Q 31 - Calculus

Question

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

Change $r=2$ to the rectangular and spherical systems 

Step-by-Step Solution

Answer

The equation in rectangular system is $x^2+y^2=4$

The equation in spherical system is $\rho =2csc\varphi$

Step 1:Given information

The given expression is $r=2$

Step 2: Simplification

The equation $r=2$ represents a right circular cylinder on z-axis

Now to convert it into rectangular system

Let us consider a point  in ${\mathrm{ℝ}}^3$ with the rectangular coordinates$\left(x,y,z\right)$ and cylindrical coordinates $\left(r,\theta ,z\right)$

Relation between them is 

$r=\sqrt{x^2+y^2}$

$\tan \theta =\frac{y}{x}$

$z=z$

Now substitute $r=\sqrt{x^2+y^2}$ in $r=2$

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

$\Rightarrow x^2+y^2=4$

Hence, the equation in rectangular system is $x^2+y^2=4$ 

Step 3: Simplification

To convert $r=2$ in spherical system, let us consider a point P in ${\mathrm{ℝ}}^3$ with the cylindrical coordinates $\left(r,\theta ,z\right)$and the spherical coordinates

The relation between them is 

$r=\rho \sin \varphi$

$\theta =\theta$

$z=\rho \cos \varphi$

substituting $r=\rho \sin \varphi$ in $r=2$

$\rho \sin \varphi =2$

$\Rightarrow \rho =\frac{2}{\sin \varphi }$

$\Rightarrow \rho =2csc\varphi$

Hence,the equation in spherical system is $\rho =2csc\varphi$