Mathematical Physics Vol 1

Chapter 2. Vector analysis

52

Figure 2.9: Möbius strip

Thus, from the very definition of an orientable surface, it follows that an orientable surface S has two distinct sides. Namely, if a normal line is observed at an arbitrary point in the surface, then two directions can be distinguished: the upward direction if the normal makes an acute angle with the Oz axis and the downward direction if the normal and the axis form an obtuse angle (Fig. 2.10).

Figure 2.10: Orientable surface.

For the upper part of the surface S the upward direction can be chosen, and for the lower part, the downward direction, in which case the normal is called external. If the surface is given by the equation z = f ( x , y ) , then the cosine of the angle that the normal forms with the positive part of the Oz -axis, is equal to:

   −

1 q 1 + f ′ 2 1 q 1 + f ′ 2

− for the upper part

′ 2

x + f

y

cos γ =

(2.72)

− for the lower part ,

′ 2

x + f

y

where

∂ f ∂ x

∂ f ∂ y

f ′ x =

f ′ y =

,

.