 Solution

ID: 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/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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