Example 7-25. The parametric equations
x=acos usin v, y =asin usin v, z =acos v
with 0 ≤u≤ 2 π and 0 ≤v≤π, represent the surface of a sphere of radius a. These
parametric equations were obtained from the geometry of figure 7-18.
Figure 7-18. Surface of a sphere of radius a.
The position vector of a point on the surface of this sphere can be represented
by the vector
r (u, v ) = acos usin vˆe 1 +asin usin vˆe 2 +acos vˆe 3.
For u 0 and v 0 constants, the curves r (u 0 , v ), 0 ≤v≤π, are meridian lines on the
sphere while the curves r (u, v 0 ), 0 ≤u≤ 2 π, are circles of constant latitude. The
tangent vectors to these curves are found by taking the derivatives
∂r
∂u =−asin usin v
ˆe 1 +acos usin vˆe 2
∂r
∂v
=acos ucos vˆe 1 +asin ucos veˆ 2 −asin vˆe 3.
From these tangent vectors, a normal vector to the surface is constructed by taking
a cross product and
N =∂r
∂v
×∂r
∂u
.
