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Section 12.13 : Spherical Coordinates
5. Convert the equation written in Spherical coordinates into an equation in Cartesian coordinates.
\[{\rho ^2} = 3 - \cos \varphi \]Show All Steps Hide All Steps
Start SolutionThere really isn’t a whole lot to do here. All we need to do is to use the following conversion formulas in the equation where (and if) possible
\[\begin{array}{c}x = \rho \sin \varphi \cos \theta \hspace{0.5in}y = \rho \sin \varphi \sin \theta \hspace{0.5in}z = \rho \cos \varphi \\ {\rho ^2} = {x^2} + {y^2} + {z^2}\end{array}\] Show Step 2To make this problem a little easier let’s first multiply the equation by \(\rho \). Doing this gives,
\[{\rho ^3} = 3\rho - \rho \cos \varphi \]Doing this makes recognizing the right most term a little easier.
Show Step 3Using the appropriate conversion formulas from Step 1 gives,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}} = 3\sqrt {{x^2} + {y^2} + {z^2}} - z}}\]