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. 

           http://matrix.skku.ac.kr/cal-lab/cal-14-2-1.html 

 

 




 2. 

 

 




3. 

 

 

     .




4. 

 

      Answer: .




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. 

            http://matrix.skku.ac.kr/cal-lab/cal-14-2-7.html 

 

 




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. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-17.html 

 

 




 18. 

            http://matrix.skku.ac.kr/cal-lab/cal-14-2-18.html 

 

 




 19.        

            http://matrix.skku.ac.kr/cal-lab/cal-14-2-19.html 

 

 




 20. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-20.html 

 

 




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

 

 21.       

            http://matrix.skku.ac.kr/cal-lab/cal-14-2-21.html 

 

 




 22. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-22.html 

 

 




 23. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-23.html 

 




 24. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-24.html 

 

 




 25. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-25.html 

 

 




 26. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-26.html 

 

 




 27. 

             http://matrix.skku.ac.kr/cal-lab/cal-14-2-27.html 

 

 




 28. 

            http://matrix.skku.ac.kr/cal-lab/cal-14-2-28.html 

 

 




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