Q 30 - Calculus

Question

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

Change $z=x+y$ to the cylindrical and spherical systems. 

Step-by-Step Solution

Answer

The cylindrical equation is $r=\frac{z}{\cos \theta +\sin \theta }$

And the spherical equation is $cot\varphi =\left(\cos \theta +\cos \varphi \right)$

Step 1:Given information

The given expression is $z=x+y$

Step 2:Simplification

Consider the equation $z=x+y$

To change $z=x+y$ into cylindrical system,we can use

$x=r\cos \theta ,y=r\sin \theta$

so,

$z=x+y$

$\Rightarrow z=r\cos \theta +r\sin \theta$

$\Rightarrow r=\frac{z}{\cos \theta +\sin \theta }$

Therefore the cylindrical system is $r=\frac{z}{\cos \theta +\sin \theta }$

Now,

to change $z=x+y$ into spherical system,we can use

$x=\rho \sin \varphi \cos \theta ,y=\rho \sin \varphi \cos \varphi ,z=\cos \varphi$

so,

$z=x+y$

$\Rightarrow \rho \cos \varphi =\rho \sin \varphi \cos \theta +\rho \sin \varphi \cos \varphi$

$\Rightarrow \rho \cos \varphi =\rho \sin \varphi \left(\cos \theta +\cos \varphi \right)$

$\Rightarrow \frac{\rho \cos \varphi }{\rho \sin \varphi }=\left(\cos \theta +\cos \varphi \right)$

$\Rightarrow cot\varphi =\left(\cos \theta +\cos \varphi \right)$

Therefore the equation in spherical system is $cot\varphi =\left(\cos \theta +\cos \varphi \right)$