Cylindrical And Spherical Coordinates

(requires JavaScript)

1. Plot the points with cylindrical coordinates $\left(2,\pi /4,1\right)$ and $\left(4,-\pi /3,5\right)$ and find the Cartesian coordinates of these points.
   $\left(\sqrt{2},\sqrt{2},1\right)$, $\left(2,-2\sqrt{3},5\right)$
2. Change points $\left(1,-1,4\right)$ and $\left(-1,-\sqrt{3},2\right)$ from rectangular to cylindrical coordinates.
   ${\displaystyle \left(\sqrt{2},7\pi /4,4\right)}$, ${\displaystyle \left(2,4\pi /3,2\right)}$
3. Plot the points with spherical coordinates $\left(5,\pi ,\pi /2\right)$ and $\left(4,3\pi /4,\pi /3\right)$ and find the Cartesian coordinates of these points.
4. Change points $\left(1,\sqrt{3},2\sqrt{3}\right)$ and $\left(0,-1,-1\right)$ from rectangular to spherical coordinates.
   $\left(4,\pi /3,\pi /6\right)$, $\left(\sqrt{2},3\pi /2,3\pi /4\right)$
5. Describe in words the surface given by equation ${\displaystyle \phi =\pi /3}$.
   Half of a cone.
6. Identify the surface whose equation is $r=2\cos <!--FUNCTION\; APPLICATION-->\left(\theta \right)$.
   Circular cylinder with radius $1$ and axis parallel to the $z\text{-axis}$.
7. Identify the surface whose equation is $r^2-2z^2=4$.

8. Sketch the solid described by the inequalities

   $r^2\le z\le 2-r^2$

9. Sketch the solid described by the inequalities

   $-\pi /2\le \theta \le \pi /2$

   $0\le \phi \le \pi /6$

   ${\displaystyle 0\le \rho \le \frac{1}{\cos <!--FUNCTION\; APPLICATION-->\left(\phi \right)}}$

10. Use spherical coordinates to find

    ${\displaystyle \underset{\left(x,y,z\right)\to \left(0,0,0\right)}{\lim }<!--FUNCTION\; APPLICATION-->\frac{xyz}{x^2+y^2+z^2}}$

    $0$