Q 72. - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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

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

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

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

$r=b\csc \varphi \csc \theta$

Hence proved.