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Equation 1 · Part 9 · From Origins to Frontier: A History of Mathematics, Proof, and Scientific Computation

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

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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