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

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- Solution b

Solution

Differentiate the polar form of a complex number using the product rule and chain rule:

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

A second differentiation gives

\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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