Paul's Online Notes
Home / Calculus II / 3-Dimensional Space / Spherical Coordinates
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 12.13 : Spherical Coordinates

8. Identify the surface generated by the given equation : $$\rho = - 2\sin \varphi \cos \theta$$

Show All Steps Hide All Steps

Start Solution

Let’s first multiply each side of the equation by $$\rho$$ to get,

${\rho ^2} = - 2\rho \sin \varphi \cos \theta$ Show Step 2

We can now easily convert this to Cartesian coordinates to get,

\begin{align*}{x^2} + {y^2} + {z^2} & = - 2x\\ {x^2} + 2x + {y^2} + {z^2} & = 0\end{align*}

Let’s complete the square on the $$x$$ portion to get,

\begin{align*}{x^2} + 2x + 1 + {y^2} + {z^2} & = 0 + 1\\ {\left( {x + 1} \right)^2} + {y^2} + {z^2} &= 1\end{align*} Show Step 3

So, it looks like we have a sphere with radius 1 that is centered at $$\left( { - 1,0,0} \right)$$.