 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/f/f2/b/i/solution

Put

Q = 1 + x^{2} + y^{2} > 0

Q^{- 1}

gives

\nabla f (x, y) = \frac{- 2}{( 1 + x ^{2} + y ^{2} ) ^{2}} (x y) .

The positive denominator also shows that the function and these derivatives are smooth throughout

R^{2}

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