Calculus-Sec-14-4-Solution

14.4    Cylindrical Coordinates and Spherical Coordinates

by SGLee - HSKim -SWSun

1-4. Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point.

1.

2.

3.

.

4.

5-6.Change from rectangular to cylindrical coordinates.

5.

so .

Therefore one set of cylindrical coordinates is . Another is .

As with polar coordinates, there are infinitely many choices.

6.

7-10. Plot the point whose spherical coordinates are given. Then, find the rectangular coordinates of the point.

7.

8.

.

9.

.

10.

.

11-12. Change from rectangular to spherical coordinates.

11.

so ,

so  (Note that  because .)

Therefore, spherical coordinates of the given point are .

12.

.

13-14. Change from cylindrical to spherical coordinates.

13.

. So, , and .

Therefore, rectangular coordinates are  and .

so .

, so ( because ).

Therefore, spherical coordinates of the given point are  and .

14.

.

15-16. Change from spherical to cylindrical coordinates.

15.

.

So,  since .

Therefore, rectangular coordinates are

, so .

Therefore cylindrical coordinates of the given point are .

16.

,

Therefore rectangular coordinates of the given point are .

, so .

Therefore cylindrical coordinates of the given point are .

17-20. Describe in words the surface whose equation is given.

17.

21-28. Identify the surface whose equation is given.

21.