97

Coordinate curves

on sphere.

For example, consider the unit sphere

r (u, v ) = cos usin vˆe 1 + sin usin vˆe 2 + cos vˆe 3

where 0 ≤u≤ 2 πand 0 ≤v≤π. The curves
r (u 0 , v )for

equi-spaced constants u 0 gives the coordinate curves

called lines of longitude on the sphere. The curves

r (u, v 0 ) for equi-spaced constants v 0 give the curves

called lines of latitude on the sphere.

A surface is called an oriented surface if

(i) each nonboundary point on the surface has two unit normals ˆenand −ˆen. By

selecting one of these unit normals one is said to give an orientation to the

surface. Thus, an oriented surface will always have two orientations.

(ii) The unit normal selected defines a surface orientation and this unit normal must

vary continuously over the surface.

(iii) Each nonboundary point on the oriented surface has a tangent plane.

(iv) If the surface is that of a solid, then the unit normal at each point on the surface

which is directed outward from the surface is usually selected as the preferred

orientation for the closed surface.

A surface Sis said to be a simple closed surface if the surface divides all of three

dimensional space into three regions defined by

(i) points interior to S, where the distance between any two points inside Sis finite.

(ii) points on the surface S.

(iii) points exterior to the surface S.

Asmooth surface is one where a normal vector can be constructed at each point

of the surface.

The sphere

The general equation of a sphere is

x^2 +y^2 +z^2 +αx +βy +γz +δ= 0

where α, β, γ and δare constants. This is a simple closed surface with outward normal

defining its orientation. It is customary to complete the square on the x, y and z

terms and express this equation in the form

(

x+α

2

) 2

+

(

y+β

2

) 2

+

(

z+γ

2

) 2

=α

(^2) +β (^2) +γ 2
4
−δ (7 .26)

After completing the square on the x, y and zterms, the following cases can arise.