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# Exercises

1. Find the Cartesian components of the basis vectors , , and of the cylindrical coordinate system. Verify that the vectors are mutually orthogonal. Do the same for the basis vectors , , and of the spherical coordinate system.

2. Use cylindrical coordinates to prove that the volume of a right cylinder of radius and length is . Demonstrate that the moment of inertia of a uniform cylinder of mass and radius about its symmetry axis is .

3. Use spherical coordinates to prove that the volume of a sphere of radius is . Demonstrate that the moment of inertia of a uniform sphere of mass and radius about an axis passing through its center is .

4. For what value(s) of is , where is a spherical coordinate?

5. For what value(s) of is , where is a spherical coordinate?

1. Find a vector field satisfying for . Here, is a spherical coordinate.
2. Use the divergence theorem to show that

where is a volume enclosed by a surface .
3. Use the previous result (for ) to demonstrate that the volume of a right cone is one third the volume of the right cylinder having the same base and height.

6. The electric field generated by a -directed electric dipole of moment , located at the origin, is

where , and is a spherical coordinate. Find the components of in the spherical coordinate system. Calculate and .

7. Show that the parabolic cylindrical coordinates , , , defined by the equations , , , where , , are Cartesian coordinates, are orthogonal. Find the scale factors , , . What shapes are the and surfaces? Write an expression for in parabolic cylindrical coordinates.

8. Show that the elliptic cylindrical coordinates , , , defined by the equations , , , where , , are Cartesian coordinates, and , , are orthogonal. Find the scale factors , , . What shapes are the and surfaces? Write an expression for in elliptical cylindrical coordinates.

Next: Ellipsoidal Potential Theory Up: Non-Cartesian Coordinates Previous: Spherical Coordinates
Richard Fitzpatrick 2016-03-31