Lemma
math, backwards
journey · 7 days · physics → finance

Change and accumulation in 7 days

Calculus is two operations — change and its accumulation — and the same pair runs both physics and finance. Distance and present value are the same shape, with different rates plugged in. This path walks the pair across both pillars, in order, with a deliberate stop in the middle to admit they really are the same machine.

Lemma is a graph, not a sequence. This page is one author’s pick of one cross-pillar path through it: physics on days 1–5, finance on day 6, and a final review that lets the reader see, in one frame, the same accumulator at work in distance and present value.

the path · 0/7 · 0%
  1. 1
    application·day 1·→ next
    /physics/projectile-motion
    Start with the falling ball — read once without any modules. Note where you reach for v(t) or y(t) and feel the gap.
    open →
  2. 2
    module·day 2
    /modules/derivatives
    The change side. Position differentiated once is velocity; twice is acceleration. The whole projectile arc 1 is exactly this ladder.
    open →
  3. 3
    module·day 3
    /modules/integration
    The accumulation side — the pair of the derivative. Reverse the ladder: from acceleration back to velocity, from velocity back to position. Note Arc 6 — the same accumulator runs finance too, with no edits to the math.
    open →
  4. 4
    review·day 4
    /physics/projectile-motion
    Re-read with both tools in hand. The y(t) = ½gt² formula now reads as ∫₀ᵗ g·s ds — the triangle area you read off geometry on day 1, derived this time from the antiderivative.
    open →
  5. 5
    application·day 5
    /physics/terminal-velocity
    Change isn't free anymore — drag bends velocity toward an asymptote. Distance is still ∫₀ᵗ v(s) ds; the curve just isn't a triangle. Same accumulator, harder shape.
    open →
  6. 6
    application·day 6
    /finance/present-value
    Same accumulator, finance noun. PV = ∫₀ᵀ c(t) e^(−rt) dt — discount each future moment, sum them. Money over time is structurally identical to velocity over time.
    open →
  7. 7
    review·day 7
    /modules/integration
    Re-read Arc 6 with all four consumers (projectile, terminal, oscillator, present-value) in hand. The cross-pillar admission is now firsthand — *same machine, different nouns*.
    open →