Question 1.128

Find the spherical coordinates of the Cartesian point \((\sqrt{2}, -\sqrt{6}, -2\sqrt{2})\).

Question 1.129

Find the spherical coordinates of the Cartesian point \((\sqrt{6}, -\sqrt{2}, -2\sqrt{2})\).

Question 1.130

• (a) The following points are given in cylindrical coordinates; express each in rectangular coordinates and spherical coordinates: \((1,45^\circ,1),\) \((2, \pi /2, -4),\) \((0, 45^\circ, 10),\) \((3, \pi /6 ,4),\) \((1, \pi /6, 0),\) and \((2,3 \pi /4, -2)\). (Only the first point is solved in the Study Guide.)
• (b) Change each of the following points from rectangular coordinates to spherical coordinates and to cylindrical coordinates: \((2,1, -2), (0,3,4), (\sqrt{2},1,1)\), \((-2\sqrt{3}, -2,3)\). (Only the first point is solved in the Study Guide.)

Question 1.131

Describe the geometric meaning of the following mappings in cylindrical coordinates:

• (a) \((r, \theta, z) \mapsto (r, \theta, -z)\)
• (b) \((r, \theta, z) \mapsto (r, \theta+ \pi, -z)\)
• (c) \((r, \theta, z) \mapsto (-r, \theta- \pi/4, z)\)

Question 1.132

Describe the geometric meaning of the following mappings in spherical coordinates:

• (a) \((\rho, \theta, \phi) \mapsto (\rho, \theta + \pi, \phi)\)
• (b) \((\rho, \theta, \phi) \mapsto (\rho, \theta, \pi - \phi)\)
• (c) \((\rho, \theta, \phi) \mapsto (2\rho, \theta + \pi/2, \phi)\)

Question 1.133

Sketch the following solids:

• (a) \(r \in [0,1], \ \theta \in [0,\pi], \ z \in [-1, 1]\)
• (b) \(r \in [0,2], \ \theta \in [0,\pi/2], \ z \in [0, 4]\)
• (c) \(\rho \in [0,1], \ \theta \in [0,2\pi], \ \phi \in [0, \pi/4]\)
• (d) \(\rho \in [1,2], \ \theta \in [0,2\pi], \ \phi \in [0, \pi/2]\)

Question 1.134

Sketch the following surfaces:

• (a) \(z=r^2\)
• (b) \(\rho = 4 \csc \phi \sec \theta\)
• (c) \(r=4\sin \theta\)
• (d) \(\rho \sin\phi=2\)

Question 1.135

• (a) Describe the surfaces \(r= \hbox{constant}, \theta= \hbox{constant}\), and \(z = \hbox{constant}\) in the cylindrical coordinate system.
• (b) Describe the surfaces \(\rho =\hbox{constant}, \theta = \hbox{constant}\), and \(\phi = \hbox{constant}\) in the spherical coordinate system.

Question 1.136

Show that to represent each point in \({\mathbb R}^3\) by spherical coordinates, it is necessary to take only values of \(\theta\) between 0 and \(2 \pi\), values of \(\phi\) between 0 and \(\pi\), and values of \(\rho \ge 0\). Are coordinates unique if we allow \(\rho \le 0\)?

Question 1.137

Describe the following solids using inequalities. State the coordinate system used.

• (a) A cylindrical shell 8 units long, with inside diameter 2 units and outside diameter 3 units
• (b) A spherical shell with inside radius 4 units and outside radius 6 units
• (c) A hemisphere of diameter 5 units
• (d) A cube of side length 2

Question 1.138

Let \(S\) be the sphere of radius \(R\) centered at the origin. Find the equation for \(S\) in cylindrical coordinates.

Question 1.139

Using cylindrical coordinates and the orthonormal (orthogonal normalized) vectors \({\bf e}_r, {\bf e}_\theta\), and \({\bf e}_z\) (see Figure 1.86),

• (a) express each of \({\bf e}_r, {\bf e}_\theta\), and \({\bf e}_z\) in terms of \({\bf i,j,k}\) and \((x,y,z)\); and
• (b) calculate \({\bf e}_\theta \times {\bf j}\) both analytically, using part (a), and geometrically.

Question 1.140

Using spherical coordinates and the orthonormal (orthogonal normalized) vectors \({\bf e}_\rho, {\bf e}_\theta\), and \({\bf e}_\phi\) [see Figure 1.86],

• (a) express each of \({\bf e}_\rho, {\bf e}_\theta\), and \({\bf e}_\phi\) in terms of \({\bf i,j,k}\) and \((x,y,z)\); and
• (b) calculate \({\bf e}_\theta \times {\bf j}\) and \({\bf e}_\phi \times {\bf j}\) both analytically and geometrically.

Question 1.141

Express the plane \(z=x\) in (a) cylindrical, and (b) spherical coordinates.

Question 1.142

Show that in spherical coordinates:

• (a) \(\rho\) is the length of \(x {\bf i} + y {\bf j} + z {\bf k}\).
• (b) \(\phi = \cos^{-1}\, ( {\bf v \,{\cdot}\, k} /\|{\bf v}\|)\), where \({\bf v} = x {\bf i} + y {\bf j} + z {\bf k}.\)
• (c) \(\theta = \cos^{-1}\, ( {\bf u} \,{\cdot}\, {\bf i} / \|{\bf u}\|),\) where \({\bf u}= x {\bf i} + y {\bf j}.\)

Question 1.143

Two surfaces are described in spherical coordinates by the two equations \({\rho= f ( \theta, \phi)}\) and \(\rho = - 2 f ( \theta, \phi)\), where \(f ( \theta, \phi)\) is a function of two variables. How is the second surface obtained geometrically from the first?

Question 1.144

A circular membrane in space lies over the region \(x^2 + y^2 \le a^2\). The maximum \(z\) component of points in the membrane is \(b\). Assume that \((x,y,z)\) is a point on the membrane. Show that the corresponding point \((r, \theta, z)\) in cylindrical coordinates satisfies the conditions \(0 \le r \le a, 0 \le \theta \le 2 \pi, |z| \le b\).

Question 1.145

A tank in the shape of a right-circular cylinder of radius 10 ft and height 16 ft is half filled and lying on its side. Describe the air space inside the tank by suitably chosen cylindrical coordinates.

Question 1.146

A vibrometer is to be designed that withstands the heating effects of its spherical enclosure of diameter \(d\), which is buried to a depth \(d/3\) in the earth, the upper portion being heated by the sun (assume the surface is flat). Heat conduction analysis requires a description of the buried portion of the enclosure in spherical coordinates. Find it.

Question 1.147

An oil filter cartridge is a porous right-circular cylinder inside which oil diffuses from the axis to the outer curved surface. Describe the cartridge in cylindrical coordinates, if the diameter of the filter is 4.5 inches, the height is 5.6 inches, and the center of the cartridge is drilled (all the way through) from the top to admit a \({\frac{5}{8}}\)-inch-diameter bolt.

Question 1.148

Describe the surface given in spherical coordinates by \(\rho = \cos 2 \theta\).

Question 1.149

• (a) Find all points \(\textbf{p} \in \mathbb{R}^3\) that have the same representation in both Cartesian and spherical coordinates.
• (b) Find all points \(\textbf{p} \in \mathbb{R}^3\) that have the same representation in both Cartesian and cylindrical coordinates.