Every Finite Change Has a Corner

Three additions around the original area

Use x for the first length and x squared for the second, starting at two. Grow an input step of three tenths. The reference rectangle stays two by four while two strips and a corner appear around it.

The first strip changes only the first dimension. Its width is delta f and its height is the old g. That makes its area g times delta f. Here four times three tenths gives one point two.

The second strip changes only the second dimension. Its width is the old f and its height is delta g. Here two times one point two nine gives two point five eight. Neither strip uses both new dimensions.

The corner uses both changes, delta f and delta g. Its area is their product, zero point three eight seven. Only after adding this third contribution have we accounted for the complete finite increase.

An exact identity before any limit

View all three contributions together. Expand f plus delta f times g plus delta g. The four terms are the old product, the two strips, and the corner. Subtracting the old product leaves exactly those three added regions.

Focus on the first strip again. Its height is the old g. If we used the new height, g plus delta g, this strip would already include the corner. Adding that corner again would count it twice.

Now focus on the corner itself. It touches both changes and belongs in the calculation once. A partition is useful because it states exactly which region each term counts.

The exact identity does not require a tiny input step. Increase the step to four tenths. The strips and corner all change size, but their three-term sum still equals the difference between the new and old complete areas.

Read a second complete calculation

At a four-tenths input step, the new first length is two point four. The new squared length is five point seven six, so its change is one point seven six. The old area is still eight.

The first strip contributes four times four tenths, or one point six. Read its height from the original rectangle and its width from the changed part of the first side.

The second strip contributes two times one point seven six, or three point five two. The larger change in the squared length makes this the larger of the two strips.

The corner contributes four tenths times one point seven six, or zero point seven zero four. The total increase is five point eight two four. Adding eight gives thirteen point eight two four, which is two point four cubed.

Equal strips still need their corner

Switch to two copies of x, both starting at two. Grow a step of three tenths. The two side changes are now equal. This symmetry gives two equal strips, but it does not remove their corner.

Each strip has area two times three tenths, or six tenths. Together they contribute one point two. The complete square grows from four to five point two nine, so the strips alone are short by nine hundredths.

The missing nine hundredths is this three-tenths by three-tenths corner. The exact identity for a square is two x times delta x, plus delta x squared. The extra squared term describes an actual region.

Reduce the step to one tenth. Each strip becomes two tenths, and the corner becomes one hundredth. The area increase is forty-one hundredths, not forty. The corner shrinks faster, but it has not disappeared at a finite step.

Halving a step changes strips and corners differently

Use two linear factors, both beginning at two, with a step of four tenths. Each strip has area eight tenths and the corner sixteen hundredths. The complete increase is one point seven six. These values will give us a clear comparison when the input step is halved.

Halve the step to two tenths. Each strip now has area four tenths, half its old value. The corner becomes four hundredths, one quarter of its old value. Both of the corner’s dimensions halved, whereas only one dimension of each strip changed. That dimensional difference explains the different scaling.

Halve the step once more, to one tenth. Each strip becomes two tenths, and the corner one hundredth. The full increase is forty-one hundredths. The ratio of corner area to total added area has decreased, but the corner remains a nonzero part of this finite square.

Hold that one-tenth step and inspect the corner. Omitting it gives forty hundredths instead of the correct forty-one. A relatively small error is still an error in an exact finite calculation. We may later explain an approximation or a limit, but neither permits silently relabeling the two-strip sum as the exact changed area.

Divide the corner by the same input step

Return to a four-tenths step for the two linear factors. The corner area is sixteen hundredths. Dividing by four tenths gives a corner contribution of four tenths to the average rate. We are now comparing area change per input change, so the denominator matters as much as the shrinking numerator.

At a two-tenths step, the corner is four hundredths. Divide by two tenths and obtain two tenths. Halving the step quartered the corner area but only halved its rate contribution. The numerator and denominator changed together; looking at area alone would not give this rate comparison.

At a one-tenth step, the corner is one hundredth and its rate contribution one tenth. For these equal linear factors, the corner area is the step squared. Dividing by the nonzero step leaves the step itself, which approaches zero as the step shrinks. That is an explicit algebraic reason for the limiting disappearance.

Show all three contributions. Each strip divided by the step contributes the unchanged side length two. Their sum is four, and the extra corner rate is the current step one tenth. The complete finite quotient is therefore four point one. Its limit is four, although the finite value still includes the corner.

A two-part exact calculation must absorb the corner

Return to the unequal factors x and x squared at input two with step three tenths. The first strip has area one point two, the second two point five eight and the corner zero point three eight seven. Together they make four point one six seven. We can regroup these regions without changing that sum.

Focus on the second strip. Its displayed width is the old first side, two. If we combine this strip with the adjacent corner, their combined width becomes two point three. Multiplying that new width by the second-side change one point two nine gives two point nine six seven. The highlighted old strip is only part of that combined quantity.

Return to all three regions and add the first strip one point two to that combined two point nine six seven. The result is the exact four point one six seven. A two-term finite expression can be exact when one term uses a new side length and thereby includes the corner. The term definitions have changed.

We could instead attach the corner to the first strip. Three tenths times the new second length five point two nine gives one point five eight seven. Add the unchanged second-strip area two point five eight and recover four point one six seven again. Either regrouping works, provided the corner belongs to exactly one combined region. Giving both strips their new opposite dimensions would count it twice.

Compare that regrouping with adding only the two old-side strips. Those total three point seven eight, short by the corner zero point three eight seven. The number of written terms does not tell us whether an identity is exact. We must identify the dimensions and regions counted by each term, including where the corner has gone.

A smaller corner and a limiting contribution

Return to x times x squared with a three-tenths step. The corner area is zero point three eight seven. Divide by the input step to compare contributions to an average rate. The corner rate is one point two nine.

Reduce the step to one tenth. The corner area becomes zero point zero four one. Dividing by one tenth gives zero point four one. Its rate contribution has also decreased, which is the relevant comparison for a derivative.

Shrink the step further, to the native minimum. The corner remains part of the finite geometry. In the limit, differentiability makes one change divided by delta x approach a finite rate while the other change approaches zero. Their product approaches zero.

So the finite identity has three terms, while the derivative has two. The corner was never an error to erase from a finite calculation. Its limiting rate contribution vanishes for a specific reason, leaving the two original-side strip rates.