Cylindrical & Spherical Coordinates

  1. Given these points expressed in rectilinear coordinates, write down their location expressed in cylindrical coordinates. So that you may verify your calculations, these have been designed to have nearly whole-number cylindrical coordinates. Once you have both the rectilinear and cylindrical coordinates for each point, briskly plot the location of the point in space using both coordinate systems to practice working with them.
    \((-0.415, 0.909, 3)\)
    \((1.081, 1.683, -3)\)
    \((-2.912, -6.363, 4)\)
    \((-2.075, 4.545, 0)\)
    \((-0.989, 0.141, 314)\)
  2. Equipping space with a cylindrical coordinate system, each of these equations describes a surface in space. Determine what that surface must look like.
    \(r = 42\)
    \(\theta = \frac{\pi}{4}\)
    \(z = -31\)
    \(r = z\)
    \(r = z^2\)
    \(r = \theta\)
  3. Recall that cylindrical coordinates describing the location of a point in space are not unique. Express the location of the point with cylindrical coordinates \(\left(5,\frac{\pi}{6}, 7\right)\) in two other ways: one in which the first coordinate is negative, and another in which the second coordinate is negative.
  4. Given these points expressed in rectilinear coordinates, write down their location expressed in spherical coordinates. So that you may verify your calculations, these have been designed to have nearly whole-number spherical coordinates. Once you have both the rectilinear and spherical coordinates for each point, briskly plot the location of the point in space using both coordinate systems to practice working with them.
    \((0.454, 0.707, 0.540)\)
    \((2.727, 0, -1.245)\)
    \((2.270, 3.536, 2.700)\)
    \((0, 0, 5)\)
    \((-1.535, -0.218, -10.879)\)
  5. Equipping space with a spherical coordinate system, each of these equations describes a surface in space. Determine what that surface must look like.
    \(\rho = 17\)
    \(\theta = 1\)
    \(\phi = \frac{\pi}{5}\)
    \(\rho = \theta\)
    \(\rho = \phi\)
  6. Recall that spherical coordinates describing the location of a point in space are not unique. Express the location of the point with spherical coordinates \(\left(2,\frac{\pi}{3}, \frac{\pi}{4}\right)\) in three other ways: one in which the first coordinate is negative, another in which the second coordinate is negative, and another in which the third coordinate is negative.
  7. Converting from rectilinear coordinates to cylindrical or spherical coordinates requires that you either (1) use the two-parameter atan2 function, or (2) that you use the \(\arctan\) function and manually assign a \(\pm\) sign to determine the azimuthal angle \(\theta.\) Regarding the latter case, how do you determine the sign of \(\theta?\) Be able to describe how to do this as if you needed to explain it to a peer in the class who doesn’t understand.
  8. We’ve developed formulas for converting between rectilinear and cylindrical coordinates, and between rectilinear and spherical coordinates, Now write down formulas that convert directly between cylindrical and spherical coordinates.