Q 37 - Calculus

Question

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems. 

. Change $\varphi =\frac{\mathrm{\pi }}{2}$ to the rectangular and cylindrical systems. 

Step-by-Step Solution

Answer

The equation in rectangular coordinates $z=0$

The equation in cylindrical  coordinates $z=0$

Step 1:Given information

The given expression is $\varphi =\frac{\mathrm{\pi }}{2}$

Step 2 Simplification

The objective is to convert the equation $\varphi =\frac{\mathrm{\pi }}{2}$ into rectangular system.

Here,width="49" style="max-width: none; vertical-align: -33px;" $\varphi =\frac{\pi }{2}$

width="217" style="max-width: none; vertical-align: -22px;" $\Rightarrow {\cos }^{-1}\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)=\frac{\pi }{2}$

width="200" style="max-width: none; vertical-align: -22px;" $\Rightarrow \frac{z}{\sqrt{x^2+y^2+z^2}}=\cos \left(\frac{\pi }{2}\right)$

$\Rightarrow \frac{z}{\sqrt{x^2+y^2+z^2}}=0$

$\Rightarrow z=0$

Hence, The given  equation in rectangular coordinates is $z=0$

Now,

To convert to the equation $\varphi =\frac{\mathrm{\pi }}{2}$ into  cylindrical system.

Substituting $\varphi ={\tan }^{-1}\left(\frac{r}{2}\right)$ in the equation $\varphi =\frac{\mathrm{\pi }}{2}$

So $\varphi =\frac{\pi }{2}$

width="114" style="max-width: none;" $\Rightarrow {\tan }^{-1}\frac{r}{z}=\frac{\mathrm{\pi }}{2}$

width="97" style="max-width: none;" $\Rightarrow \frac{r}{z}=\tan \frac{\mathrm{\pi }}{2}$

$\Rightarrow \frac{r}{z}=\frac{1}{0}$

$\Rightarrow z=0$

Hence,The equation in cylindrical  coordinates  $z=0$