원기둥좌표와 구면좌표

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $\varphi$ in radians rounded to four decimal places.

[T] $\left(1,\frac{\pi }{4},3\right)$

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $\varphi$ in radians rounded to four decimal places.

[T] $(5,\pi ,12)$

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $\varphi$ in radians rounded to four decimal places.

$\left(3,\frac{\pi }{2},3\right)$

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $\varphi$ in radians rounded to four decimal places.

$\left(3,-\frac{\pi }{6},3\right)$

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

$\left(2,-\frac{\pi }{4},\frac{\pi }{2}\right)$

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

$\left(4,\frac{\pi }{4},\frac{\pi }{6}\right)$

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

$\left(8,\frac{\pi }{3},\frac{\pi }{2}\right)$

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

$\left(9,-\frac{\pi }{6},\frac{\pi }{3}\right)$

The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length $a,$ where $a>0$

A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of $a$ and $b,$ respectively, where $b>a>0$

For the following exercises, find the most suitable system of coordinates to describe the solids.

A solid inside sphere ${x}^{2}+{y}^{2}+{z}^{2}=9$ and outside cylinder ${\left(x-\frac{3}{2}\right)}^{2}+{y}^{2}=\frac{9}{4}$

A cylindrical shell of height $10$ determined by the region between two cylinders with the same center, parallel rulings, and radii of $2$ and $5,$ respectively

[T] Use a CAS to graph the region between elliptic paraboloid $z={x}^{2}+{y}^{2}$ and cone ${x}^{2}+{y}^{2}-{z}^{2}=0.$ Then describe the region in cylindrical coordinates.

[T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ and elliptical cone ${x}^{2}+{y}^{2}-{z}^{2}=0.$

Washington, DC, is located at $39^\circ$ N and $77^\circ$ W (see the following figure). Assume the radius of Earth is $4000$ mi. Express the location of Washington, DC, in spherical coordinates.

San Francisco is located at $37.78^\circ \text{N}$ and $122.42^\circ \text{W}.$ Assume the radius of Earth is $4000$ mi. Express the location of San Francisco in spherical coordinates.

Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are $(4000,-43.17^\circ ,102.91^\circ ).$

Find the latitude and longitude of Berlin if its spherical coordinates are $(4000,13.38^\circ ,37.48^\circ ).$

[T] Consider the torus of equation ${({x}^{2}+{y}^{2}+{z}^{2}+{R}^{2}-{r}^{2})}^{2}=4{R}^{2}({x}^{2}+{y}^{2}),$ where $R\ge r>0.$

a. Write the equation of the torus in spherical coordinates. b. If $R=r,$ the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is $\rho =2R\,\sin \,\varphi .$ c. Use a CAS to graph the horn torus with $R=r=2$ in spherical coordinates.

[T] The “bumpy sphere” with an equation in spherical coordinates is $\rho =a+b\,\cos (m\theta )\sin (n\varphi ),$ with $\theta \in [0,2\pi ]$ and $\varphi \in [0,\pi ],$ where $a$ and $b$ are positive numbers and $m$ and $n$ are positive integers, may be used in applied mathematics to model tumor growth.

a. Show that the “bumpy sphere” is contained inside a sphere of equation $\rho =a+b.$ Find the values of $\theta$ and $\varphi$ at which the two surfaces intersect. b. Use a CAS to graph the surface for $a=14,$ $b=2,$ $m=4,$ and $n=6$ along with sphere $\rho =a+b.$ c. Find an equation of the intersection curve of the surface at b. with the cone $\varphi =\frac{\pi }{12}.$ Graph the intersection curve in the plane of intersection.