8.E: Exercises
- Page ID
- 14547
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the following, polar coordinates \((r,θ)\) for a point in the plane are given. Find the corresponding Cartesian coordinates.
- \((2,\pi /4)\)
- \((-2, \pi/4)\)
- \((3, \pi/3)\)
- \((-3, \pi/3)\)
- \((2,5\pi /6)\)
- \((-2, 11\pi /6)\)
- \((2,\pi /2)\)
- \((1,3\pi /2)\)
- \((-3, 3\pi /4)\)
- \((3, 5\pi /4)\)
- \((-2, \pi /6)\)
Consider the following Cartesian coordinates \((x, y)\). Find polar coordinates corresponding to these points.
- \((-1,1)\)
- \((\sqrt{3},-1)\)
- \((0,2)\)
- \((-5,0)\)
- \((-2\sqrt{3},2)\)
- \((2,-2)\)
- \((-1,\sqrt{3})\)
- \((-1,-\sqrt{3})\)
The following relations are written in terms of Cartesian coordinates \((x, y)\). Rewrite them in terms of polar coordinates, \((r,\theta )\).
- \(y=x^2\)
- \(y=2x+6\)
- \(x^2+y^2=4\)
- \(x^2-y^2=1\)
Use a calculator or computer algebra system to graph the following polar relations.
- \(r=1-\sin (2\theta ),\:\theta\in [0,2\pi ]\)
- \(r=\sin (4\theta ),\:\theta\in [0,2\pi ]\)
- \(r=\cos (3\theta )+\sin (2\theta ),\: \theta\in [0,2\pi]\)
- \(r=\theta,\:\theta\in [0,15]\)
Graph the polar equation \(r = 1+\sinθ\) for \(θ ∈ [0, 2π]\).
Graph the polar equation \(r = 2+\sinθ\) for \(θ ∈ [0, 2π]\).
Graph the polar equation \(r = 1+2 \sinθ\) for \(θ ∈ [0, 2π]\).
Graph the polar equation \(r = 2+\sin(2θ)\) for \(θ ∈ [0, 2π]\).
Graph the polar equation \(r = 1+\sin(2θ)\) for \(θ ∈ [0, 2π]\).
Graph the polar equation \(r = 1+\sin(3θ)\) for \(θ ∈ [0, 2π]\).
Describe how to solve for \(r\) and \(θ\) in terms of \(x\) and \(y\) in polar coordinates.
This problem deals with parabolas, ellipses, and hyperbolas and their equations. Let \(l\), \(e > 0\) and consider \[r=\frac{l}{1\pm e\cos\theta}\nonumber\] Show that if \(e = 0\), the graph of this equation gives a circle. Show that if \(0 < e < 1\), the graph is an ellipse, if \(e = 1\) it is a parabola and if \(e > 1\), it is a hyperbola.
The following are the cylindrical coordinates of points, \((r,θ,z)\). Find the Cartesian and spherical coordinates of each point.
- \((5,\frac{5\pi}{6},-3)\)
- \((3,\frac{\pi}{3},4)\)
- \((4,\frac{2\pi}{3},1)\)
- \((2,\frac{3\pi}{4},-2)\)
- \((3,\frac{3\pi}{2},-1)\)
- \((8,\frac{11\pi}{6},-11)\)
The following are the Cartesian coordinates of points, \((x, y,z)\). Find the cylindrical and spherical coordinates of these points.
- \((\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},-3)\)
- \((\frac{3}{2},\frac{3}{2}\sqrt{3},2)\)
- \((-\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},11)\)
- \((-\frac{5}{2},\frac{5}{2}\sqrt{3},23)\)
- \((-\sqrt{3},-1,-5)\)
- \((\frac{3}{2},-\frac{3}{2}\sqrt{3},-7)\)
- \((\sqrt{2},\sqrt{6},2\sqrt{2})\)
- \((-\frac{1}{2}\sqrt{3},\frac{3}{2},1)\)
- \((-\frac{3}{4}\sqrt{2},\frac{3}{4}\sqrt{2},-\frac{3}{2}\sqrt{3})\)
- \((-\sqrt{3}1,2\sqrt{3})\)
- \((-\frac{1}{4}\sqrt{2},\frac{1}{4}\sqrt{6},-\frac{1}{2}\sqrt{2})\)
The following are spherical coordinates of points in the form \((ρ,φ,θ)\). Find the Cartesian and cylindrical coordinates of each point.
- \((4,\frac{\pi}{4},\frac{5\pi}{6})\)
- \((2,\frac{\pi}{3},\frac{2\pi}{3})\)
- \((3,\frac{5\pi}{6},\frac{3\pi}{2})\)
- \((4,\frac{\pi}{2},\frac{7\pi}{4})\)
- \((4,\frac{2\pi}{3},\frac{\pi}{6})\)
- \((4,\frac{3\pi}{4},\frac{5\pi}{3})\)
Describe the surface \(φ = π/4\) in Cartesian coordinates, where \(φ\) is the polar angle in spherical coordinates.
Describe the surface \(θ = π/4\) in spherical coordinates, where \(θ\) is the angle measured from the positive \(x\) axis.
Describe the surface \(r=5\) in Cartesian coordinates, where \(r\) is one of the cylindrical coordinates.
Describe the surface \(\rho =4\) in Cartesian coordinates, where \(\rho\) is the distance to the origin.
Give the cone described by \(z=\sqrt{x^2+y^2}\) in cylindrical coordinates and in spherical coordinates.
The following are described in Cartesian coordinates. Rewrite them in terms of spherical coordinates.
- \(z=x^2+y^2\)
- \(x^2-y^2=1\)
- \(z^2+x^2+y^2=6\)
- \(z=\sqrt{x^2+y^2}\)
- \(y=x\)
- \(z=x\)
The following are described in Cartesian coordinates. Rewrite them in terms of cylindrical coordinates.
- \(z=x^2+y^2\)
- \(x^2-y^2=1\)
- \(z^2+x^2+y^2=6\)
- \(z=\sqrt{x^2+y^2}\)
- \(y=x\)
- \(z=x\)