Homework 1: Coordinates

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Graded Problems

Consider the surface in $\mathbb {R}^3$ described by the equation $(r-2)^2+z^2=1$ in cylindrical coordinates.

(a): Sketch the intersection of this surface with the $xy$ -plane.
(b): Given that $\theta$ does not appear in the equation, what can you say about the surface?
(c): Sketch the surface.

(a): Suppose we have a curve in $\mathbb {R}^2$ defined by the polar equation $r=f(\theta )$ , for some function $f$ . What is the relationship between this curve and the curve defined by $r=f(\theta +\pi /4)$ in polar coordinates?
(b): Suppose we have a surface in $\mathbb {R}^3$ defined by the cylindrical equation $r=g(\theta , z)$ , for some function $g$ . What is the relationship between this surface and the surface defined by $r=g(\theta + \pi /4, z)$ in cylindrical coordinates?
(c): Suppose we have a surface in $\mathbb {R}^3$ defined by the spherical equation $r=h(\theta , \phi )$ , for some function $h$ . What is the relationship between this surface and the surface defined by $r=h(\theta + \pi /4, \phi )$ in spherical coordinates?
(d): Again, suppose we have a surface in $\mathbb {R}^3$ defined by the spherical equation $r=h(\theta , \phi )$ , for some function $h$ . What is the relationship between this surface and the surface defined by $r=h(\theta ,\phi +\pi )$ in spherical coordinates? Hint: it’s not a rotation.

Professional Problem

Consider the solid below, which is obtained by taking the portion of a solid sphere of radius $2$ which is outside of the (double) cone $z^2 = x^2+y^2$ .

(a): Is this region easier to describe in spherical coordinates or in cylindrical coordinates? Justify your answer.
(b): Describe this region in either spherical coordinates or cylindrical coordinates (based on your answer to (a)).

Completion Packet

Consider the point $P=\left (\frac {\pi }{4}, \frac {\pi }{4}, \frac {\pi }{4}\right )$ , given in crylindrical coordinates. What are the Cartesian coordinates of $P$ ?

Consider the point $P=\left (\frac {\pi }{2}, \frac {\pi }{2}, \frac {\pi }{2}\right )$ , given in spherical coordinates. What are the spherical coordinates of $P$ ?

Consider the point $P = (2,1,3)$ , given in Cartesian coordinates. Find all possible ways to write $P$ in cylindrical coordinates.

Consider the point $P = (2, 1, 2)$ , given in Cartesian coordinates. Find all possible ways to write $P$ in spherical coordinates.

Sketch the graph of the surface in $\mathbb {R}^3$ defined by the equation $\rho = \cos (\phi )$ in spherical coordinates, for $0\leq \phi \leq \pi$ .

Consider the surface in $\mathbb {R}^3$ defined by the equation $2x^2+2y^2+z^2=1$ in Cartesian coordinates. Convert the equation to cylindrical coordinates. Convert the equation to spherical coordinates. Sketch the surface.

Sketch the region defined by the inequalities $\pi /4\leq \theta \leq \pi$ , $0\leq r\leq 2$ , and $0\leq z\leq r$ in cylindrical coordinates.

Sketch the region defined by the inequalities $0\leq \phi \leq \pi$ , $0\leq \theta \leq \pi$ , and $0\leq \rho \leq \theta$ .

Consider the region defined by the inequalities $0\leq r\leq 1$ , $0\leq \theta \leq 2\pi$ , and $0\leq z\leq 2$ in cylindrical coordinates.

(a): Sketch this region.
(b): Describe this region in spherical coordinates.