Calculus Small Changes, Big Ideas

Section 6.3 Polar coordinates

• Some curves are better understood in polar coordinates.

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• Converting from polar to rectangular:

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  \begin{align*} x \amp = r\cos\theta\\ y \amp = r\sin\theta \end{align*}
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• Converting from rectangular to polar:

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  \begin{align*} r^2 \amp = x^2+y^2\\ \tan\theta \amp = \frac{y}{x} \end{align*}
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• Some graphs:

  • Rose curves: \(r=a\sin n\theta\text{,}\) \(r=a\cos n\theta\)

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  • Limaçon: \(r=a+b\sin\theta\text{,}\) \(r=a+b\cos\theta\)

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  • Circle: \(r=a\text{,}\) \(r=a\sin\theta\text{,}\) \(r=a\cos\theta\)

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  • Lemniscate: \(r^{2}=a^{2}\sin 2\theta\text{,}\) \(r^{2}=a^{2}\cos 2\theta\)

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• Derivative:

  \begin{equation*} \dv{y}{x} =\frac{\dd(r\sin\theta)}{\dd(r\cos\theta)}=\frac{r\cos\theta\dd{\theta}+\sin\theta\dd{r}}{-r\sin\theta\dd{\theta}+\cos\theta\dd{r}}=\frac{r\cos\theta+r'\sin\theta}{-r\sin\theta+r'\cos\theta} \end{equation*}

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