Practice Problems: Coordinate Systems

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Online Problems

Find the Cartesian coordinates of each point, which is given in polar coordinates.

$(r,\theta ) = (2, \pi /6)$ $(x,y) = \answer {(\sqrt {3},1)}$ $(r,\theta ) = (\sqrt {2}, 3\pi /4)$ $(x,y) = \answer {(-1,1)}$ $(r,\theta ) = (1, 0)$ $(x,y) = \answer {(1,0)}$ $(r,\theta ) = (3, \pi )$ $(x,y) = \answer {(-3,0)}$

Find the polar coordinates of each point, which is given in Cartesian coordinates. Your answers should satify $0\leq r$ and $0\leq \theta < 2\pi$ .

$(x,y) = (\sqrt {3},0)$ $(r,\theta ) = \answer {(\sqrt {3},0)}$ $(x,y) = (0,2)$ $(r,\theta ) = \answer {(2,\pi /2)}$ $(x,y) = (-1,-1)$ $(r,\theta ) = \answer {(\sqrt {2},5\pi /4)}$ $(x,y) = (1,\sqrt {3})$ $(r,\theta ) = \answer {(2,\pi /3)}$

Find the Cartesian coordinates of the point $(\pi /2, \pi , 2)$ , given in cylindrical coordinates. $(x,y,z) = \answer {(0, -\pi /2, 2)}$

Find cylindrical coordinates for the point $\left (0, -1, 3\right )$ , written in Cartesian coordinates. Your answer should satisfy $0\leq r$ and $0\leq \theta <2\pi$ . $(r, \theta , z) = \answer {(1, \pi , 3)}$

Consider the surface described in Cartesian coordinates by $2z^2 = x^2 +y^2.$ Describe this surface with an equation in cylindrical coordinates, of the form $0=f(r,\theta , z)$ . $0=\answer {r^2-2z^2}$ FIGURE OUT HOW TO HANDLE THIS!!! What type of shape is this?

Plane Cylinder Sphere Cone Other

Consider the following region in $\mathbb {R}^3$ .

This region is the set of points $(r, \theta , z)$ , in cylindrical coordinates, satisfying the inequalities

$\begin{align*} &\answer{1}\leq r\leq \answer{2}\\ &\answer{\pi/2}\leq\theta\leq\answer{\pi}\\ &\answer{-1}\leq z\leq\answer{1} \end{align*}$

Find the Cartesian coordinates of each point, given in cylindrical coordinates.

$(r,\theta ,z) = (1,1,1)$ $(x,y,z) = \answer {(\cos (1), \sin (1), 1)}$ $(r,\theta ,z) = (\pi ,\pi ,\pi )$ $(x,y,z) = \answer {(-\pi , 0, \pi )}$ $(r,\theta ,z) = (2,4\pi /3,-2)$ $(x,y,z) = \answer {(-\sqrt {3}, -1, -2)}$

Find the cylindrical coordinates of each point, given in Cartesian coordinates. Your answers should satisfy $r\geq 0$ and $0\leq \theta < 2\pi$ .

$(x,y,z) = (1,1,1)$ $(r,\theta ,z) = \answer {(\sqrt {2}, \pi /4, 1)}$ $(x,y,z) = (\pi ,\pi ,\pi )$ $(r,\theta ,z) = \answer {(\sqrt {2}\pi , \pi /4, \pi )}$ $(x,y,z) = (2,2\sqrt {3},-2)$ $(r,\theta ,z) = \answer {(4, \pi /6, -2)}$

Find the Cartesian coordinates of the point $(2, \pi , \pi /2)$ , given in spherical coordinates. $(x,y,z) = \answer {(-2,0,0)}$

Find spherical coordinates for the point $\left (-\sqrt {2}, \sqrt {2}, 2\sqrt {3}\right )$ , written in Cartesian coordinates. Your answer should satisfy $0\leq \rho$ , $0\leq \theta \leq 2\pi$ , and $0\leq \phi \leq \phi$ . $(\rho , \theta , \phi ) = \answer {(4, 3\pi /4, \pi /6)}$

Consider the following region in $\mathbb {R}^3$ .

This region is the set of points $(\rho ,\theta ,\phi )$ , in spherical coordinates, satisfying the inequalities

$\begin{align*} &\answer{0}\leq\rho\leq \answer{2}\\ &\answer{0}\leq\theta\leq\answer{pi/2}\\ &\answer{0}\leq\phi\leq\answer{pi} \end{align*}$

For each of the following equations in spherical coordinates, select the type of shape they define.

FIGURE OUT CORRECT ANSWERS

$\rho = \cos \phi$

plane cylinder sphere other

$\rho = \sin \theta$

plane cylinder sphere other

$\rho \cos \theta \sin \phi = 1$

plane cylinder sphere other

Find the spherical coordinates of each point, given in Cartesian coordinates. Your answers should satisfy $0\leq r$ , $0\leq \theta < 2\pi$ , and $0\leq \phi \leq \pi$ .

$(x,y,z) = (1,1,1)$ $(\rho ,\theta ,\phi ) = \answer {(\sqrt {3}, \pi /4, \pi /4)}$ $(x,y,z) = (1, -1, -1)$ $(\rho ,\theta ,\phi ) = \answer {(\sqrt {3}, 3\pi /4, 3\pi /4)}$ $(x,y,z) = (1,\sqrt {3},0)$ $(\rho ,\theta ,\phi ) = \answer {(\sqrt {4}, \pi /6, \pi /2)}$

Find the Cartesian coordinates of each point, given in spherical coordinates.

$(\rho ,\theta ,\phi ) = (\pi ,\pi ,\pi )$ $(x,y,z) = \answer {(0, 0, -\pi )}$ $(\rho ,\theta ,\phi ) = (3,\pi /2,\pi /4)$ $(x,y,z) = \answer {(0, 3\sqrt {2}/2, 3\sqrt {2}/2)}$ $(\rho ,\theta ,\phi ) = (2,7\pi /6,3\pi /4)$ $(x,y,z) = \answer {(-\sqrt {6}/2, -\sqrt {2}/2, -\sqrt {2})}$

Written Problems

For several values of the constant $a$ , sketch the graph of the curve in $\mathbb {R}^2$ given by the polar equation $r = a\sin \theta .$ What do you notice about these curves?

Which points in $\mathbb {R}^2$ have the same coordinates when written in Cartesian and polar coordinates? (That is, for what points do we have $x=r$ and $y=\theta$ ?)

Consider the surface described by $(r-3)^2 +z^2 = 1$ in cylindrical coordinates, with the restriction $r\geq 0$ .

(a): Sketch the intersection of the surface with the half-plane $\theta = 0$ .
(b): Sketch the intersection of the surface with the half-plane $\theta = \frac {\pi }{2}$ .
(c): Sketch the intersection of the surface with the plane $z = 0$ .
(d): Sketch the surface.

Sketch the region in $\mathbb {R}^3$ with cylindrical coordinates satisfying the inequality $r\leq z \leq 4-2r$

Convert the following equation, given in cylindrical coordinates, into Cartesian coordinates. $r = 0$

Sketch the region in $\mathbb {R}^3$ given by $\begin{align*} &r\leq z\leq 3\\ &0\leq\theta \leq\pi/2 \end{align*}$

in cylindrical coordinates.

Sketch the region in $\mathbb {R}^3$ given by $r^2-2\leq z\leq 2-r^2$ in cylindrical coordinates.

Which points in $\mathbb {R}^3$ have the same coordinates when written in Cartesian and cylindrical coordinates?

(a): Given a function $f$ , consider the graphs of the equations $r = f(\theta )$ and $r = 2f(\theta )$ , in polar coordinates. How are these graphs related?
(b): Given a function $f$ , consider the graphs of the equations $\rho = f(\theta , \phi )$ and $\rho = 2f(\theta , \phi )$ , in spherical coordinates. How are these graphs related?
(c): Given a function $f$ , consider the graphs of the equations $r = f(\theta )$ and $r = -f(\theta )$ , in polar coordinates. How are these graphs related?
(d): Given a function $f$ , consider the graphs of the equations $\rho = f(\theta , \phi )$ and $\rho = -f(\theta , \phi )$ , in spherical coordinates. How are these graphs related?

Sketch the surface in $\mathbb {R}^3$ given by the equation $1-\cos \phi$ in spherical coordinates.

Consider the surface in $\mathbb {R}^3$ given by the equation $\rho \sin \phi \sin \theta = 1$ in spherical coordinates.

Convert this equation to Cartesian coordinates and cylindrical coordinates, and sketch the surface.

Consider the region in $\mathbb {R}^3$ consisting of points whose spherical coordinates satisfy $1\leq \rho \leq 3$ Sketch this region.

Consider the region in $\mathbb {R}^3$ consisting of points whose spherical coordinates satisfy $\begin{align*} &0\leq \phi\leq \pi/2\\ &0\leq \rho\leq 1 \end{align*}$

Sketch this region.

Consider the region in $\mathbb {R}^3$ consisting of points whose spherical coordinates satisfy $\cos \phi \leq \rho \leq 2$ Sketch this region.

Which points in $\mathbb {R}^3$ have the same coordinates when written in Cartesian and spherical coordinates?