7.7: Cylindrical and Spherical Coordinates
- Page ID
- 143918
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Plot the point \((r, \theta, z) = \left(1, \dfrac{\pi}{2}, 1\right)\) expressed in cylindrical coordinates.
- Plot the point \((r, \theta, z) = \left(3, \dfrac{5\pi}{4}, -2\right)\) expressed in cylindrical coordinates.
- Plot the point \((r, \theta, z) = \left(2, \dfrac{5\pi}{6}, 3\right)\) expressed in cylindrical coordinates.
- Plot the point \((\rho, \theta, \phi) = \left(1, \dfrac{3\pi}{2}, \dfrac{\pi}{2}\right)\) expressed in spherical coordinates.
- Plot the point \((\rho, \theta, \phi) = \left(4, \dfrac{7\pi}{6}, \dfrac{\pi}{3}\right)\) expressed in spherical coordinates.
- Plot the point \((\rho, \theta, \phi) = \left(2, 0, 0\right)\) expressed in spherical coordinates.
- Consider the point \((r, \theta, z) = \left(1, \dfrac{\pi}{4}, 0\right)\) expressed in cylindrical coordinates. Find the rectangular coordinates \((x, y, z)\) of the point.
- Consider the point \((r, \theta, z) = \left(3, \dfrac{2\pi}{3}, -4\right)\) expressed in cylindrical coordinates. Find the rectangular coordinates \((x, y, z)\) of the point.
- Consider the point \((r, \theta, z) = \left(2, \dfrac{11\pi}{6}, \dfrac{5\pi}{4}\right)\) expressed in cylindrical coordinates. Find the rectangular coordinates \((x, y, z)\) of the point.
- Consider the point \((x, y, z) = \left(0, 2, 5\right)\) expressed in rectangular coordinates. Find the cylindrical coordinates \((r, \theta, z)\) of the point.
- Consider the point \((x, y, z) = \left(-1, -1, -1\right)\) expressed in rectangular coordinates. Find the cylindrical coordinates \((r, \theta, z)\) of the point.
- Consider the point \(x, y, z) = \left(\sqrt{3}, 3, \pi\right)\) expressed in rectangular coordinates. Find the cylindrical coordinates \((r, \theta, z)\) of the point.
- Consider the point \((\rho, \theta, \phi) = \left(1, \pi, \dfrac{\pi}{2}\right)\) expressed in spherical coordinates. Find the rectangular coordinates \((x, y, z)\) of the point.
- Consider the point \((\rho, \theta, \phi) = \left(2, \dfrac{5\pi}{6}, \dfrac{\pi}{3}\right)\) expressed in spherical coordinates. Find the rectangular coordinates \((x, y, z)\) of the point.
- Consider the point \((\rho, \theta, \phi) = \left(4, \dfrac{3\pi}{2}, \dfrac{3\pi}{4}\right)\) expressed in spherical coordinates. Find the rectangular coordinates \((x, y, z)\) of the point.
- Consider the point \((x, y, z) = \left(1, 1, \sqrt{2}\right)\) expressed in rectangular coordinates. Find the spherical coordinates \((\rho, \theta, \phi)\) of the point.
- Consider the point \((x, y, z) = \left(-2, -2\sqrt{3}, -4\sqrt{3}\right)\) expressed in rectangular coordinates. Find the spherical coordinates \((\rho, \theta, \phi)\) of the point.
- Consider the point \((x, y, z) = \left(-3, 0, 0\right)\) expressed in rectangular coordinates. Find the spherical coordinates \((\rho, \theta, \phi)\) of the point.
- Graph the surface \(r = 2\) given in cylindrical coordinates.
- Graph the surface \(\theta = \dfrac{\pi}{4}\) given in cylindrical coordinates.
- Graph the surface \(z = -1\) given in cylindrical coordinates.
- Graph the surface \(\rho = 2\) given in spherical coordinates.
- Graph the surface \(\theta = \dfrac{\pi}{3}\) given in spherical coordinates.
- Graph the surface \(\phi = \dfrac{\pi}{4}\) given in spherical coordinates.
- Consider the surface \(x^2 + y^2 + z^2 = 4\) given in rectangular coordinates. Find the equation of the surface in cylindrical coordinates and the equation of the surface in spherical coordinates.
- Consider the surface \(x^2 + y^2 = 4\) given in rectangular coordinates. Find the equation of the surface in cylindrical coordinates and the equation of the surface in spherical coordinates.
- Consider the surface \(z^2 = x^2 + y^2\) given in rectangular coordinates. Find the equation of the surface in cylindrical coordinates and the equation of the surface in spherical coordinates.
- Consider the surface \(z = r^2\) given in cynlindrical coordinates. Find the equation of the surface in spherical coordinates and the equation of the surface in rectangular coordinates.
- Consider the surface \(r = 3\sin \theta\) given in cynlindrical coordinates. Find the equation of the surface in spherical coordinates and the equation of the surface in rectangular coordinates.
- Consider the surface \(r = 2\sec \theta\) given in cynlindrical coordinates. Find the equation of the surface in spherical coordinates and the equation of the surface in rectangular coordinates.
- Consider the surface \(\rho = 2\cos \phi\) given in spherical coordinates. Find the equation of the surface in cylindrical coordinates and the equation of the surface in rectangular coordinates.
- Consider the surface \(\rho = 4\csc \phi\) given in spherical coordinates. Find the equation of the surface in cylindrical coordinates and the equation of the surface in rectangular coordinates.
- Consider the surface \(\rho = 3\csc \phi \sec \theta\) given in spherical coordinates. Find the equation of the surface in cylindrical coordinates and the equation of the surface in rectangular coordinates.