 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/13/d/solution

Evaluate the line integral along the three coordinate segments, treating coordinates not changing along a segment as constants. On the first segment,

y = z = 0

, so

F_{x} = 0

and the contribution is zero. On the second segment,

\int_{0}^{y} F_{y} (x, η, 0) d η = \int_{0}^{y} (x^{2} - 2 η) d η = x^{2} y - y^{2} .

On the final segment,

\int_{0}^{z} F_{z} (x, y, ζ) d ζ = \int_{0}^{z} (- 3 x ζ^{2} - 1) d ζ = - x z^{3} - z .

Φ (x, y, z) = x^{2} y - y^{2} - x z^{3} - z + C .

Taking its gradient verifies each component of the field. The constant is

C = Φ (0, 0, 0)

; the field specifies only potential differences because the gradient of every constant is zero.

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