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Equation 1 · Part 8 · From Origins to Frontier: A History of Mathematics, Proof, and Scientific Computation

derivative

∂ζ∂t+J(ψ,ζ)=0,ζ=∇2ψ,\frac{\partial \zeta}{\partial t} + J(\psi, \zeta) = 0, \qquad \zeta = \nabla^2 \psi,
derivative

What this part means

This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.

Its job in the formula

This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.

The passage around this formula

A model of the barotropic vorticity equation used in that 1950 work can be written, in simplified form, as ∂ζ∂t+J(ψ,ζ)=0,ζ=∇2ψ\frac{\partial \zeta}{\partial t} + J(\psi, \zeta) = 0, \qquad \zeta = \nabla^2 \psi. where ψ\psi is the stream function describing the horizontal flow, ζ\zeta is the relative vorticity, and J(ψ\psi,ζ\zeta) is the Jacobian term expressing advection of vorticity by the flow itself. This is stated here not to reproduce the numerics of the original run but because the equation exposes exactly what made the problem hard to do by hand and tractable by machine: it is a single nonlinear evolution equation, and marching it forward requires repeating the same arithmetic over many small time steps across many grid points — precisely the kind of repetitive, exact,…

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Learn the underlying idea

A derivative describes how quickly one quantity changes as another changes. It is the slope of a curve at a particular point.

Open the illustrated derivatives: instantaneous rate of change guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.