Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 7

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- Solution 7

Solution

The chain rule gives

d (e^{x y}) = e^{x y} (y d x + x d y)

, so the given total differential is exact. Thus

f = e^{x y} + C

; two such functions differ by a constant because their gradient difference vanishes on the connected plane. The assigned value at the origin forces

C = - 1

f (x, y) = e^{x y} - 1 .

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