Question

Let $b$be a constant. Prove that the equation of the plane $y=b$is$r=bcsc\phi csc\theta$in spherical coordinates.

Short answer

This is proved using relation$x=r\sin \varphi \cos \theta ,y=r\sin \varphi \sin \theta ,z=r\cos \varphi$.

Step-by-step solution

Given Information

Equation of rectangular coordinates is given as $y=b$ ($b$is constant).

Simpification

We know that

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

For conversion of given equation of plane to spherical coordinates, use $y=r\sin \varphi \sin \theta$in given equation

$r\sin \varphi \sin \theta =b$

$\Rightarrow r=\frac{b}{\sin \varphi \sin \theta }$

$=b\left(\frac{1}{\sin \varphi }\right)\left(\frac{1}{\sin \theta }\right)$

role="math" localid="1652391295106" $r=b\csc \varphi \csc \theta$

Hence proved.