Lyra paused. At the center ( r \to 0 ), velocity couldn’t be infinite (no whirlpool tears a hole in reality). So ( C = 0 ). The true function was clean and smooth:
[ r v = \int 3r^3 , dr = \frac{3}{4} r^4 + C ]
The city was saved. And Lyra learned that differential equations describe how things change, but integrals measure what has changed. Together, they hold the power to calm any storm. Integral calculus including differential equations
[ v(r) = \frac{3}{4} r^3 ]
Lyra, a young apprentice, faced her final trial: to tame the , a rogue whirlpool deep beneath the city that pulsed with erratic, destructive energy. If she failed, Aethelburg would be torn apart by the year's first monsoon. Lyra paused
[ \frac{d}{dr}(r v) = 3r^3 ]
The left side was a perfect derivative:
Kael nodded grimly. "That’s the energy. If you release a counter-vortex with exactly that integrated strength, shaped like ( u(r) = 48 - \frac{3}{4}r^3 ), the sum of the two integrals will be zero. The Churnheart will still itself."