Q 71. - Calculus

Question

Let $a$ be a constant. Prove that the equation of the plane $x = a$ is  $\rho = a csc \varphi sec \theta$ in spherical coordinates.

Step-by-Step Solution

Answer

Conversion is done using $x=\rho \sin \varphi \cos \theta ,y=\rho \sin \varphi \sin \theta ,z=\rho \cos \varphi$.

Step 1: Given Information

Equation of plane in rectangular coordinates is given by $x=a$ ($a$ is constant).

Step 2: Simplification

The spherical coordinates are given by following mathematical relation

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

For the transformation of given plane to spherical coordinates, use $x=\rho \sin \varphi \cos \theta$ in $x=a$.

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

$\Rightarrow \rho =\frac{a}{\sin \varphi \cos \theta }$

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

Hence, $\rho =a\csc \varphi \sec \theta$