Answer to Question #348390 in Calculus for ;)x

Question

Identify the surfaces of the following equations by converting them into equations in the Cartesian form.

ρ = sin ϕ sin θ

Answer 1

The equation $ρ = sin ϕ sin θ$ is in spherical coordinates.

To convert from spherical coordinates to rectangular coordinates there are the next equations:

$x=ρsin\phi cosθ,$

$y=ρsin\phi sinθ,$

$z=ρcos\phi,$

$ρ^2=x^2+y^2+z^2.$

From the second equation we have: $sin ϕ sin θ=\frac{y}{\rho},$

and we can rewrite the equation of the surface:

$\rho=\frac{y}{\rho},$

$\rho ^2=y,$

$x^2+y^2+z^2=y,$

$x^2+y^2-y+z^2=0,$

$x^2+(y-\frac{1}{2})^2-{\frac{1}{4}}+z^2=0,$

$x^2+(y-\frac{1}{2})^2+z^2= \frac{1}{4}.$

This is the equation of a sphere with a center in $(0, \frac{1}{2}, 0)$ and radius $\frac{1}{2}.$