 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/11/b/solution

\overset{z}{˙} = (\overset{r}{˙} + i r \dot{θ}) e^{i θ} .

\overset{z}{¨} = [\overset{r}{¨} + i (\overset{r}{˙} \dot{θ} + r \ddot{θ}) + i \dot{θ} (\overset{r}{˙} + i r \dot{θ})] e^{i θ} = [(\overset{r}{¨} - r \dot{θ}^{2}) + i (2 \overset{r}{˙} \dot{θ} + r \ddot{θ})] e^{i θ} .

The two contributions to

2 \overset{r}{˙} \dot{θ}

arise separately from differentiating

r \dot{θ}

and from differentiating the moving exponential factor.

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