Cylindrical and spherical coordinates practice problems
Converting points between rectangular, cylindrical and spherical coordinates. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Convert the rectangular point \( \displaystyle \left(0, 5, -3\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).
Convert the spherical point \( \displaystyle \left(5, \frac{7 \pi}{4}, \frac{3 \pi}{4}\right) \) to cylindrical coordinates.
Convert the spherical point \( \displaystyle \left(3, 0, \frac{\pi}{6}\right) \) to cylindrical coordinates.
Convert the spherical point \( \displaystyle \left(1, \frac{3 \pi}{4}, \frac{\pi}{6}\right) \) to cylindrical coordinates.
Convert the spherical point \( \displaystyle \left(6, \frac{\pi}{4}, \frac{\pi}{4}\right) \) to cylindrical coordinates.
Convert the rectangular point \( \displaystyle \left(0, -5, -1\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).
Convert the rectangular point \( \displaystyle \left(- \frac{5 \sqrt{2}}{2}, \frac{5 \sqrt{2}}{2}, -4\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).
Convert the rectangular point \( \displaystyle \left(-2, 2 \sqrt{3}, 0\right) \) to spherical coordinates (\( \displaystyle \rho > 0 \), \( \displaystyle 0 \le \theta < 2\pi \), \( \displaystyle 0 \le \varphi \le \pi \)).
Convert the spherical point \( \displaystyle (\rho, \theta, \varphi) = \left(5, \pi, \frac{\pi}{4}\right) \) to rectangular coordinates.
Convert the rectangular point \( \displaystyle \left(0, -6, 5\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).