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(Hints: (1) Designate the point P as north pole and confine P to the zero
0
1
meridian. With this choice, the coordinates of the vertices are given by:
P = (θ = / 2,φ = 0)
π
0 0 0
P = (,θφ = 0)
1 1 1
P = (θφ )
,
2 2 2
P = (θφ )
,
3 3 3
(2) From symmetry, the optimal tetrahedron will have a base (P , P , P ) that
3
2
1
is an equilateral triangle in a plane parallel to the equatorial plane. The lati-
tude of (P , P , P ) is θ, while their longitudes are (0, 2π/3, –2π/3), respec-
1
2
3
tively. (3) The area of the tetrahedron is the sum of the areas of the four
triangles (012), (023), (031), (123), where we are indicating each point by its
subscript. (4) Express the area as function of θ. Find the value of θ that maxi-
mizes this quantity.)

7.6 Vector Valued Functions
As you may recall, in Chapter 1 we described curves in 2-D and 3-D by para-
metric equations. Essentially, we gave each of the coordinates as a function of
a parameter. In effect, we generated a vector valued function because the
position of the point describing the curve can be written as:
r
Rt() = x t ê() + y t ê() + zt ê() (7.53)
1 2 3
r
If the parameter t was chosen to be time, then the tip of the vector Rt() would
be the position of a point on that curve as a function of time. In mechanics,
r
finding Rt() is ultimately the goal of any problem in the dynamics of a point
particle. In many problems of electrical engineering design of tubes and other
microwave engineering devices, we need to determine the position of elec-
trons whose motion we control by a variety of electrical and magnetic fields
geometries. The following are the kinematics variables of the problem. The
dynamics form the subject of mechanics courses.
To help visualize the shape of a curve generated by the tip of the position
r
vector Rt() , we introduce the tangent vector and the normal vector to the
curve and the curvature of the curve.
The velocity vector field associated with the above position vector is
defined through:

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