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.

Short answer

The equation in rectangular coordinates$z=0$

The equation in cylindrical coordinates$z=0$

Step-by-step solution

Step 1:Given information

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

Simplification 

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

Here,width="49">ϕ=π2

width="217">⇒cos-1zx2+y2+z2=π2

width="200">⇒zx2+y2+z2=cosπ2

localid="1653033836537" $\Rightarrow \frac{z}{\sqrt{x^2+y^2+z^2}}=0$

$\Rightarrow z=0$

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">⇒tan-1rz=π2

width="97">⇒rz=tanπ2

localid="1653034873552" $\Rightarrow \frac{r}{z}=\frac{1}{0}$

$\Rightarrow z=0$

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