EXERCISES
61
8.
Derive the
hi’s
for the cylindrical and spherical systems. Also derive the expres-
sions given in the chapter for the gradient, divergence, and curl operators in these
systems.
9.
Redo Exercise
8
of Chapter
2
using spherical coordinates.
10.
A Tokamak fusion device has a geometry that takes the shape of a doughnut
or torus. Calculations for such a device are sometimes done with the toroidal
coordinates shown in the figure below.
1
-
Ma.jor Axis
/’
/
Minor Axis
,/>‘
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The “major axis” of the device is a vertical, straight line that forms the toroidal
axis of symmetry. The “minor axis” is a circle of fixed radius
R,
that passes
through the center of the doughnut. The position
of
a point is described
by
the
coordinates
(r,
8,+).
The coordinates
r
and
8
are similar to a two-dimensional
polar system, aligned perpendicular to the minor axis. The coordinate
+
measures
the angular position along the minor axis.
(a)
Make a sketch of the unit
basis
vectors for this toroidal system. Are these
(b)
Obtain expressions relating the toroidal coordinates to a set of Cartesian
(c)
Obtain the
hi
scale factors for the toroidal system.
(d)
Write expressions for the displacement vector
dF,
a differential surface area
(e)
Write expressions for the gradient
v@,
divergence
V
*
A,
and curl
v
X
(f)
Laplace’s equation written in vector notation is
V2@
=
0.
What
is
Laplace’s
vectors orthogonal?
Do
they form a right-handed system?
coordinates.
dV,
and a differential volume
d7
in this system.
operations
in
this system.
equation expressed in these toroidal coordinates?
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