Calculus-III (Section 13.7)

Spherical Spherical Coordinate SystemCoordinate System

Conversion from Spherical to Conversion from Spherical to Cylindrical Coordinate SystemCylindrical Coordinate System

RecallRecall

Cylindrical to Cartesian Spherical to Cylindrical

Spherical to CartesianSpherical to Cartesian

Also note that since we know that                             we get,

Example -1Example -1Convert the point from Cartesian to Spherical coordinates.

( , , ) (1,1,1)x y z

1,1,1

1,1,0

1,1,1

1,1,0

1

4

2 2 2 2 2 21 1 1 3x y z

1

1cos3

1cos3

z

1

Therefore, (1,1,1) in cartesian coordinate system corresponds

1to 3, ,cos in spherical coordinate system.4 3

Example -2Example -2Convert the point from Cartesian to Spherical coordinates.

( , , ) 1,1, 2x y z

The first thing that we’ll do here is find     .

Now we’ll need to find  

Finally, we will find Finally, we will find

You find it

Since the points with coordinates x = -1 and y = 1 lies in the second quadrant,

3 .4

The spherical coordinates of the

point are then            .     

2 2 2 2

2 2

x y z a

aa

2 2

2 2 2 2 2 2

2 2

cos sin cos sin sin

cos sin cos sincos sin

tan 134

z x y

Identify the surface 3

Identify the surface

This is a vertical plane that makes an angle of with the positive x-axis.

3

3

3

                                                                              

                                                                                                                                                                                                                                                      

                                                       

d sinrd d