Question

Let $a$be a constant. Prove that the equation of the plane $x=a$isrole="math" localid="1652390612497" $\rho =acsc\varphi sec\theta$ in spherical coordinates.

Short answer

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

Step-by-step solution

Given Information

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

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