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I was skimming through one of my favorite textbooks, which goes through elementary and vector calculus entirely in terms of differential forms, and remembered a passage from it that’s been niggling at me ever since I first read it.

First, it gives a fairly standard statement of the fundamental theorem of calculus:

Let F(t) be a function for which the derivative F’(t) exists and is a continuous function for t in the interval {a ≤ t ≤ b}. Then

Nothing new here. Then in the exercises there are some problems designed to explore the physical interpretations of the FTC. The last one, however, is a little strange:

Let ƒ(t) be a given function and let A_[a, b] be the area under the curve, that is, the area between the graph (t, ƒ(t)) of the function and the interval [a, b] of the t-axis, counting area as negative if the curve lies below the t-axis. Fix t₀ and consider A_[t0, b] = F(b) as a function of b. Give a geometrical interpretation of the 1-form F’(t) dt, and hence of the function F’(t). Describe each side of [the equation in the FTC] as an expression for A_[a, b]. (This is an extremely awkward interpretation of the Fundamental Theorem. See Exercise 12, §3.2.)

That last comment is a little odd, considering that most textbooks use a geometric “area under the curve” argument to prove the FTC. That explains why this particular textbook took a different approach, instead decomposing F(b) − F(a) into [F(b) − F(t_n - 1)] + … + [F(t₁) − F(a)] and arguing that it approaches ∫_a^b F’(t) dt as the length of each subinterval Δt goes to 0.

Exercise 12, §3.2. goes into more detail into this point of view:

Let S be a curve in the plane parametrized by the coordinate x, S = {(x, ƒ(x)): a ≤ x ≤ b} where ƒ is a differentiable function. Use d(y dx) = dy dx = −dx dy to interpret ∫_S y dx as an oriented area. Express this integral in terms of x.
(This is the interpretation of the integral as ‘the area under a curve’. The widespread idea that an integral ‘is’ the area under a curve is very unfortunate because it completely obscures the meaning of the Fundamental Theorem. An integral ‘is’ the limit of a sum, and area ‘is’ a double integral.)

My initial reaction was to dismiss this point of view, as thinking of the integral as the area under the curve—that is, the limit of approximating the area with a set of thin rectangular slabs, dx and ƒ(x) being the width and height of an “infinitely thin” rectangular slab and ∫ being the operator that sums up all “infinity” of these slabs—is so useful as a practical and visualization tool, and thinking of it instead as, say, a limit that could be interpreted the work done by a one-dimensional force doesn’t yield any new insights. But then, mulling it over for a while, I realized that it didn’t yield any new insights for me because I was already familiar with thinking of the area integral as simply ∫_R dA instead of the usual form and of the analogous higher-dimensional integrals. Furthermore, I started getting used to the idea of the integral over an interval being an operation only on the interval, or perhaps on the curve (x, ƒ(x)), and the connection to the area under the curve being a happy coincidence of the integral vanishing on the other three sides of the border.

A similar situation can be found with the trigonometric functions. Usually they’re taught first in terms of the ratios of the sides of right triangles; this is a useful and practical way of thinking about them, but is ultimately a limited viewpoint. Inevitably one who works with them for long realizes that they have more to do with periodic behavior and that the relationship to triangles is just the trivial fact that a point on the unit circle (itself only slightly more illuminating than the right triangle formulation) forms a right triangle with the origin and its projection on the x-axis.

In fact, a fairly recent book attempts to reformulate trigonometry—and ultimately geometry—without reference to the usual trigonometric functions, instead claiming that they are best learned in the context of periodic motion. While I do not agree with the author’s assertion that his “rational trigonometry” should be taught in place of “classical trigonometry” in secondary school, I can see more advanced students supplementing their study of geometry with his material.