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Concept

Fundamental theorem

The result naming the central fact of a subject: for calculus, that integration and differentiation undo each other. The calculus one is what makes areas computable by antidifferentiation, and it is why the two halves of the subject are one subject.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

analysis · The integral Area is the undoing of slope. Above, a positive function with the area from 0 to 1.80 shaded. Below, that area plotted against where it stops. The lower curve's slope at 1.80 is 1.129, which is exactly the upper curve's height there.

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

analysis · The integral The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

algebra · Polynomial roots The area that names the number. The curve 1/x with the area under it from 1 to 2.7183 shaded, measuring 1.0000.

The area that names the number

The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.

analysis · The exponential Integration by parts is a rectangle. The increasing curve v = u²/4 between u = 1 and u = 3. The region under it is shaded one way and the region between it and the vertical axis another; together they fill the rectangle from the origin to (3, 2.25) minus the rectangle to (1, 0.25).

A rectangle cut by a curve

Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.

analysis · The integral

Named alongside it

The objects these essays reach for when they reach for this one.

AreaAntiderivativeContinuityDerivativeIntegralLimitLogarithmRiemann sumAccumulationClosed curveComplex numbersConvergence

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