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- 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.

- 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
.

- 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
.

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

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

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

where
is a volume enclosed by a surface
.
- 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.

- 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
.

- 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.

- 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
2016-01-22