Question

find an equation in cylindrical coordinates for the surface represented by the rectangular equation. y = x^2

Explanation:

Step1: Recall cylindrical - rectangular conversion

In cylindrical coordinates, $x = r\cos\theta$, $y = r\sin\theta$, and $z = z$.

Step2: Substitute $x$ and $y$ into the given equation

Given $y=x^{2}$, substitute $x = r\cos\theta$ and $y = r\sin\theta$. We get $r\sin\theta=(r\cos\theta)^{2}$.

Step3: Simplify the equation

$r\sin\theta = r^{2}\cos^{2}\theta$. Since $r
eq0$ (if $r = 0$, the equation is also satisfied in a trivial sense, and we can divide both sides by $r$ for non - zero $r$ cases), we have $\sin\theta=r\cos^{2}\theta$, or $r=\frac{\sin\theta}{\cos^{2}\theta}=\tan\theta\sec\theta$.

Answer:

$r = \tan\theta\sec\theta$