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

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- Solution 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 .

Thus this exact differential form has

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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