 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/f/f2/c/i/solution

D G (x, y) = (f_{x} (x, y) f_{x} (x, y^{2}) f_{y} (x, y) 2 y f_{y} (x, y^{2})), D G (0, 0) = (1110) .

- 1

, so it is invertible. The map

G

is smooth and

G (0, 0) = (0, 0)

U

of the origin for which

G (U)

is open, contains the origin, and

G : U \to G (U) is a diffeomorphism .

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