Q 29 - Calculus

Question

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

Change $x=4$ to the cylindrical and spherical systems 

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is $r=4sec\theta$

And the equation in spherical coordinates is $\rho =4csc\varphi sec\vartheta$

Step 1:Given information

The equation is $x=4$

Step 2:Simplification

The equation $x=4$represents the plane parallel to $yz-$plane

To convert $x=4$ to cylindrical system

substituting $x=r\cos \theta$,then

$$\begin{gathered} r\cos \theta =4 \\ \Rightarrow r=4sec\theta \end{gathered}$$

Therefore the equaion in cylindrical coordinates is $r=4sec\theta$

Now,

To convert $x=4$ to spherical system

substituting $x=\rho \sin \varphi \cos \theta$

$\Rightarrow \rho =\frac{4}{\sin \varphi \cos \theta }$

$\Rightarrow \rho =4csc\varphi sec\theta$

Therefore the equation in spherical coordinates is $\rho =4csc\varphi sec\vartheta$