 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/18/b/ii/solution

Let

w = x + i y

. Then

y - i x = - i w

sin (- i w) = - i sinh w

gives

sinh (x + i y) d x - sin (y - i x) d y = sinh w (d x + i d y) = d [cosh (x + i y)] .

Thus the form is exact, with complex-valued scalar potential

F (x, y) = cosh (x + i y)

. Differentiating that potential with respect to the two real variables checks both coefficients individually.

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