the rule (dB)^2 = dt
The heuristic identity (dB)^2 = dt is the single rule that distinguishes stochastic calculus from ordinary calculus. In ordinary calculus, (df)^2 is of order (dt)^2 and is dropped in any first-order expansion. For Brownian motion the squared increment is NOT negligible: (dB)^2 contributes at first order in dt, and ignoring it gives wrong answers. The rule is shorthand for a precise, provable theorem about the quadratic variation of Brownian motion, and it is exactly the ingredient that forces the second-order correction term in Ito's formula.
Made precise: the quadratic variation of Brownian motion over [0, t] is the limit, as the partition mesh tends to 0, of the sum of squared increments sum (B_{t_{k+1}} - B_{t_k})^2, and this limit equals t (almost surely along refining partitions, and in L^2 along any sequence of partitions whose mesh goes to 0). We write [B]_t = t. The companion entries of the 'Ito multiplication table' are dB * dt = 0 and (dt)^2 = 0, because time has zero quadratic variation and finite (in fact linear) total variation. The L^2 calculation is transparent: each squared increment has mean (t_{k+1} - t_k) and variance 2 (t_{k+1} - t_k)^2, so the sum has mean t and variance 2 sum (t_{k+1} - t_k)^2 <= 2 * mesh * t, which vanishes as the mesh shrinks; hence the random sum concentrates on its deterministic mean t. The deterministic limit (no randomness left) is what makes the rule usable as if it were an algebraic identity.
Why it is the engine: when you Taylor-expand f(B_t) you get f'(B) dB + (1/2) f''(B) (dB)^2 + ...; in ordinary calculus the second term is discarded, but (dB)^2 = dt keeps it as (1/2) f''(B) dt, which is precisely the Ito correction. The same rule explains why integral B dB = (B^2 - t)/2 instead of B^2/2. The honest caveats: first, (dB)^2 = dt is a statement about the L^2 / almost-sure limit of squared increments, not a pathwise pointwise equality of differentials — it is a clean mnemonic for a limit theorem, and the almost-sure version requires the partitions to be nested (refining) or summably fine. Second, this is special to Brownian motion (and to continuous martingales via their own [M]); a process of bounded variation has zero quadratic variation, and a jump process picks up jump contributions (dB)^2 -> (Delta X)^2 in the table. The rule (dB)^2 = dt is the continuous-Brownian special case of the general identity that the quadratic variation of a continuous local martingale is its [M].
Apply the multiplication table to d(B_t^2). Taylor: d(B^2) = 2B dB + (dB)^2. Ordinary calculus would set (dB)^2 = 0 and give 2B dB; the rule (dB)^2 = dt gives d(B^2) = 2B dB + dt instead. Integrating, B_t^2 = 2 integral_0^t B dB + t, i.e. integral_0^t B dB = (B_t^2 - t)/2 — the extra +t is precisely the accumulated quadratic variation.
The +dt that ordinary calculus would drop is exactly the quadratic variation accumulating at rate 1.
(dB)^2 = dt is a mnemonic for the limit of summed squared increments, not a pathwise differential identity; the squared increment of a single sample over [t, t+dt] is random with mean dt, and only the SUM over a fine partition concentrates on its deterministic mean.