14.2 Cylindrical Coordinates and Spherical Coordinates


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

 Sol)

(You may do it with Sage in http://math1.skku.ac.kr/ . Open resources in http://math1.skku.ac.kr/pub/)

 

T = Cylindrical('height', ['radius', 'azimuth'])

T.transform(radius=2, azimuth=- pi/3, height=4)

  (1, -sqrt(3), 4)


2. .

 Sol) .


3. .  

 Sol) .


4. .

 Sol) .


5-6.Change from rectangular to cylindrical coordinates.


5. .

 Sol) , 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

 

 

 

 Sol)

 

T = Spherical('radius', ['azimuth', 'inclination'])

T.transform(radius=3, azimuth=pi/6, inclination=pi/6)

  (3/4*sqrt(3), 3/4, 3/2*sqrt(3))


8. .

 Sol) .


9. .

 Sol) .


10. .

 Sol) .


11-12. Change from rectangular to spherical coordinates.


11. .

 Sol) , so ,

      so (Note that because .)

     Therefore, spherical coordinates of the given point are .


12. .

 Sol) .


13-14. Change from cylindrical to spherical coordinates.


13. .

 Sol) , , , .

      So, , and .

      Therefore, rectangular coordinates are and .

      , so .

      , so ( because ).

      Therefore, spherical coordinates of the given point are and .


14. .

 Sol) .


15-16. Change from spherical to cylindrical coordinates.


15. .

 Sol) , , , .

      So, , since , .

      Therefore, rectangular coordinates are

      , , so .

      Therefore cylindrical coordinates of the given point are .


16. .


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

 

17. .

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

 Sol)

 

S=Cylindrical('radius', ['azimuth', 'height'])

theta, z=var('theta, z')

plot3d(4, (theta, 0, 2*pi), (z, -2, 2), transformation=S)


 


18. .

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

 Sol)

 

S=Spherical('radius',['azimuth','inclination']);

var('p,theta')

plot3d(4,(p,0,10),(theta,0,2*pi),transformation=S)


 

19. .       

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

 Sol)

 

S=Spherical('inclination', ['radius', 'azimuth'])

r,theta=var('r,theta')

plot3d(2*pi/3, (r,0,3), (theta, 0, 2*pi), transformation=S)


 


20. .

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

 Sol)

 

S=Spherical('azimuth', ['radius', 'inclination'])

r, phi=var('r, phi')

plot3d(pi/3, (r,0, 10), (phi, 0, pi) , transformation=S)



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


21. .       

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

 Sol)

 

S=Cylindrical('height', ['radius', 'azimuth'])

r, theta=var('r, theta')

plot3d(2*r^2, (r, 0, 2), (theta,0,2*pi), transformation=S)


22. .

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

 Sol)

 

S=Cylindrical('radius', ['azimuth', 'height'])

theta,z=var('theta, z')

plot3d(2*sin(theta), (theta,0,2*pi), (z, -2, 2), transformation=S)



23. .

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

 Sol)

 

S=Spherical('radius', ['azimuth', 'inclination'])

theta, phi=var('theta, phi')

plot3d(1/(2*cos(phi)), (theta, 0, 2*pi), (phi, 0, pi), transformation=S)


 

 

24. .

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

 Sol)

 

T = Cylindrical('height',['radius','azimuth']);

var('r,t,z');

implicit_plot3d(r*sin(z)==2,(r,0,3),(t,0,2*pi),(z,-2,2),transformation=T)


 


25. .

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

 Sol)

 

T=Cylindrical('radius', ['azimuth', 'height'])

theta, z=var('theta, z')

plot3d(3*cos(theta), (theta, 0, 2*pi), (z, -2, 2), transformation=T)


 


26. .

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

 Sol)

 

T = Cylindrical('height',['radius','azimuth']);

var('r,t,z');

implicit_plot3d(r==3*cos(z),(r,0,3),(t,0,2*pi),(z,-2,2),transformation=T)

 


27. .

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

 Sol)

 

T=Cylindrical('height', ['radius', 'azimuth'])

r, theta=var('r, theta')

p1=plot3d((9-r^2)^(1/2), (r, -3, 3), (theta, 0, 2*pi), transformation=T)

p2=plot3d(-(9-r^2)^(1/2), (r, -3, 3), (theta, 0, 2*pi), transformation=T)

show(p1+p2, aspect_ratio=1)


 


28. .

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

 Sol)

 

T = Cylindrical('height',['radius','azimuth']);

var('r,t,z');

implicit_plot3d(r^2*(sin(z)^2+4*cos(z)^2)==2,(r,0,3),(t,0,2*pi),(z,-2,2),transformation=T)