7.7: Cylindrical and Spherical Coordinates
- Page ID
- 143918
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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.