 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-2/section-b/6/b/solution

Integrating

ψ_{x} = - (2 x y + z)

with respect to

x

gives

ψ = - x^{2} y - x z + h (y, z)

. Matching

ψ_{y} = - (x^{2} + 2 y)

requires

h = - y^{2} + j (z)

, and matching

ψ_{z} = - x

requires

j

constant. A scalar potential with the requested sign is

ψ = - x^{2} y - x z - y^{2} + C .

R^{3}

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