Polar coordinates in the cartesian plane.

Question

${dy}/{dx} = {dy}/{d\theta}$ divided by $dx/d\theta$ where $x$ and $y$ are in the Cartesian plane and $\theta$ is in the polar plane and $x = r\cos( \theta), \ y = r \sin (\theta)$.

If $dy/dx = 0$ why does that have to mean that $dy/d\theta = 0$? Why couldn't $d\theta/dx = 0$ instead?

Answer 1

$$\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\;\dfrac{\mathrm{d}y}{\mathrm{d}\theta}\;}{\dfrac {\mathrm{d}x}{\mathrm{d}\theta}}$$

$$y_x(x) = \dfrac{y_\theta(\theta)}{x_\theta(\theta)}=\dfrac{r \cos \theta}{-r \sin \theta}$$

So, if the LHS equals $0$, then the numerator on the RHS must equal $0$, or else the denominator must approach infinitude ($\infty$).

In this particular case the denominator is $- r\sin\theta$ which is never infinite.

Answer 2

By the chain rule you have

$\frac{dy}{d\theta}=\frac{dy}{dx}\cdot \frac{dx}{d\theta}$

It follows that

$\frac{dy}{dx}=0$ $\Rightarrow$ $\frac{dy}{d\theta}=0$