 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/12/a/iii/solution

The coefficients follow by differentiating the same monomial:

d (x^{2} y^{3} z^{4}) = 2 x y^{3} z^{4} d x + 3 x^{2} y^{2} z^{4} d y + 4 x^{2} y^{3} z^{3} d z .

f (x, y, z) = x^{2} y^{3} z^{4} + C .

Exhibiting a global scalar potential proves exactness directly, without needing an additional domain argument.

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