Everything in this volume so far has happened on a line: y as a function of x, one derivative at a time. In three dimensions the chain rule's companion is a theorem about flux: how much of something crosses a closed surface. Its best example is the one the reader already asked about, the expanding universe. That example also answers a question the flat calculus cannot even pose: how fast is the visible universe growing?
Loomis and Sternberg state it in its general form as Theorem 6.1 of §10.6 (p. 421), “The divergence theorem”. Take a region D with a boundary, and a vector field X, such as the velocity of a fluid. Integrating the field's divergence, its local rate of spreading out, over the inside gives the same number as integrating its outward component over the boundary. In the Euclidean form they give on p. 423:
The left side counts sources inside; the right side counts what crosses the surface. The flat calculus has one number at each point. This has a volume and its skin, and a law connecting them CITED.
On large scales galaxies recede from us, and from each other, with velocity proportional to distance: v = H r. That field has the same divergence everywhere, div v = 3H, one H for each dimension. So the divergence theorem says that the volume flowing out through any sphere of radius R equals 3H times the volume inside it:
The script checks it by brute force. It sums v·n over 320,000 patches of a sphere and compares the total with 3H × volume; they agree to one part in 10,000 COMPUTED. Any region that expands with the galaxies therefore grows at the fixed fractional rate 3H. That is why volume goes as the cube of the scale factor and matter density falls as its inverse cube.
With the Planck collaboration's H0 = 67.36 km s−1 Mpc−1 (Table 1, p. 15):
| quantity | value |
|---|---|
| Hubble time, 1/H0 | 14.52 billion years |
| Hubble radius, c/H0: where recession reaches the speed of light | 14.52 billion light-years |
| fractional growth of any comoving volume, 3H0 | 2.07 × 10−10 per year |
| volume doubling time at today's rate, ln 2 / 3H0 | 3.35 billion years |
| distance doubling time at today's rate, ln 2 / H0 | 10.06 billion years |
At today's rate, every volume of the universe doubles in about three and a third billion years COMPUTED. Distances take three times as long, because volume grows in three directions at once. That factor of three is the divergence. The rate is falling, so the true doubling will take longer. MODEL
The sphere we can see is not a region that expands with the galaxies. Its edge, the particle horizon, is the farthest distance light has had time to cover since the beginning. The script integrates the standard flat model with Planck's parameters and finds that edge today at a radius of 46.2 billion light-years (14.15 Gpc). That is larger than 13.8 billion because the space the light crossed has stretched since. The bright wall of last scattering, whose light we now see as the microwave background, lies at 45.2 billion light-years, within 2% of the horizon COMPUTED.
A surface that moves needs one extra term in the flux law (in fluid mechanics, the Reynolds transport theorem). The horizon moves outward at the expansion speed plus the speed of light, because light keeps arriving:
The first term is the galaxies' flux. The second is new light arriving through the skin. Their ratio is c/(H0D) = 0.315, so the visible universe grows about 30% faster than the expansion alone would make it: about 1.1 × 1023 cubic light-years a year COMPUTED. Here the flat calculus has nothing to offer. The question needs a volume, a moving boundary and the theorem that ties them together.
The divergence theorem does not replace the chain rule; it rests on it. Loomis and Sternberg prove it in charts, and every change of chart is a composite map whose Jacobian enters as a determinant. That is Chapter 2's d(G∘F) = dG∘dF, now scaling volumes instead of lines. The “3” in 3H is the trace of the linear map r ↦ Hr, and the trace is how a determinant changes to first order. The flat calculus of this volume is the one-dimensional shadow of that calculus.