Q 22. - Calculus

Question

From Example 1, recall that $x^2+y^2=1$ is the equation of the cylinder with radius $1$, whose axis of symmetry is the $z$-axis. Show that the equation of this cylinder in spherical coordinates is $\rho = csc \varphi$ .

Step-by-Step Solution

Answer

It is solved by substituting the value of $x,y$ in terms of spherical coordinates.

Step 1: Given Information

The equation of cylinder is $x^2+y^2=1$ with radius $1$ unit and axis of symmetry as $z$ axis.

Step 2: Simplification

We know the relation:

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

$y=\rho \sin \varphi \sin \theta$

$z=\rho \cos \varphi$

Using values of $x,y$ in equation of cylinder,

$(\rho \sin \varphi \cos \theta {)}^2+(\rho \sin \varphi \sin \theta {)}^2=1$

${\rho }^2{\sin }^2\varphi {\cos }^2\theta +{\rho }^2{\sin }^2\varphi {\sin }^2\theta =1$

${\rho }^2{\sin }^2\varphi \left({\cos }^2\theta +{\sin }^2\theta \right)=1$

${\rho }^2{\sin }^2\varphi =1$

Therefore, we get

${\rho }^2=\frac{1}{{\sin }^2\varphi }$

${\rho }^2={\csc }^2\varphi$

Hence

$\rho =\csc \varphi$

Hence, proved.