 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/12/b/solution

(\nabla \times G)_{x} = x^{2} - x^{2} = 0, (\nabla \times G)_{y} = (2 x y - 1) - (2 x y - 1) = 0, (\nabla \times G)_{z} = (3 + 2 x z) - (3 + 2 x z) = 0.

An exact differential is a differential one-form equal to

d ϕ = ϕ_{x} d x + ϕ_{y} d y + ϕ_{z} d z

for a single-valued scalar function. This field is smooth on the simply connected domain

R^{3}

, so its vanishing curl is sufficient for exactness. More concretely,

G \cdot d r = d (3 x y + x^{2} yz - x z) .

The explicit potential proves exactness globally, rather than relying on curl-free behavior without checking the domain.

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