Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 19 / a

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Solution

In the Cartesian coordinate system, the Laplacian of a scalar field is the sum of its three unmixed second partial derivatives. Each squared coordinate here contributes one, so

\nabla^{2} ψ = ψ_{xx} + ψ_{yy} + ψ_{zz} = 1 + 1 + 1 = 3.

Equivalently

\nabla ψ = (x, y, z)

and its divergence is three.

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