? We are actually rarely interested in an actual square-mile area — Archimedes solved the problem of determining the volume of arbitrary bodies a while ago. Instead, remember how the derivative of position is speed, and thus the integral of speed over time tells you how far you went. For vanilla Path and Surface integrals, the answer is simple: the height of the curve determines the per-unit area (or volume). That ‘per-unit’ statement is important, though! In our usual computations, where we integrate over intervals of the real line, every unit is worth the same, but that doesn’t have to be the case. For example, if you tallied up (integrated) the 2021 Georgia special senatorial vote percent by county, where f(x) has units “Warnock vote percentage”, would we show that Sen. Warnock outpaced his opponent?

County-level results for the 2021 Georgia Special Senate Election (Source: Politicsfan4 — Own work)

Surely not! However, this is not how we compute the popular vote margin; it is not:

The total average county result in the Georgia special election — this is not what we want!

This has the wrong units — instead, we have to weight by the population of the counties!

The weighted average of county results in the Georgia special election — this is the raw vote total!

This is exactly why our parameterizations have the slightly more complicated form they have. We simply do not know, and thus have to correct for, what shenanigans our parameterization might be causing. You hopefully see the resemblance between the ║c’(t)║ in the Path integral formula and the ║dΦ/du × dΦ/dv║ in the Surface integral — these are the same “correction” factor. In our county example, these derivatives have the same units of “total votes per county”. Just as we integrate “miles per hour” across time to get a distance of miles, here we need a term with units “total votes per county” to turn the units we were given — “Warnock vote percentage” — into the units we want — “Warnock votes per county”.

Now, for the Line integral and the Flux integral, we want to ask ourselves again, what does the height mean? The problem here, though, is that our function F is vector-valued — there isn’t a “height” to speak of! The first thing we need to do is convert these vectors into scalars, and to do that we have to ask ourselves “what are we actually trying to measure here?” With Line and Flux integrals, the answer is: we want to know “how much” of the vector field is “blowing through” the region. One important note, however, is that I’m being purposefully ambiguous: for a Line integral, I want to know “blowing along”, i.e. parallel to the curve, while for a Flux integral, I want to know “blowing into”, i.e. orthogonal to the surface. For simplicity I’m go to call both of these “blowing through”.

This means that what we really want is the component of F either parallel to the curve c (for Line integrals) or parallel to the normal vector du× dv (for Flux integrals). To do this, we compute projections:

Now in order to compute these integrals, we simply integrate these the way we want to, using the Path and Surface integral formulas. The important takeaway is that the Line and Flux integrals are just Path and Surface integrals of how much the vector field is blowing through the surface. The denominators in the projection formulas cancel out with the correction term of the Path and Surface integrals, giving the expressions we saw at the beginning.

The final type of integral on the list is the Curl integral. This is the exact same story: the scalar quantity we want to measure from our vector field is “how much does my vector field spin in the surface?” Since the curl measures the axis of rotation at a given point, we can reformulate this as “how much does my curl vector align with that normal vector?” This question is computed by projecting the curl onto the normal vector:

Again, computing the Surface integral of this quantity yields the answer we want, with the same cancellation of the correction term.

So, even though we’ve got five differently named integrals floating around, I think it is easier if you remember these two facts:

• The Path and Surface integrals are the same concept, the One True Integral, just over different-dimensional domains.
• The Line, Flux, and Curl integrals are just Integrals of certain scalar functions of the vector field.