Rewrite the following equation in spherical coordinates. x^2 + y^2 + z^2 = 4.


Rewrite the following equation in spherical coordinates.

{eq}x^2 + y^2 + z^2 = 4. {/eq}

Spherical Coordinates:

Along with rectangular coordinates and cylindrical coordinates, spherical coordinates round out the most common three systems that we use. Recall that spherical coordinates are an extension to the old polar coordinates where we introduce a new angle. We measure this new angle downward from the positive {eq}z {/eq}-axis, and we call it {eq}\phi {/eq}. Recall the following relations:

{eq}x=\rho \cos \theta \sin \phi {/eq}

{eq}y = \rho \sin \theta \sin \phi {/eq}

{eq}z = \rho \cos \phi {/eq}

{eq}\rho^2 = x^2+y^2+z^2 {/eq}

{eq}\begin{align*} \rho^2 \sin^2 \phi &= x^2 + y^2 \end{align*} {/eq}

{eq}dV = \rho^2 \sin \phi \ d\rho \ d\phi \ d\theta {/eq}

Answer and Explanation: 1

Our surface is a sphere; this coordinate system was literally built for such surfaces. Ours is

{eq}\begin{align*} x^2 + y^2 + z^2 &= 4 \\ \rho^2 &= 4 \\ \boldsymbol \rho &= \boldsymbol 2 \end{align*} {/eq}

We could have just as easily recognized this surface immediately; it is one of the most recognizable general forms for a surface that there is.

Learn more about this topic:

Cylindrical & Spherical Coordinates: Definition, Equations & Examples


Chapter 13 / Lesson 10

In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. Both cylindrical and spherical coordinates use angles to specify the locations of points, a feature they share with 2-D polar coordinates.

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