 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/14/c/solution

\frac{\partial ( x , y )}{\partial ( r , θ )} = (cos θ sin θ - r sin θ r cos θ), J = r cos^{2} θ + r sin^{2} θ = r .

For

r > 0

, the corresponding area element is

d x d y = r d r d θ

. The determinant vanishes at

r = 0

, where the polar angle does not give a unique coordinate representation.

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